Practice Material on Thermodynamics - Carnot
Engine
Question: A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K
and a cold reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of
heat from the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
A Carnot engine operates between a hot reservoir at temperature 𝑇𝐻=500 K and a cold
reservoir at temperature 𝑇𝐶=300 K. If the engine absorbs 𝑄𝐻=2000 J of heat from
the hot reservoir per cycle, calculate:
(a) The efficiency of the Carnot engine.
(b) The work done by the engine per cycle.
(c) The heat rejected to the cold reservoir per cycle.
Solution:
(a) The efficiency 𝜂 of a Carnot engine is given by:
𝜂=1−𝑇𝐶
𝑇𝐻
Substitute 𝑇𝐻=500 K and 𝑇𝐶=300 K:
𝜂=1−300
500=1−0.6=0.4 (40%)
(b) The work done 𝑊 by the engine per cycle is:
𝑊=𝜂𝑄𝐻=0.4×2000 J=800 J
(c) The heat rejected 𝑄𝐶 to the cold reservoir per cycle is:
𝑄𝐶=𝑄𝐻−𝑊=2000 J−800 J=1200 J
Quantum Mechanics - Particle in a Box
Question: Consider a particle of mass 𝑚 in a one-dimensional box of length 𝐿. The
particle is in the ground state. Find:
(a) The energy of the ground state.
(b) The wavelength of the particle in the ground state.
Solution:
(a) The energy levels 𝐸𝑛 of a particle in a one-dimensional box are given by:
𝐸𝑛=𝑛2𝜋2ℏ2
2𝑚𝐿2
For the ground state 𝑛=1:
𝐸1=𝜋2ℏ2
2𝑚𝐿2
(b) The wavelength 𝜆 of the particle in the ground state can be found using the de
Broglie relation:
𝜆=2𝐿
𝑛
For 𝑛=1: 𝜆=2𝐿
Electromagnetism - Magnetic Field of a Solenoid
Question: A solenoid has 𝑁 turns, length 𝐿, and carries a current 𝐼. Derive the
expression for the magnetic field inside the solenoid.
Solution:
The magnetic field 𝐵 inside a solenoid is given by Ampere’s law:
∮B⋅𝑑l=𝜇0𝐼enc
For a solenoid, the enclosed current 𝐼enc =𝑁𝐼:
𝐵⋅𝐿=𝜇0𝑁𝐼
Thus, the magnetic field inside the solenoid is:
𝐵=𝜇0𝑁𝐼
𝐿
Relativity - Time Dilation
Question: An astronaut travels at a speed of 0.8𝑐 relative to Earth to a star 8 light-years
away. Calculate:
(a) The time taken for the journey as measured by an observer on Earth.
(b) The time taken for the journey as measured by the astronaut.
Solution:
(a) The time taken 𝑡 for the journey as measured by an observer on Earth is:
𝑡=distance
speed =8 light-years
0.8𝑐 =10 years
(b) The time taken 𝑡′ for the journey as measured by the astronaut is given by time
dilation:
𝑡′=𝑡√1−𝑣2
𝑐2
Substitute 𝑣=0.8𝑐:
𝑡′=10 years√1−(0.8)2=10 years√0.36=10×0.6=6 years
Statistical Mechanics - Partition Function
Question: Calculate the partition function 𝑍 for a system of non-interacting particles
with two energy states, 0 and 𝜖, at temperature 𝑇.
Solution:
The partition function 𝑍 is given by:
𝑍=∑𝑒−𝛽𝐸𝑖
𝑖
where 𝛽= 1
𝑘𝐵𝑇.
For the two energy states 𝐸1=0 and 𝐸2=𝜖:
𝑍=𝑒−𝛽⋅0+𝑒−𝛽𝜖 =1+𝑒−𝜖/(𝑘𝐵𝑇)
