Physics

Homework Questions

1. Assume the muscle is 37oC and is separated from the outside air by layers of fat and skin. The layer of fat, at a particular location on the skin, is 2 mm thick and has a conductivity of 0.16 W m-1 K-1. Above this is a 1.1 mm dermis layer with a thermal conductivity of 0.53 W m-1 K-1. Finally, the outermost epidermal layer is 0.1 mm thick with a thermal conductivity of 0.21 W m-1 K-1. How much heat is lost per unit area and unit time if the ambient air temperature is 0oC?

2. Graphene consists of a 2D network of carbon atoms just one atom thick. This simple network of atoms results in remarkable physical properties: graphene has higher strength, better thermal conductivity, and greater intrinsic mobility than any other material known. These properties translate into huge technological potential, for example, in transparent electrodes for flatscreen TVs, in fast, energy-efficient transistors, and in ultrastrong composite materials. Hence an enormous global effort is focused on understanding and controlling graphene's properties with the aim of tailoring them for specific applications.

Imagine the atoms are charged with a charge q, calculate the electric field in the center of a hexagonal unit. Clearly state any assumptions made.

Calculate an expression for the electric potential energy of 6 equally charged particles distributed on the vertices of a hexagon.

3. Takao Someya has developed flexible conductors from polymers and carbon nanotubes. These materials can be stretched up to 35% without losing structural integrity. This opens up the possibility for futuristic therapeutic devices, such as instrumented surgical gloves.

When mixing two conducting species the overall conductivity is limited by the Voigt-Reuss bounds. The maximum conductivity is given by

The minimum is given by

The exact relationship depends on how the two materials are mixed, but must be in between these two bounds. Is the volume fraction of nanotubes, is the volume fraction of polymer. The conductivity of the nanotubes is 1000 times greater than the polymer.

At what volume fraction of nanotubes is the difference between the two bounds the greatest?

At what volume fraction of nanotubes is the average between the two bounds 10 times greater than the conductivity of the polymer?

4. In the circuit shown, calculate the current through the resistive load (the 100 Ohms resistor).

What is the voltage difference between the two circles, across the 40 Ohm resistor?

5. Consider the figure below which shows a conducting bar being pulled upward, whilst maintaining electrical contact with the conductive rails. Find the current as a function of time induced in the bar assuming that the velocity of the bar is constant (and known), and that the magnetic field magnitude is also known. Assume the rails have negligible resistance relative to the resistance in the bar.

6. Consider the following charged rod. It is of length L and the charge per unit length increase from zero at one end to at the other

where x is a spatial coordinate starting at one end of the rod. Obtain an expression for the electric field as a function of the spatial coordinates in a two-dimensional plane containing the rod.

7. Assuming that the index of refraction of water is 4/3 and that raindrops are spherical, show that the location of a rainbow is approximately 42o above the line from the sun to you. Hint: Start by ray tracing the light inside the water droplet. Then minimize the deviation angle (the deviation of the outgoing light to the incident light). This is when the light bunches up and causes a bright spot.

8. A spherically symmetric charge distribution results in an electric potential of the form

What is the charge distribution? Hint: consider the difference in electric field between two values of r.

9. Solve for the 7 currents in following circuit.

10. Dowsing is a form of divination used to locate hidden water wells, buried metals, gemstones, or other objects. A scientific explanation was proposed for this phenomena claiming that moving ones arm in the earths magnetic field could induce an emf along the arm of the dowser and cause muscles to twitch and the dowsing rod to move. The earths magnetic field could be slightly higher over water or other conducting materials, causing muscles to twitch over these objects (almost like a human metal detector).

Experimentally investigate this phenomena. Use the casing from two old pens and place bent pieces of wire inside the casing. Do you see any affect as you walk around?

Use your knowledge of electricity and magnetism to theoretically investigate this phenomena. Useful information for this question will include the action potential (voltage) required for muscle contraction which is 10 mV and the earths magnetic field (in Pittsburgh) is 55 T. Are you able to generate this potential in your arms by moving them in the earth's magnetic field?

11. Superconductivity was discovered on April 8, 1911 by Heike Kamerlingh Onnes, who was studying the resistance of solid mercury at cryogenic temperatures using the recently-produced liquid helium as a refrigerant. In an experiment carried out by S. C. Collins between 1955 and 1958, a current was maintained in a superconducting lead ring for 2.50 yr with no observed loss. If the inductance of the ring was 3.14 10-8 H and the sensitivity of the experiment was 1 part in 10 9, what was the maximum resistance of the ring?