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Magnetic Fields and Magnetic Forces Page 1 of 6

Magnetic Fields and Magnetic Forces

NAME: ____________________________________

NAME: ____________________________________

NAME: ____________________________________

RECITATION SECTION: __________________________

INSTRUCTOR: __________________________

DATE: __________________________

This activity is based on the following concepts:

 All currents create a magnetic field. We can calculate this field using two different approaches: the Biot-Savart Law and Ampère’s Law (today we focus on the latter)

 Ampère's Law is useful in cases of obvious symmetry and relates the integral of the

magnetic field B 

around a closed loop C to the total current I passing through the area

bounded by the loop: IsdB C

0 

 A magnetic field B 

is defined in terms of the force BF 

acting on a test particle with

charge q moving with velocity v 

via: BvqF 



 Recall the rules for the cross product of vectors:

o The magnitude of the resultant vector is given by sinA B A B   where θ is the

angle between vectors A and B . o The direction of the resultant vector is given by the right-hand rule. One version of

this rule states that if you align the fingers of your right hand with the direction of the

first vector and orient your wrist such that you can curl your fingers into the direction

of the second vector, your thumb will point in the direction of the resultant vector.

o A useful mnemonic for remembering how to write the components of the cross- product comes from matrix algebra. We can write the cross-product of two vectors as

the determinant of the following matrix:

ˆˆ ˆ

x y z

x y z

i j k

A B A A A

B B B

 

o It’s also often helpful to know that ˆ ˆ ˆ i j k . If you rotate that equation (e.g. to

ˆ ˆˆ j k i the sign is still positive, but if you flip it you change the sign: ˆ ˆ ˆ  j i k

Magnetic Fields and Magnetic Forces Page 2 of 6

Exercise 1

The figure below shows the cross-sectional view of a coaxial cable that consists of a “core” wire

of radius a that carries a uniform current i out of the page, whilst the “shielding” is a very thin

cylindrical metal shell of radius b that carries a current i into the page. Your aim is to use

Ampère’s Law to determine the magnetic field as a function of the distance r from the cylinder

axis. The exercise is divided into three distinct regions as indicated below. In each case, you

should show the following steps:

(a) Draw an appropriate closed loop for carrying out the Ampere Law integral, clearly indicating the direction of integration;

(b) Set up the integral, clearly indicating what is the current enclosed by the loop and how you selected the “sign” of the current involved;

(c) Solve the integral to yield an expression for B(r), and check what values it gives at boundaries (i.e. what are B(r) for r = a and r = b?);

(d) If the magnetic field is non-zero, indicate its direction (e.g. counter-clockwise or clockwise).

Q1. Region I: r > b

Q2. Region II: a < r < b

Core radius a

current i out of page

Shielding radius b

current i into page

Magnetic Fields and Magnetic Forces Page 3 of 6

Q3. Region III: r < a

Q4. Sketch below the variation of B(r).

r

B(r)

r = a r = b

Magnetic Fields and Magnetic Forces Page 4 of 6

Exercise 2:

The following equations describe five situations in which a proton with velocity v 

is moving

through a magnetic field B 

.

I. ĵ3î4 v 

m/s and k̂2B 

T

II. ĵ4î3 v 

m/s and k̂5B 

T

III. k̂4ĵ3 v 

m/s and î5B 

T

IV. k̂5ĵ4î3 v 

m/s and k̂10ĵ8î6 B 

T

V. k̂5ĵ4î3 v 

m/s and k̂5ĵ4î3 B 

T

Q5. For each of the five situations above, make a sketch the velocity and magnetic field vectors

in a way that clearly shows the relative orientation of the two vectors in 3D, being sure to label

both vectors. On the same sketch, then indicate the direction of the magnetic force exerted on the

proton. Your sketches need not be perfect; the purpose of this exercise is to become more

comfortable with visualizing the orientation of the vectors.

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Q6. Now, rank the five situations given above in order of increasing magnetic force on the

proton. Note that there may be more than one way to correctly determine this ranking. For

example, based on your sketches, are the velocity and magnetic field perpendicular to each

other? Are they parallel? If they are either perpendicular or parallel, can you use this to calculate

the magnitude of the force, based only on your knowledge of the magnitudes of the vectors and

the angle between them? If not, how must you determine the magnitude of the force? Be sure to

show the calculations which support your ranking.

Exercise 3:

The figure below shows the path of a proton (mass mp) as it passes through two regions of space

(labeled 1 and 2) containing uniform magnetic fields of magnitudes B1 and B2, respectively. Both

fields are oriented perpendicular to the page. The proton’s path in each region is a semi-circle of

radius R1 and R2, respectively. In answering the questions that follow, make sure you defend

each of your answers with a well-reasoned argument.

Region 1

Region 2

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Q7. Does the proton move faster in one region than the other? Why or why not?

Q8. Is the magnetic field stronger in one region than the other? If so, which magnetic field is

stronger and how can you tell?

Q9. What are the directions of the two magnetic fields?

Q10. Does the proton spend more time in region 1 or region 2?

Q11. If we replaced the proton with an electron (mass me=5.44×10 –4

mp) while leaving the rest

of the experiment the same (i.e., same initial velocity, same magnetic fields, same starting point,

etc.), how would the path followed by the electron be different?