Consider a charged particle traveling at = 30◦ with respect to a uniform magnetic field of strength B = 2.6 mT as shown below. The particle has speed v= 3.0×106 m/s.

MagneticFieldsManual8.pdf

Magnetic Fields

Pre-Lab: Magnetic Fields

Name: Section:

1. Consider a charged particle traveling at = 30◦ with respect to a uniform magnetic field of strength B = 2.6 mT as shown below. The particle has speed v= 3.0×106 m/s.

B v

a) If the particle is a proton, what is the magnitude and direction (into or out of the page) of the force exerted on the particle by the magnetic field?

b) If the particle is an electron instead, how would your answer change?

2. What direction does Earth’s magnetic field point in Toledo? Does the direction change as you move North? South? How about right at the (magnetic) North pole? Hint: Earth’s magnetic field is similar to the field produced by a bar magnet.

3. Briefly summarize the procedures you will follow in this lab. Write one or two sentences for each activity.

4. List any part (or parts) of the lab that you think may suffer from non-trivial experimental error, or may otherwise cause you trouble. How might this affect your results?

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Electricity is actually made up of extremely tiny particles called electrons, that you cannot see with the naked eye unless you have been drinking.

Dave Barry

Objectives

• To understand how magnetic field is defined in terms of the force experi- enced by a moving charge.

• To understand the mathematical basis for predicting that a charged particle moving perpendicular to the magnetic field lines in a uniform magnetic field will travel in a circular orbit.

• To use an understanding of the forces on an electron moving in a magnetic field to measure the ratio of its charge to its mass, e/m.

Overview

As children, all of us played with small magnets and used compasses. Magnets exert forces on each other. The small magnet that comprises a compass needle is attracted by the earth’s magnetism. Magnets are used in electrical devices such as meters, motors, and loudspeakers. Magnetic materials are used in cassette tapes and computer disks. Large electromagnets consisting of current-carrying wires wrapped around pieces of iron are used to pick up whole automobiles in junkyards.

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100 Magnetic Fields

From a theoretical perspective, the fascinating characteristic of magnetism is that it is really an aspect of electricity rather than something separate. In the next two units you will explore the relationship between magnetic forces and electri- cal phenomena. Permanent magnets can exert forces on current-carrying wires and vice versa. Electrical currents can produce magnetic fields and changing magnetic fields can, in turn, produce electrical fields. In contrast to our earlier study of electrostatics, which focused on the forces between resting charges, the study of magnetism is at heart the study of the forces acting between moving charges.

Part I Magnetic Forces and Fields

Let’s try something unusual. Let’s see if a magnet can exert forces on elec- trical charges that are moving. In the front (or back) of the room there is an oscilloscope. With your magnet poles (labeled N and S), perform a qualitative investigation of the force exerted on the beam of electrons. To do this you will need:

• Oscilloscope • Bar magnet

Activity 1.1 The Magnetic Force Exerted on Moving Charges

1. Make a qualitative sketch of the B-field around a bar magnet, making sure that you show the direction of the field at both the north pole and the south pole.

2. Move the north pole of your magnet parallel and then perpendicular to the electron beam in an oscilloscope. What is the direction of the displacement (and hence the force on the beam) in each case? Sketch vectors showing

Part I. Magnetic Forces and Fields 101

Comment:

The force that

results from the

movement of charge

in a magnetic field is

known as the

Lorentz force. This

leads to a rather

backwards

mathematical

definition of

magnetic field B.

The magnetic field B

is defined as that

vector which, when

crossed into the

product of charge

and its velocity,

leads to a force F

given by the cross

product

�F = q�v�B .

the direction of the magnetic field, the direction of motion of the original electron beam before it was deflected, and the direction of the resultant force on the beam.

3. If you could perform quantitativemeasurements, you would find that when the bar magnet is perpendicular to the un-deflected beam, the actual mag- nitude of its magnetic force is F = qvB where q is the charge, v is the speed, and B is the magnetic field. As you should have found, the force depends on the orientation of B relative to v. Assuming that this depen- dence is a simple sine or cosine dependence, determine which function it is by making further observations. Explain your reasoning.

4. Show that the vector cross product �F = q�v×�B properly describes your observations (at least qualitatively) in terms of the relative directions of the three vectors. Hint: Don’t forget that q is negative in the case of an electron beam.

102 Magnetic Fields

Part II Forces and Fields Caused by Currents

A charged particle moving in a magnetic field experiences a Lorentz force per- pendicular to its velocity. In the next activity you are going to study a famous piece of apparatus used to measure the charge on the electron known as the “e/m apparatus”. The reason for the name is that it can be used to determine the ra- tio of e/m, but not the value of e or m individually. This may seem like small potatoes to you, but back when physicists were trying to determine these actual quantities that we all take for granted today, it was difficult to come up with exact values for either e or m for such a tiny fundamental particle as the electron! In an e/m apparatus, a beam of electrons is accelerated by a large potential difference to a fairly high velocity. The beam is then passed into a magnetic field that is perpendicular to the direction of motion so that it experiences a Lorentz force given by the now familiar equation

�F =−ev×�B , where −e is the charge of the electron. The way in which the electron beam bends in a known magnetic field allows us to determine the ratio of electron charge to the electron mass e/m experimentally.

This exploration of the motion of electrons in a uniform magnetic field will involve several activities:

1. First, you’ll predict what kind of path an electron will follow if it is shot into a uniform magnetic field.

2. You’ll figure out how you can measure e/m if an electron beam of known velocity is bent by a uniform magnetic field into a circle of measurable radius.

3. You’ll determine that current-carrying coils can produce a magnetic field that is fairly uniform.

4. You’ll use a classic e/m tube placed in a magnetic field to measure the ratio of charge to mass, e/m, for the electron.

For these activities you will need:

• e/m apparatus • Small compass • Ruler

Part II. Forces and Fields Caused by Currents 103

Activity 2.1 Magnetic Force on an Electron

Consider an electron that is shot with velocity v from left to right in the presence of a uniform magnetic field B that is into the paper. This is indicated by the x’s in the diagram below.

×

�Bin− �v

××

×××

×××

1. Using a qualitative argument based on the Lorentz force law, what is the direction of the force on the electron? You can use the terms up, down, left, right, in, or out to describe the direction of the force. Sketch a force vector on the charge in the diagram above.

2. In the next moment after it is launched, will the electron still be traveling in the same straight line? Why or why not? Sketch where it might be in the next moment on the diagram above.

3. If the Lorentz force is perpendicular to the direction of motion of the elec- tron in the first moment, is it still perpendicular in the second moment? In the third moment? Why or why not?

104 Magnetic Fields

4. If the force is always perpendicular to the direction of motion is any work done on the particle as it moves in a curved path? Recall that the formal definition of work for a small displacement s is given by the equation: W = F (in the direction of motion) times s.

5. If no work is done on the electron as it moves, does its speed (i.e., the magnitude of its velocity) change or remain the same?

6. The displacement of an electron bending in a magnetic field is shown in the diagram below for the first two moments. Complete the diagram and thus show the shape of the path of the electron in the magnetic field.

× �Bin

× ×

× ×× ×

× × ×

7. Suppose, like Mother Nature, you broke the path above up into a huge number of tiny steps. What would the shape of the path be? How might it change if you increase the magnitude of the magnetic field?

Part II. Forces and Fields Caused by Currents 105

Activity 2.2 Derivation of the e/m Equation

Now let’s put all the pieces together. Assume that we shoot a beam of electrons at velocity v perpendicular to a uniform magnetic field B and that we then observe that the electrons move in a circle of radius r. What is the theoretical equation you would use to calculate a value of e/m from your measurements of v, B, and r?

The magnitude of the centripetal force needed to keep a mass moving in a circle of radius r depends on the square of the speed and is given by the equation

Fc = mv2

r

At this point you have all the equations you need for the derivation of e/m.

1. Since the Lorentz force always acts inward toward the center of the circu- lar path taken by an electron in a magnetic field, find the equation of e/m in terms of v, r, and B by setting the centripetal force equal to the Lorentz force and solving the resultant equation for e/m.

2. What is the equation for the kinetic energy of a mass m moving at a speed v?

3. The electrons in an e/m tube are accelerated to a speed v by “falling” through a potential difference of V before being shot into the magnetic field. What is the equation for the potential energy lost by a charge e falling through a potential difference V?

106 Magnetic Fields

4. Set the potential energy lost by the electron equal to the kinetic energy gained by it as a result of its acceleration. Then show that e/m is given by the expression

e

m =

2V

r2B2 (8.1)

Activity 2.3 The Magnetic Field Inside the Coils

The only remaining task is to figure out how to determine the magnitude of the magnetic field that the beam of electrons moves through. In the typical apparatus used to measure e/m an approximately uniform magnetic field is produced by two large current-carrying coils called Helmholtz coils. For now, you should just accept on faith that the magnetic field between the Helmholtz coils can be calculated from well-accepted equations if the current in the coils is known.

For a pair of Helmholtz coils of radius R and spacing R, it can be shown that the magnitude of the magnetic field B in the region between the two coils is given by

B= 8√ 125

0NI

R , (8.2)

where 0 = 4 ×10−7 N/A2, N is the number of turns in the coil, R is the radius of each coil and the spacing between them, and I is the current through the coils.

The experiment should already be setup by the instructor. Do not turn on the power supply until your setup has been inspected by you lab instructor! You have the following controls:

Plate Controls the strength of the electron beam. Don’t exceed the volt- age listed on the tape on the coil apparatus!!

Part II. Forces and Fields Caused by Currents 107

Grid Focuses the electron beam. The effect is generally minor.

Field Controls the strength of the magnetic field. Adjust this to bend the electron beam.

1. Turn up the Plate voltage until you see the electron beam. Don’t exceed the voltage listed on the tape on the coil apparatus! Adjust the grid voltage if necessary.

2. Turn up the Field voltage until the electron beam begins to bend. Place the compass between the coils. Are the magnetic field lines parallel or perpendicular to the plane of the coils? What is the evidence for your answer?

3. Does the field inside the coils seem fairly uniform? Cite evidence for your answer.

4. Adjust the Field voltage until the electron beam hits the outer ring (4 cm diameter). Fill in the following table with your measurements. Reverse the�magnetic�field�direction�and�repeat�the�measurement�once�more.

Measurement Symbol [Units] Trial 1 Trial 2 Plate voltage V [V]

Radius of beam path r [m] 0.01 0.01 Field current I [A]

Number of turns of coil wire N Radius of Helmholtz coil R [m]

108 Magnetic Fields

5. Refer to Equation (8.2) above relating the magnetic field inside the coils to N, I, and R. Compute the value of the magnetic field B in Teslas for the two trials.

Activity 2.4 Determining e/m Experimentally

1. Use the values for the magnetic field B, the radius of the electron beam r, and the recorded value of the accelerating voltage V to calculate the value of e/m in Equation (8.1). Be sure to include units. Average the results from your two trials.

2. The accepted values of e/m is 1.76×1011 C/kg. What is the percent dis- crepancy between your measured value of e/m and the accepted value of e/m?