Physics 16, Winter 1997

Lab 5 - Magnetic Field Mapping

Theory

Faraday's law states that if you have a coil of wire of n turns through which there is a changing

magnetic flux, then there will be an induced emf (measured in volts) in the coil of magnitude

ε = - n dφdt

(1)

where φ is the magnetic flux (φ = ∫∫ B . dA) through the loop. In this laboratory we will make use of

Faraday's law to map the magnitude of the magnetic fields created in a single coil of conduction

wire and in a matched pair of coils (Helmholtz coils) when an alternating current is passed through

them.

The Single Coil

The formula for the magnetic field on the axis of a circular loop of wire given in your text. If

this formula is modified to apply to a coil of N turns and radius R, and if the distance from the

center of the coil to the filed point in question is Z, then it becomes,

Figure 1

B = µoNIR2

2 R2 + Z2 3/2. (2)

In this experiment, the current through the coils will be alternating and can be expressed as I(t) = Io

sin ωt where ω = (2π) (60). Since both the magnitude and direction of the current will be varying

sinusodially at a rate of 60 Hz, the magnitude and direction of the magnetic field at a point along the

axis of the coil will also be varying sinusodially at 60 Hz. To reflect this time dependence, equation

(2) can be rewritten as B(t) = Bo sin ωt where

Bo = µoNIoR

2

2 R2 + Z2 3/2. (3)

If Io is in amperes and all length measurements are in meters, Bo will be in tesla.

To check the validity of equation (3) and to map the magnitude of the magnetic field along the

axis, we need an independent way to measure Bo. Consider then a small search coil consisting of n

turns of conducting wire and cross sectional area A placed along the axis of the large coil so that the

normal to A lies along the axis of the large coil as shown in figure 2. The flux φ through this small

search coil is

Figure 2

φ = B t . dA = ABosin ωt .

Since φ is varying sindusodially there will be an induced emf in the search coil; and an application

of Faraday's law yields an expression for its magnitude

ε(t) = - n dφdt

= - nABod(sin ωt)

dt

or

ε(t) = - n ωABocos ωt . (4)

If the ends of the search coil are connected to the input of an oscilloscope and ε(t) displayed on the

screen, the amplitude of ε(t),

εo = nωABo , (5)

can be measured. Since n, ω and A can also be easily measured, equation (5) provides us with an

independent way of measuring Bo. If εo is in volts and A is in m2, then Bo will be in tesla. We will

use this method to measure Bo in the laboratory.

Note that if the axis of the search coil is rotated so that it makes an angle φ with the axis of the

large coil, the amplitude of ε(t) becomes

εo = ωnBoAcos φ . (6)

Note also that the current in the large coil and the induced emf are 90o out of phase.

The Helmholtz Coils

Helmholtz coils are a matched pair (same number of coils, same radii, etc.) spaced a distance

apart equal to their radii. The field on the axis at the midpoint between them (point P in figure 3)

can be found from equation (2) by multiplying by 2 (there are two coils each producing a field at P)

and setting Z = 1/2 R (since the field point is at the same distance in front of one and behind the

other). Thus, B(t) = Bo sin ωt where

Figure 3

Bo = µoNIo

(1.4) R. (7)

We will use the same method employed with the single coil to check equation (7) and to map the

magnetic field. Helmholtz coils provide a very uniform field over a relatively large volume in the

space between the coils and are, therefore, a convenient and valuable apparatus for experiments

which call for a uniform magnetic field.

References

The sections in Bueche which are pertinent to this laboratory are:

(1) Chapter 22 sections 22.1 - 22.6

(2) Chapter 23 sections 23.1 - 23.8

(3) Chapter 24 sections 24.1 and 24.7.

They should be read before coming to lab.

Experimental Purpose

The specific experimental purposes of this laboratory are the following:

1. to experimentally measure the magnitude of the magnetic field, Bo, at the center of a single

coil and to compare that value to the value predicted by equation (3);

2. to experimentally measure the magnitude of the magnetic field, Bo, at a point on the axis

midway between the two coils of a set of Helmholtz coils and to compare that value to the

value predicted by equation (7);

3. to map the magnetic field strengths of a single coil and of a set of Helmholtz coils along both

their transverse and longitudinal axes by measuring Bo at various positions along those axes;

4. to confirm that the emf induced in the search coil is 90o out of phase with the current in the

coils creating the magnetic fields;

5. to experimentally confirm that the emf induced in a search coil which has its longitudinal

axis at an angle φ to the direction of a uniform magnetic filed is given by equation (6);

6. to make a rough estimate of the volume within a given set of Helmholtz coils in which the

magnetic field in uniform; and

7. to map the shape of the magnetic field of a single coil in a plane perpendicular to the plane of

the coil and passing through the center of the coil.

In all cases, the measurement of Bo will be made with the use of a search coil and equations (5) and

(6).

Procedure

The apparatus is illustrated in figure 4 on the next page. An oscilloscope is used to measure the

amplitude of the sinusodial voltages. The current in the large coils comes from a voltage reducing

transformer which is plugged into the line voltage. The voltage output from the transformer is

approximately 0.5 volts at 60 Hz and is taken from the terminals marked A. The terminals marked

B give the voltage across a 0.1 Ω resistor which is in series with the coils. The current in the coils

can be determined from the voltage across this 0.1Ω resistor and Ohm's law. The search coil has

20 turns and a cross-sectional area of 1 cm2. The Helmholtz coils have 60 turns and are

approximately 13.5 cm in diameter. The actual dimensions of the large coils should be measured.

A scale is mounted in the middle of the coils so as to measure distances along the axes of the coils.

1. Connect the positive terminal of the transformer output (the red jack of terminals A) to the red

jack on the Helmholtz coils. Connect the negative terminal of the transformer output to the bare

wire joining the wire in the front coil to the rear coil. Connect the B terminals of the transformer

to channel 2(Y) of the oscilloscope. Connect the search coil to channel 1(X) of the

oscilloscope. In this configuration only one coil will have a current passing through it.

Channel 1(X) of the oscilloscope can be used to measure the induced emf and channel 2(Y)

can be used to measure the current in the coil. Plug in the transformer and turn on the

oscilloscope. Place the scale along the axis of the coils making it level with its top edge 5 mm

below the midline (the approximate radius of the search coil).

2. Keeping the plane of the search coil parallel with the plane of the coil, hold it at the center of the

coil carrying the current and measure the amplitude of the induced emf. Then measure the

current in the coils. Use equation (3) to calculate the magnitude of Bo from Io and use equation

(5) to calculate the magnitude of Bo from εo. Compare these two values with each other and

with the given calibrated value.

3. Adjust the vertical gain controls on channels 1(X) and 2(B) so that the two are about equal.

What is the phase difference? Set the scope to the XY mode and observe the Lissajous

figure.Note what you observe and give a brief explanation.

4. Measure Bo at 5 mm intervals along the longitudinal axis starting 3 cm on one side of the coil

and ending 3 cm from the other side. Make a plot of B as a function of position along the axis.

5. Place the scale along the transverse axis and measure Bo along this line at 5 mm intervals

starting at one edge of the coil and ending at the other edge. Note that the curvature of the field

along the two directions (transverse and longitudinal) is opposite in the two directions so that

the center point is neither an absolute maximum nor an absolute minimum. Make a plot of B as a

function of position along this axis.

6. Connect the positive terminal of the transformer output to the positive jack of the Helmholtz

Figure 4

coils. Connect the negative terminal of the transformer output to the negative jack of the

Helmholtz coils. Repeat the measurement of steps 2-5. Make a rough estimate of the volume

within the coils where B is constant to ± 5%. Compare the homogeneity of the field of a single

coil with that obtainable from the Helmholtz arrangement.

7. Measure the change in the induced emf that occurs when the search coil is held at an angle to the

field so that the flux is reduced by a factor cos φ in equation (6). To do this, place a piece of

polar graph paper under the coils with the origin at the center. Hold the search coil at the center

and looking down form the top bring the stick out at 45o to the axis of the coils. Repeat for

angles of 30 o and 60 o. Note that this could be used to find the direction of the field at any

point in space around the coils. How? Set up the single coil again. Use this method to find the

shape of the magnetic field in a plane parallel to the table top which intersects the center of the

coils.

Lab Report

Follow the usual lab notebook format. Your lab report should include the answers to all of the

questions asked in the introduction or procedure, all raw and derived data, and an estimate of the

magnitude and sources of error in any data recorded. When answering any question or when giving

any comparison or explanation, always refer to specific data to support your statements. Be sure to

include the following in your notebook:

1. all raw data;

2. samples of all calculations done;

3. the results of the comparisons in step (2);

4. the results of the observation in step (3);

5. plots of B vs position for the single coil and Helmholtz coils along their transverse and

longitudinal axes;

6. a comparison of the field homogeneity obtainable from a single coil with that of a pair of

Helmholtz coils;

7. an estimate of the volume within the Helmholtz coils where B is constant within ±5%;

8. the results of the calculations and mapping done in step 7;

9. a discussion of the sources of error in this experiment with an estimate of their effect on the

results.