FARADAY'S LAW
References
Crummett and Western Physics: Models and Applications , Sec. 30.1
Halliday, Resnick, and Walker, Fundamentals of Physics (5th ed.), Sec. 31-3
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec. 26-2
Introduction :
The purpose of this experiment is to explore some aspects of Faraday's law of electromagnetic induction. According to Faraday's law, when the magnetic flux Φ
M
through a loop of wire changes with time, a potential difference or electromotive force (emf

) is induced in the loop whose magnitude is given by

= d  dt
M
(1)
In this experiment, the current in a large coil (it will be called the "field coil") creates a magnetic field. When this current is changing in time, the flux through a smaller "search coil" at the center of the field coil changes, inducing an electromotive force, given by (1), in the search coil.
Suppose the small search coil is in a magnetic field B , as indicated in Figure 1. Vector A in the figure is a vector, directed perpendicular to the plane of the coil , whose magnitude is the area of the coil. The magnetic flux through each turn of the search coil is given by
 ( each)
M
=

B


A = B A cos (

) (2)
If we think of the field strength B as the "density of lines" of the magnetic field, then (2) is the total
Figure 1 field
Search coil in Bnumber of lines passing through each turn of the coil. When θ is equal to zero, the magnetic field is perpendicular to the plane of the coil, and Φ = BA ; when θ is π /2 (that is, 90

), the field lines are in the plane of the coil, and the flux is zero. (Note that this definition of magnetic flux has the same form as the definition of the flux of the electric field used, for example, in Gauss's law.)
(2)
(1)
For a coil of N turns, the total magnetic flux through the entire coil is

M
= N  (each)
M
= N B A cos (

) (3)
If the magnetic field B is changing in time, then by (1) and (3),
4-37

4-38 FARADAY'S LAW

= d
 dt
= N s
A s cos (

) dB dt where N
S
and A
S
refer to the search coil.
The magnetic field B through the search coil is generated by the current I flowing in the larger field
(4) coil. If the field coil is circular, with N
F
turns of radius a , the magnetic field at its center is given by
B =

0
2
N a
F
I where μ
0

4 π 
10 -7 Tm/A is the magnetic force-law constant. When the voltage across the field coil
(5) changes, so does the current I = V / R flowing in it and, consequently, so does the magnetic flux through the search coil. Putting (5) into (4) gives for the emf induced in the search coil

=

0
N
F
N
2 a R
S
A
S cos (

) dV dt
In this experiment, you will measure d V /d t (the rate at which the voltage across the field coil is changing ), and the emf induced in the search coil, using the oscilloscope display, and use these measurements in several ways to check Eq. (6) above — and therefore Faraday's law.
Equipment oscilloscope signal generator field and search coils digital multimeter frequency meter leads
Procedure and Analysis
(1) Before you continue, give a complete derivation of Eq.(6) from Eq.(2)-(5), filling in any missing steps, in your notebook.
(2) Measure the inner and outer radius of each of your coils, and calculate an average radius for each.
Calculate the effective area A
S
of the search coil, using its average radius. Measure the total resistance R of the larger (field) coil and the resistor (roughly 1200 Ω ) in series with it. The number of turns in the coils are N
F
= 200 in the field coil, and N
S
= 4000 in the search coil.
(3) Set the Wavetek signal generator for a triangular output waveform (the signal you are applying to the field coil is something like what's shown in Figure (2), and set its frequency to around 500 Hz
(6)

FARADAY'S LAW 4-39
Figure 2 Triangular voltage waveform using the dial. Measure the frequency, f, using the frequency meter . Do not change the frequency setting of the signal generator after you make this measurement!
(4) Now connect the high impedance output of the signal generator to the Y input of the oscilloscope. Make sure that voltage (VOLTS/DIV) and time-base
(SEC/DIV) controls are in their
"calibrated" positions.
Make sure that these controls stay in this position throughout the experiment ; otherwise you will not know what the voltage and time readings that you take from the oscilloscope mean.
(5) Set the VOLTS/DIV to 5 V/cm and the time base to 0.5 msec/cm (you may also want to try the autoscale feature on the scope to optimize the display). These set (respectively) the vertical and horizontal scales of the oscilloscope display. Adjust the amplitude of the signal generator output to produce about a 15 V rise and fall on the oscilloscope display. Make sure that the frequency and amplitude settings on the signal generator are not changed during the rest of the experiment.
Measure the frequency of your signal using the oscilloscope to see whether it is consistent, within the 3% tolerance of the oscilloscope, with the frequency you measured before.
(6) Determine the rate of change of voltage with time (d V /d t ) in V/s, for both rising and falling voltages, by means of measurements made directly from the oscilloscope display. (The point of using the triangular output from the signal generator is that d V /d t is constant during each part of the cycle.)
(7) Now you know the value of d V /d t for the output of the signal generator. Connect this output to the field coil. Make sure that the oscilloscope input is set for DC coupling and that the flat oscilloscope trace, with no signal at the Y input, is along the central horizontal axis. Place both coils horizontally. In this arrangement, the angle θ between the axis of the search coil and the magnetic field produced by the field coil is 0, so cos( θ ) = 1.
Connect the search coil to the Y input of the oscilloscope. The oscilloscope display should look something like Figure 3. When the input voltage is rising at a constant rate d V /d t , the magnetic field produced by the field coil, and thus the flux through the search coil, will likewise be increasing at a constant rate. According to Faraday's law, there will then be a constant emf, V
A
, induced in the search coil; see Eq. (6). When the input voltage is decreasing at a constant rate, a constant emf of the opposite sense ( V
B
) is produced in the search coil. Measure the search coil emf's, V
A and V
B
, from the oscilloscope display.
(8) Calculate the search coil emf from Eq.(6) and your other measurements. Do your calculated values agree, within experimental error, with the values that you measured?

4-40
(9) Now rotate the plane of the search coil so that it makes a
45

angle with the plane of the field coil. Once again, measure
V
A
and V
B
. Try to make as precise a measurement as possible; change the
VOLTS/CM scale on the
FARADAY'S LAW oscilloscope if it will help. Each partner should make an individual measurement of V
A and V
B
.
Figure 3 Search coil signal
(10) Repeat the measurement
above for angles of 60

, 75

and 90

. As part of your report, make a graph of the induced emf
(the average of the absolute values of V
A
and V
B
) as a function of cos( θ ) . Explain what you see.
Is this graoh consistent with the theory?
(11) Change the signal generator output to a sine wave (do NOT change the frequency), and connect its output to the Y input of the oscilloscope. Measure the peak-to-peak voltage value of the sinusoidal signal produced by the generator. Each lab partner should make independent measurements. One half this peak-to-peak voltage value is the voltage amplitude V o
of the sinusoidal signal:
V = V
0 sin (
 t +

) where ω = 2 π f is the angular frequency of the signal. You can calculate d V /d t for this input signal just from taking the derivative of (7); its maximum value is
(7) dV dt  max
=

V
0
(8)
(12) Connect the signal generator output to the field coil, and the search coil to the oscilloscope input, with both coils horizontal. From (6) and (8), the amplitude of the emf induced in the search coil should be
 max
=

V
0

0
N s
N
2 a R
F
A s cos (

) (9)
Measure this maximum induced emf on the oscilloscope display, each partner making an independent reading, for θ = 0

and again for θ = 45

. Estimate the uncertainty in your measured maximum induced emf, and compare your measured value, with uncertainty, to what you calculate from Eq.(9).
Faradays law.doc last update 6/2002