phy

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FARADAY’S LAW

Introduction

In this experiment, you will observe induced electromotive forces in a solenoid and compare

their time dependences with those predicted by Faraday’s Law.

The beginnings of modern electrical science and technology can be traced to two very

important discoveries in the early Nineteenth Century. The first, seen first in many different

contexts by several scientists, but generally called Ampere’s Law, recognizes that electrical

currents give rise to magnetic fields and gives the mathematical relationship between them.

Thus, for example, using Ampere’s Law (and Gauss’ theorem for magnetic fields, ∮ ���� ∙ ����� = 0), one can deduce expressions for the magnetic fields associated with a conducting solenoid. The

solenoid used in this experiment is a long, tightly wound, circular coil of insulated wire. If the

solenoid has N1 loops, or “turns” over a total length L1, then when it is conducting a current I1, the

interior magnetic field is directed along the solenoid’s axis (in the direction of net current flow)

and has the magnitude

� = � ����

�� (1)

where µ0 is the permeability of free space: µ0 = 4π × 10 -7

Tesla·meter/amp.

The second great discovery relating electrical and magnetic phenomena was the work of

Faraday, Henry and Lenz. Its mathematical expression is called Faraday’s Law.

Their experiments can be summed up as follows: an electromotive force is induced in a

conductor whenever there is a change in the magnetic flux going through the conductor. This

change of flux can be produced by changing the magnetic field; by changing the effective area of the loop or both at the same time. The key to the induction process is change.

Qualitatively, Faraday’s Law says that a time-varying magnetic field induces electromotive

force, or non-electrostatic voltage drop, that can, among other things, cause currents to flow.

These currents will themselves generate additional magnetic fields which, according to Lenz’s

Law, will in general oppose the changes in the original magnetic fields, but this effect will have

very little bearing on your particular experimental results.

In order to fully understand how induction occurs in coils and to explain the different phenomena

the following set of basic concepts, that describe the relationship between the magnetic field and

charge, are summarized below:

A stationary charge does not generate a magnetic field. Only an electric field is generated. In

addition, a magnet has no effect on a stationary charge.

Charges moving in a specific direction at a constant speed will generate a constant magnetic

field in a given point of the space. It will generate also a constant electric field. However, the two

fields are uncoupled. If the stream of charges (or current line) is alternating in direction and

varying in strength over time, then so will be the generated magnetic and electric fields. In this

case both fields will be coupled.

The magnetic flux, �, is given by

� = ���� ∙ ���� = ∙ � ∙ ���(�) (2)

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where B is the intensity of the magnetic field crossing the coil, in Tesla (T), A is cross-sectional

area of the coil in square meter (m 2 ) and θ is the angle between a normal to the plane of the coil

and the magnetic field vector. The flux is measured in Webers (Wb), (1Wb = 1T·1m 2 ). In this lab,

magnetic field is perpendicular to the area A of the pick-up coil (parallel to the axis of the coil),

so B·A·cos(0°) = B·A.

The emf induced in a coil is found to depend on several factors: the number of turns N in the coil,

the change in the magnetic flux dΦ through the loop, and the time dt required to produce this

change. The average induced electromotive force Vemf is given by

���� = −! "#

"$ (3)

The induced emf is proportional to the rate of change in the magnetic flux ΦΦΦΦ the loop is

exposed to and the number of turns N in the loop. If the magnetic flux Φ is in units of Webers,

time t in seconds, the induced voltage Vemf will be in Volts.

We can generate a time-dependent magnetic field B1(t) inside our solenoid by varying the

current through the solenoid. B1’s time dependence will very nearly the same as I1(t)’s. Then,

according to Faraday’s Law, a small wire probe placed in the solenoid should experience an

induced electromagnetic force - emf. Indeed, if the probe itself a coil of N2 turns and effective

loop area, A2, and if its axis is parallel to the axis of the solenoid, Faraday’s Law gives us the

quantitative expression for the induced voltage across the probe:

���� = −!%�% "&� "$

(4)

The right-hand side of this expression is just the (negative of) the time derivative of the total

magnetic flux, Φ2, linking the probe.

The induced emf in the probe will generate a voltage signal that can be displayed, along with

the original signal voltage, on an oscilloscope.

Apparatus

Your team is provided with the following apparatus:

1. A signal generator that can provide three wave forms: square wave, triangular and sinusoidal.

2. An oscilloscope. 3. A large solenoid with N1 = 550 turns (or 770 turns: check with instructor). 4. A probe coil with N2 = 315 turns. 5. A resistance box and various connection cables. The resistance box should be set at the

value far exceeding the resistance of the solenoid.

6. Rulers and Vernier Calipers.

The solenoid should be connected through the resistance box and the signal generator as

shown in the diagram below. Connect both the solenoid and the probe to the oscilloscope as

indicated. Make sure that the CAL knobs on the oscilloscope are set into the calibrated position.

Please note that the probe and the solenoid connectors are fragile. Handle the equipment with

care.

Familiarize yourselves with the apparatus by observing the effects of adjusting frequency,

amplitude and resistance controls on the signal generator’s and the probe’s waveform displays.

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Procedure

You have the possibility of testing Faraday’s Law for two different periodic time-

dependencies of the current in the solenoid (sinusoidal and triangular). The parameters you can

adjust are the amplitude and the frequency; the geometry and numbers of turns in the coils are

fixed.

You should begin by considering in turn each of the input signal wave forms at your disposal

and attempting to predict the behavior of the output signal from the probe with respect to the

parameters you can vary. Thus, for example, how do you expect the time-dependence of the

output signal to be related to that of the input? How should the amplitude of the output depend on

the amplitude and, or, the frequency of the input? In your analysis (as long as R is much larger

than the internal impedance of the solenoid), you may use the relationship '� = ()*

+ where R is

the resistance box setting and Vsg is the signal generator voltage.

Be aware that because of the alternating current principles the magnitude of the current in the

circuit (hence the magnetic field) depends on the frequency. For better results you may want to

make some small adjustments in the output signal amplitude to keep the current amplitude

constant. In Graphical Analysis make the graphs Vemf. meas. (V) vs. frequency f (Hz) and Vemf. meas. (V) vs. Vr,m (input amplitude). Are the graphs linear? What does that mean?

Hints: A Sample Experimental Procedure

If your team is having difficulty getting started, this sample procedure is provided for the

sinusoidal wave-form.

We switch the signal generator wave-form selector to ‘sinusoidal’ and suppose that the

voltage signal picked up across the resistor is

�, (-) = �,,� sin(234-),

where Vr,m is the amplitude and f is the frequency. Both can be measured on the oscilloscope

screen if we adjust the amplification and sweep to appropriate values.

With this expression for Vr(t), we have

Signal Gen. To Channel #1

VR + -

R

B

N1= 550 turns N2= 315 turns

To Channel #2

- +

Pick-up Coil L1

A

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'� = (�,,�/6) sin(234-).

Then from equation (1),

� = 78!�

9�

�,,� 6

sin(234-)

To find the induced emf, we take the time-derivative of this and insert it into equation (4) to

obtain

���� = − %:� �

��+ !�!%�%�,,� cos(234-) (5)

This is the equation of the wave form we should be observing as the lower trace on the

screen. We note three important features: 1) the induced voltage is proportional to the input

signal voltage; 2) the induced voltage is proportional to the driving frequency; 3) the input signal

voltage does not affect the output frequency, but it is 90° out of phase with the induced voltage.

These statements can be checked quantitatively by adjusting the input frequency and voltage and

measuring the output. In particular, a plot of Vemf,m, which is the amplitude of the induced voltage

(equation (5)), vs. frequency f should be a straight line with slope equal to 2πµ0N1N2A2Vr,m /L1R.

Finally, since we know, or can measure, all the other parameters in equation (5), we can

predict the ratio Vemf,m /Vr,m and compare it to our observed results.