PHYS112 : Labs

8 Magnetic Fields and Faraday's Law

In addition to reading this assignment, you may need to refer to Appendix A on uncertainties and Appendix B on linear regressions.

Magnetic fields are produced by moving charges. An electromagnet produces a magnetic field by passing current through a wire loop or coil. A simple electromagnet consists of a single loop of current-carrying wire. The magnetic field at the center of a circular current loop of radius is given by

$\begin{displaymath} B = \frac{\mu_o i}{2 r} \end{displaymath}$

(22)

where $\mu_o = 4 \pi \times 10^{-7} \frac{\mathrm{Tm}}{\mathrm{A}}$ . If the loop consists of

turns of wire instead of a single loop, the magnetic field is simply

times the prediction of Eq. 22. You will have an opportunity to test this model with a multi-turn circular loop and a magnetic field probe. You will also use a magnetic field probe to investigate the magnetic field of the Earth in the lab.

Faraday's Law

The magnetic flux through a loop of area $\vec{A}$ in a magnetic field $\vec{B}$ is given by

$\begin{displaymath} \Phi^\mathrm{mag} = B A \cos \phi \end{displaymath}$

(23)

where $\phi$ is the angle between the field and area vectors as illustrated in Figure 18. The area is expressed here as a vector quantity with its direction perpendicular to the plane defined by the loop.

**Figure 18:** Illustration of the vectors involved in calculating the magnetic flux through a loop.
$\scalebox{0.7}{ \includegraphics{Faraday-flux.eps} }$

Magnetic flux is worth calculating, because any change in the magnetic flux through a conducting loop induces an emf in the loop which leads to a current. This phenomenon, known as magnetic induction is commonly applied in electric motors and generators. Faraday's law relates the emf induced in a wire loop of turns to the rate of change of the magnetic flux $\phi$ through it,

$\begin{displaymath} \mathcal{E} = - N \frac{d \Phi^\mathrm{mag}}{d t} \end{displaymath}$

(24)

Lenz's Law

Faraday's law gives the magnitude of the induced emf but not its polarity. This missing information is provided by Lenz's law:

An induced emf drives a current which produces a magnetic field opposing the change in magnetic flux.

To apply Lenz's law,

Determine the direction of a magnetic field which would oppose the change in the magnetic flux through the loop.
- If $\Phi^\mathrm{mag}$ is increasing, then a field opposing the existing field is required.
- If $\Phi^\mathrm{mag}$ is decreasing, then a field reinforcing the existing field is required.
Use the right hand rule for currents to determine the direction of a current that would produce the required magnetic field. The right hand rule for currents is illustrated in Figure 19. Hint: When applying Lenz's law, focus on the direction of the field inside the loop.
The polarity of the induced emf is such that it produces a current in the direction determined in step 2.

**Figure 19:** Illustration of the right hand rule for determining the relative directions of the current in a loop and the magnetic field $\vec{B}$ it produces.
$\scalebox{0.7}{ \includegraphics{Faraday-rhr.eps} }$

Experiments and Analysis

Testing Faraday's Law

You will test Faraday's and Lenz's laws by moving a permanent magnet through a wire coil and studying the resulting induced emf. Specifically, by measuring the speed of the magnet with a motion detector while also measuring the induced emf in the coil, you can test Faraday's prediction that the induced emf is directly proportional to the rate of change of the flux.

Plug the voltage probe into CH 1 and the motion detector into DIG/SONIC port 1 of the LabPro interface.
Connect the voltage leads to the two cylindrical metal contacts on the base of the coil (not to the wires of the coil which are painted with insulation).
Open the file Faraday.cmbl in Logger Pro.
Attach the permanent bar magnet to the meter stick with masking tape such that the north-south axis of the magnet is parallel to the stick.
Lay the heavy cylindrical coil on the lab table so that its axis is horizontal. Arrange the coil and motion detector so that you can use the meter stick to move the magnet through the coil while tracking its motion with the motion detector. For best results, hold a notebook or a textbook against the end of the meter stick to give the motion detector an easy target to track.
Click on Collect and move the magnet through the coil and back again several times at different speeds.
Make sure that the motion detector can pick up your full range of motion. It is ``far-sighted'' - it cannot track the motion of objects within about 50 cm of its front face.
Each back and forth motion should produce a particular response on the Potential vs. Time graph. Sketch this response curve and explain it using Faraday's law and Lenz's law. (You will be asked to hand this in as part of the individual assignment.) Your explanation should include a sketch of the coils and magnet showing the direction of the winding of the coils and which way the poles of the magnet are oriented.
For each passage of the magnet through the coil, use the Examine button ( $\scalebox{0.7}{\includegraphics{LoggerPro3-examine.eps}}$ ) to determine the magnitude of the maximum emf induced by the north pole of the magnet (or use the south pole - the important thing is to be consistent) and the corresponding speed. If you aren't sure how to consistently identify which pole corresponds to which peak, ask for help.
In each case, look carefully at the Distance vs. Time graph. Your motion detector may lose track of its target briefly from time to time, leading to spikes on the Distance vs. Time graph. These spikes do not reflect the actual motion, and the corresponding velocity graph is also unreliable. Throw away any data points with bad velocity measurements.
Record your maximum emf and speed values in separate columns in Excel.
Use Excel to graph the maximum induced emf vs. velocity.
Make more measurements, filling in gaps in your graph, until you feel there is enough evidence to determine whether or not your data is compatible with a linear model (see the first part of Appendix B: ``Is your data linear?''). Try to cover as wide a range of speeds as you can, and try not to leave large gaps in the graph.

The Magnetic Field of a Current Loop

Make sure the DC power supply is turned off and the voltage knob is turned all the way down (CCW).
Connect the black PASCO Scientific coil in series with the DMM, using the COM and 10 A terminals of the DMM (or 20 A, depending on the meter). Then, connect this series combination to the DC power supply. Caution! If you don't connect things properly, you could blow a fuse in the DMM. Please don't hesitate to ask for help if you are unsure of how things should be connected.
Turn the knob on the DMM to the 10 A current setting (or 20 A).
Turn the knob on the power supply until the DMM reads a current of 0.100 A.
Remove the motion detector and voltage probe from the LabPro interface.
Set the switch on the magnetic field sensor to the HIGH amplification setting ( $200 \times$ ), and plug it into CH 1 of the LabPro interface.
Run Logger Pro, and under File -> Open, select the file _Physics with Computers -> 28 Magnetic Field in Coil.
The active part of the magnetic field probe is the rectangular paddle at the end. It measures magnetic field along an axis perpendicular to the plane of the paddle. The white dot on the paddle faces the direction of positive field readings.
At the center of a current loop, the magnetic field is directed perpendicular to the plane of the loop. Place the magnetic field sensor at the center of the coil with the paddle parallel to the plane of the coil.
Turn off the DC power supply, and press the Zero button next to the Collect button in Logger Pro to zero the probe.
Turn on the DC power supply, and record the magnetic field reading.
Use repeated measurements to estimate the uncertainty in your result.
Measure the radius of the coil, and use Eq. 22 to predict the magnetic field at the center of the coil. (You will need to multiply this prediction by the number of turns in the coil, which is printed on it.)

The Magnetic Field of the Earth

Make sure that all magnets and things that might be magnetized are moved aside, and that the power supply is turned off.
Hold the Magnetic field sensor in the air.
Rotate the sensor both horizontally and vertically until you find the orientation that gives the largest magnetic field reading. (This will be a smaller field than the one produced by the coil.)
Rotate the magnetic field sensor 90 degrees along any axis so that it is perpendicular to the orientation that you just found. While holding it in this orientation, zero the probe. (Press the Zero button.)
Find the orientation giving the maximum reading again.
Record the magnitude and direction of the magnetic field. (When the probe gives its maximum positive reading, the white dot is facing in the direction of the magnetic field.) Describe the direction of the field in three dimensions, relating it somehow to landmarks in the room.

Before You Leave Lab

Discuss with your instructor preliminary answers to the questions below.

Hand In ...

... a printout of your spreadsheet and answers to the following.

Hand in a discussion of the graph of the response (Potential vs. Time) of the coil to the magnet moving through it. Include an explanation of the polarity of the signal.
Is your graph of induced emf vs. speed compatible with Faraday's law? Explain.
How does the measured magnetic field of the PASCO Scientific coil compare with the prediction of Eq. 22?
Report your measurement of the magnitude and direction of the Earth's magnetic field in the laboratory. Does the direction make sense? Explain.