Changes in EMF due to a bar magnet falling through a loop of wire

[phosgene]

Homework Statement

A bar magnet is dropped through a horizontal coil as shown below. An EMF is induced in the coil with the resultant voltage shown on the graph.

Explain the shape of the graph, in particular the differences in the regions marked as A, B and C.

Homework Equations

Magnetic flux = BAcosx, where B = magnetic field, A = area of wire loop and x is the angle between B and a line perpendicular to the face of the wire loop.

EMF = -N(change in magnetic flux)/(change in time), where N = thenumber of loops of wire.

The Attempt at a Solution

In segment A, there is an increase and decrease in the EMF because the bottom of the magnet is passing through the loop. The change stems from the fact that the angle of the field lines with respect to the face of the loop is constantly changing, yielding a non-zero change in magnetic flux.

In segment B, the field lines are perpendicular to the loop and the angle doesn't change, thus there is no net change in magnetic flux and the EMF does not increase.

Segment C shows a change in EMF for the same reason as section A, however, it is more spiked and brief. This is due to the bar magnet moving faster due to gravity...making the change in magnetic flux more rapid.

I drew a picture which shows which sections of the magnet product which changes in EMF according to the segment (picture of magnet and field lines stolen from google, hope this doesn't break any rules!):

Am I right here?

 

[technician]

You explanation is excellent.
I will add a diagram to show how the direction of any induced current can be determined.
You should note that an induced current will only occur if thecoilis part of a complete circuit. In your case it is a closed loop butin my case I have completed the circuit by adding a light bulb... that is only done to make my diagram consistent.
Lenz's law determines the direction and I have drawn the situation as the S pole enters then leaves the coil. Can you see the induced current direction tries to prevent the S pole entering then tries to prevent the S pole leaving? This gives the first pulse on your graph.
When the N pole reaches the coil the same thing will happen but theinduced current direction will reverse... I have not drawn this.
Do you know why the size of the pulse changes?
Hope this adds something to your excellent answer.

 

[phosgene]

Thanks for the reply. :) The EMF changes from positive to negative because of the change in the direction of the current...at least I think that's what you were asking me.

 

[jmblock2]

Just to clarify a few points:

Only the perpendicular component is relevant in ALL cases (the part that goes through the loop of area A). The component tangential to the loop isn't passing through the loop. That's why there is cos x in your flux equation. So the reason section B is flat is because there is no change in flux (no change in the number of field lines passing through the coil), not that the field lines are perpendicular.

Additional interesting note, the integral of the graph in A is equivalent to the integral of the graph in C by the symmetry of the magnet. You are entirely right that the distribution is purely from the acceleration. Excellent!

But can you better explain why there is an exponential curve, then a line in A where as there is a line, then an exponential curve in C? A hint would be, what would that transition region look like if the bar was perfectly perpendicular to the plane of the coil.

 

[phosgene]

Thanks for your input and the clarification :). Isn't your question the same as my initial question (as I've understood it)? There's a curve and then a flat line in A because initially the part of the magnet falling through the loop has field lines whose angle with the perpendicular lines from the face of the loop is constantly changing. The flat line afterwards is because the bottom end of the magnet has fallen through and the angle of the field lines is now no longer changing because the center of the magnet is the part falling through the loop. And finally, there is a curve in C because the field lines at the top of the magnet are curved and so their angle is again changing.

 

[RG94]

Why then is the area under the graph of both spikes equal?