2023 A-Level · Paper 2 · Question 6
Three wires between magnetic poles. One oscillates up and down; one is pushed straight through the field; one is folded in half and pushed through in the same way. Eight marks: Faraday's law, why the first e.m.f. alternates, the size of the second, and the surprising value of the third. Below is the full solution, and beside it each wire moving through the field.
State Faraday's law of electromagnetic induction.
The magnitude of the induced e.m.f. is proportional to the rate of change of magnetic flux linkage.
Watch
"An e.m.f. is induced when the flux changes" is a description of induction, not the law. The law is the proportionality, and it is to the rate of change of flux linkage.
A wire between two magnetic poles is displaced vertically and released, so that it oscillates. Explain why an alternating e.m.f. is induced across its ends.
Approach
Read the figure in three dimensions: the pole faces face each other across the wire, so the field is horizontal and perpendicular to the wire. A vertical oscillation therefore cuts the field. Then give two reasons: why there is an e.m.f. at all, and why it alternates.
As the wire moves up and down it cuts the magnetic flux, so by Faraday's law an e.m.f. is induced, of size . The speed changes continuously, so the e.m.f. changes; and each half-oscillation the wire reverses direction, so the e.m.f. reverses too.
The oscillating wire cuts the magnetic field lines, so the flux it cuts changes and an e.m.f. is induced (Faraday's law). When the wire moves up the e.m.f. is in one direction; when it moves down the direction of cutting reverses, so the e.m.f. reverses. Its size varies with the wire's speed, so the e.m.f. is alternating.
Note
The e.m.f. is largest as the wire passes through the middle (fastest) and zero at the top and bottom of the swing (momentarily at rest). It follows the velocity, not the displacement, so it is a quarter of a cycle ahead of it.
A wire moves at perpendicular to a field that exists only between poles wide. Find the induced e.m.f.
Approach
Only the part of the wire inside the field cuts flux. The rest of the wire moves through zero field and contributes nothing, so is the pole width, not the wire length.
In one second the wire sweeps an area of field, so the rate of cutting flux is :
e.m.f.
Watch
Using gives , which is wrong: of that wire is outside the field.
A wire XY is folded in half, with X just above Y, and moved through the same field at the same speed. State and explain the e.m.f. across X and Y.
Approach
Each half on its own is the wire of part (c), so each has an e.m.f. of pointing the same way in space. Now follow the wire from X to Y: you travel along one half and come back along the other.
Zero. Both halves cut the field at the same rate, so each has an e.m.f. of in the same direction in space. Going round the wire from X to Y, the second half is traversed in the opposite direction to the first, so the two e.m.f.s act in opposite senses and cancel.
Note
In flux terms: the folded wire encloses almost no area, so there is no change of flux linkage as it moves, and no e.m.f. This is why the leads to a meter are twisted together.
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