Question

fred places two circular loops of conducting wire, one large (radius R) and one small (radius r),
on a horizontal table such that they are centred at the same point. He arranges the larger loop
to carry a constant current I clockwise (when viewed from above). In order to induce a current
in the smaller loop, fred then lifts the smaller loop vertically upwards at a constant speed v,
without changing its orientation.
1 - Explain why fred is correct to expect an induced current in the smaller loop, and determine which way this current will flow.
2 - Show that the emf induced in the small loop is approximately
E = (U_03IR^2v/2z^4)*PIr^2
where z is the height above the table. You may use the result that the magnetic field
along the z-axis produced by the large loop is B = µ_0IR^2/2z^3. Make sure to explain,
with the aid of a diagram, the simplifying assumptions you need to make.
3 - Meanwhile sam, who is also in the room, happens to be levitating upwards at the same
velocity as the small loop. Thus the small loop is stationary in her frame. Explain,
from sam’s point of view, why she should also expect to see an induced emf in the small
loop. Your explanation should comment on the differences between fred’s and sam’s
description of the cause of the emf and explain how it is that they can both be correct.

Answer 1

Figure shows the circular coil of radius R through which current I is passed through to create magnetic field.

A small concentric loop of radius r is moved along the direction of axis with speed v as shown in figure.

Magnetic field produced due to the current passing through bigger loop at axis is given by

....................(1)

where z is axial distance of small loop from bigger loop as shown in figure. Since magnetic field varies with distance z , when small loop is moved along the axis with speed v as shown in figure, there is change in magnetic flux . This change in magnetic flux creates induced emf in the small loop , hence induced current flows in the small loop as shown in figure.

Direction of induced current will be opposite to the direction of current in the bigger loop that creates magnetic field.

induced EMF E = -d/dt

where = B A , is the magnetic flux, B is magnetic field induction and A is area of small loop of radius r.

hence induced EMF E = - d/dt ( B A ) = - A (dB/dt) .............(2)

we get dB/dt by differentiating eqn.(1)

................(3)

we have substituted dz/dt = v , speed of coil movement, in above equation

Hence from eqn.(2) and (3) , we get EMF E as

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(3) it appears from the question, Sam is moving bigger loop keeing small loop as stationary and observing induced emf in small loop.

If we move bigger loop we have variation of magnetic field along the axis as explained in previous part.

Hence variation of magnetic field due to movement of bigger loop creates induced emf in smaller loop , hence smaller loop gets induced current.

When fred is moving smaller loop , magnetic field is stationary and other loop is moving through varying magnetic field region . When Sam is moving bigger loop , smaller loop is statioary but smaller loop experiences varying magnetic field due to movement of bigger loop .

To get induced EMF there should be relative motion between magnetic field and conducting loop .