Figure shows a circular coil of N turns and radius a, connected to a battery of emf ε through a rheostat. The rheostat has a total length L and resistance R. The resistance of the coil is r. A smalll circular loop of radius a' and resistance r' is placed coaxially with the coil. The centre of the loop is at a distance x from the centre of the coil. In the beginning, the sliding contact of the rheostat is at the left end and then sliding contact of the rheosts is at the left end and then onwards it is moved towards right at a cinstant speed v. Find the emf induced in the small circular loop at the instant
(a) The contact begins to slide and
(b) It has slid through hals the length of the rheostat.
Magneticfield due to the coil (1) at center of (2) is
B = μ0Nia22(a2+x2)32
Flux linked with the second,
Φ=B.A′=μ0Nia2(a2+x2)32Φa′2
E.m.f. induced
= dΦdt=μ0Nia2a′2π2(a2+x2)32didt
= μ0Nπa2a′22(a2+x2)32ddte(RLx+r)
= μ0Nπ2a′22(a2+x2)32e.−1.RL.v(RLx+r)2
(a) For x = L
e = μ0Nπa2a′2RvE2L(a2+x2)32(R+r)2
(b) e = μ0Nπa2a′22(a+x2)32ERvL(RL+r)2
[forx=L2,RL,x=R2]