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Question

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.

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Solution

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)32Φa2

E.m.f. induced

= dΦdt=μ0Nia2a2π2(a2+x2)32didt

= μ0Nπa2a22(a2+x2)32ddte(RLx+r)

= μ0Nπ2a22(a2+x2)32e.1.RL.v(RLx+r)2

(a) For x = L

e = μ0Nπa2a2RvE2L(a2+x2)32(R+r)2

(b) e = μ0Nπa2a22(a+x2)32ERvL(RL+r)2

[forx=L2,RL,x=R2]


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