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Q.

A conducting wire of parabolic shape, initially y=x2  is moving with velocity V→=V0i^ in a non – uniform magnetic field  B→=B01+yLβk^, as shown in figure. If V0.B0,L and β are positive constants and Δϕ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are

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a

Δϕ=12B0V0L for β=0

b

Δϕis proportional to the length of the wire projected on the y-axis.

c

Δϕremains the same if the parabola wire is replaced by a straight wire, y=x initially, of length  2L

d

Δϕ=43B0V0L for β=2

answer is B.

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Detailed Solution

Δϕ=∫V0B01+yℓβ⋅dy⇒Δϕ=B0V0y+(y)β+1(β+1)ℓβ0ℓ⇒Δϕ=B0V0ℓ+ℓβ+1⇒Δϕ=B0V0β+2β+1ℓ If ,β=0⇒Δϕ=2β0V0ℓ If ,β=2⇒Δϕ=B0V043ℓThe length of the projection of the wire y = x of length 2L on the y-axis is L thus the answer remain unchanged.
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