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Question

A conducting wire of parabolic shape, initially y=x2, is moving with velocity V=V0^i in a non-uniform magnetic field B=B0(1+(yL)β)^k, 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:


A
|Δϕ|=43B0V0L for β=2
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B
|Δϕ| remains same if the parabolic wire is replaced by a straight wire, y=x, initially, of length 2l
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C
|Δϕ|=12B0V0L for β=0
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D
|Δϕ| is proportional to the length of wire projected on y -axis
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Solution

The correct option is D |Δϕ| is proportional to the length of wire projected on y -axis

For calculating the motional emf across the length of the wire, let us project wire such that B,v,l becomes mutually orthogonal. Thus

dε=Bv0dy=B0[1+(yL)β]V0dy
ε=L0B0(1+(yL)β)V0dy
=BoVo[y+yβ+1Lβ(β+1)]L0
=BoVo[L+Lβ+1Lβ(β+1)]
=B0V0L[1+1β+1] emf in loop is proportional to L for given value of β.
For
β=0;ε=2B0V0L
β=2;ε=B0V0L[1+13]=43B0V0L
Since the projection of length of the wire on the y-axis is L, therefore the answer remain unchanged

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