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Question:

A conducting wire of parabolic shape, y=x², is moving with velocity V→=V₀i^ in a non-uniform magnetic field B→=B₀(1+(y/L)²)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?

|ΔΦ|=4/3B₀V₀L for β=2

|ΔΦ| is proportional to the length of wire projected on y-axis

|ΔΦ|=1/2B₀V₀L for β=0

|ΔΦ| remains same if the parabolic wire is replaced by a straight wire, y=x, initially, of length 2L

Solution:

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→, becomes mutually orthogonal. Thus dε=Bv₀dy=B₀[1+(y/L)²]V₀dy
ε=∫₀ᴸB₀(1+(y/L)²)V₀dy=B₀V₀L[1+1/β+1]
emf in loop is proportional to L for given value of β.
for β=0; ε=2B₀V₀L
β=2; ε=B₀V₀L[1+1/3]=4/3B₀V₀L
The length of the projection of the wire y=x of length 2L on the y-axis is L thus the answer remain unchanged.
Therefore, answer is 1,2,4.