Conducting wire of parabolic shape, initially `y=x^(2)` is moving with velocity `vec(V)=v_(0) hati` in a non-uniform magnetic field `vec(B)=B_(0)(1+(y/L)^(beta))hatk` as shown in figure. If `V_(0) , B_(0) L` and B are +ve constant `Deltaphi` is potential difference develop between the ends of wire, then correct statements (s) is/are

Options

A. `|Deltaphi|=1/2 B_(0)V_(0)L` for `beta=0`
B. `|Deltaphi|=4/3 B_(0)V_(0)L` for `beta=2`
C. `|Deltaphi|` is proportional to the length of wire projected on y-axis
D. `|Deltaphi|` remains same if the parabolic wire is replaced by a straight wire, y=x, initially of length `sqrt(2)l`

Answer

B, C, D

Explanation

For calculating the motional emf across the length of the wire, let us projected wire such that `vec(B),vec(v),vec(l)` becomes mutually orthogonal. Thus
`depsilon=Bv_(0)dy=B_(0)[1+(y/L)^(beta)]V_(0)dy`
`epsilon=int_(0)^(L)B_(0)(1+(y/L)^(beta))V_(0) dy`
`=B_(0)V_(0)L[1+1/(beta+1)]`
emf in loop is proportional to L for given value of `beta`.
for `beta=0 , " " epsilon=2B_(0)V_(0)L`
`beta=2, " " epsilon=B_(0)B_(0)L[1+1/3]=4/3 B_(0)V_(0)L`
the length of the projection of the wire y=x of length `sqrt(2)L` on the y-axis is L thus the answer remain unchanged
Therefore, answer is B,C,D