An inverted L shaped conductor PRQ is made by joining two perpendicular conducting rods, each of length 1.5 L, at end R. This structure is moving in x-y plane containing variable magnetic field `vec(B) = -3xhate(k)` with a velocity `vhate(i) + vhat(j)`. If potential of P is `V_(p)` and that of Q is `V_(Q)`, then value of `V_(P) - V(Q)` at the instant when P is at origin as shown in Fig. Will be

Options

A. `(9vL^(2))/(8)`
B. `(27vL^(2))/(8)`
C. `-(9vL^(2))/(8)`
D. `-(27vL^(2))/(8)`

Answer

D

Explanation

`V_(R)-V_(P)=int(vec(v) xx vec(B))*dvec(e)`
`=int (v hat(i)+vhat(j)) xx (-3 xhat(k))*dxhat(i)`
`=3v int x(-hat(j)+hat(i)).dxhat(i)`
`V_(Q)-V_(R)=int (vhat(i)+v hat(j)) xx (-3(1.5L)hat(k)).dy hat(j)`
`=-4.5 vL int(-hat(j)+hat(i)).dyhat(j)`
`=4.5 vl int_(0)^(1.5L_ dy=(27)/(4) vL^(2)`
from above
`V_(P)-V_(Q)=27/8 vL^(2)-27/4 vL^(2)=(-27)/(8) vL^(2)`.