Loop `A` of radius `rgtgtR` moves toward loop `B` with a constant velocity `V` such a way that their planes are always parallel. What is the distance between the two loops `(x)` when the induced emf in loop `A` is maximum?

Options

A. (a) `R`
B. (b) `(R )/(sqrt(2)`
C. ( c) `(R )/(2)`
D. (d) `R(1 - (1)/(sqrt(2)))`

Answer

C

Explanation

( c) `phi_(A) = (mu_(0)ipiR^(2))/(2pi(R^(2) + x^(2))^(3//2))pir^(2)`
`rarr E_(A) = -(dphi)/(dt)`
`= (-mu_(0)ipi)/(2)R^(2)r^(2)(-3//2)(R^(2) + x^(2))^(-5//2)2x`
`E_(A)` is maximum when `(dE_(A))/(dx) = 0`
`rarr (d)/(dx)(x)/((R^(2) + x^(2))^(5//2)) = 0`
or `(R^(2) + x^(2))^(5//2) - (5x)/(2) (R^(2) + x^(2))^(3//2)2x = 0`
or `R^(2) + x^(2) - 5x^(2) = 0`
or, `x = (R )/(2)`