A simple pendulum with bob of mass m and conducting wire of length L swings under gravity through an angle `2theta.` The earth's magnetic field component in the direction perpendicular to swing is B. Maximum potential difference induced across the pendulum is :

Options

A. `2BL sin ((theta )/(2)) (gL) ^(1//2)`
B. `BL sin ((theta )/(2)) (gL)`
C. `BL sin ((theta )/(2)) (gL) ^(2)`
D. `BL sin ((theta )/(2)) (gL) ^(5//2)`

Answer

A

Explanation

`h = L - L cos theta `
`implies h = L ( 1 - cos theta )" "...(1)`

`therefore v ^(2 )= 2 gh = 2 g L (1- cos theta )`
`= 2g L ( 2 sin ^(2) "" (theta )/( 2))`
`implies v = sqrt2 sqrt (gL) sin ""(theta )/(2)`
Thus, maximum potential difference
`V _(max) = BvL`
`= B xx 2 sqrt (gl) sin ""(theta)/( 2) L`
`= 2 BL sin "" (theta )/(L) (gL) ^(1//2)`