Shown in the figure is a very long semi-cylindrical conducting shell of radius R and carrying a current i. An infinitely long straight current carrying conductor is lying along the axis of the semicylinder. If the current flowing through the straight wire be `i_(0)`, then the force per unit length on the conducting wire is :

Options

A. `(mu_(0) ii_(0))/(pi R^(2))`
B. `(mu_(0) i_(0)i)/(pi^(2) R)`
C. `(mu_(0) I_(0)^(2)I)/(pi^(2) R)`
D. none of these

Answer

B

Explanation

The net magnetic force on the conducting wire per unit length `= F = int 2 dF cos theta`
`implies F = int 2 [(mu_(0) (di)i_(0))/(2 pi R)] cos theta`
`implies F = (mu_(0)i_(0))/(pi R) int di cos theta`
when `di = (i)/(pi R) xx Rd theta = (id theta)/(pi)`
`implies F = (mu_(0) i_(0))/(pi R) int ((id theta) cos theta)/(pi)`
`implies F = (mu_(0) i_(0) I)/(pi^(2) R) underset(0)overset(pi//2)(int) cos theta d theta = (mu_(0) i_(0) i)/(pi^(2) R)`