Find the force experienced by a semicircular rod having a charge q as shown in fig. Radius of the wire is R, and the line of charge with linear charge density `lambda` passes through its center and is perpendicular to the plane of wire.

Options

A. `(lambda q)/(2 pi^(2)epsilon_(0)R)`
B. `(lambda q)/(pi^(2)epsilon_(0)R)`
C. `(lambda q)/(4 pi^(2)epsilon_(0)R)`
D. `(lambda q)/(4 pi epsilon_(0)R)`

Answer

B

Explanation

`F_("net")=int dq E cos theta`
`=int_(-pi//2)^(pi//2)((q)/(pi R))Rd theta (lambda)/(2 pi epsilon*R) cos theta`
`=(lambda q)/(2 pi epsilon_(0)R)int_(-pi//2)^(pi//2) cos theta d theta =(lambda q)/(2 pi epsilon_(0)R)[sin theta]_(-pi//2)^(pi//2)`
`(lambda q)/(2 pi epsilon_(0)R)[1-(-1)]=(lambda q)/(pi^(2) epsilon_(0)R)`