A half ring of radius R has a charge of `lambda` per unit length. The electric force on 1C charged placed at the center is

Options

A. zero
B. `(klambda)/R`
C. `(2klambda)/R`
D. `(kpilambda)R`

Answer

C

Explanation

As, R be the radius of the ring. Consider a small strip of length dl having charge dq lying at an angle `theta`.
`dl = Rdtheta`
Charge on dl=`lambdaRdtheta`
Force at 1C due to dl
`=(klambdaRdtheta)/(R^(2)) = (klambda)/R dtheta=dF`
We need to consider only the component `dRcostheta`, as the component `dFsintheta` will cancel out because of the symmetrical element dl.

The total force on 1 C is
`F= int_(-pi//2)^(pi//2)dFcostheta= (klambda)/Rint_(-pi//2)^(pi//2)costhetad(theta)=(klambda)/R xx 2 = (2klambda)/R`