In the figure shown an infinitely long wire of uniform linear charge density `lamda` is kept perpendicular to the plane of figure such that it extends upto infinity on both sides of the paper. Find the electrostatic force on a semicircular ring kept such that its geometrical axis coincides with the wire. The semicircular ring has a uniform linear charge density `lamda'`.

Options

A. `(lamdalamda')/(piepsilon_(0)R)`
B. `(lamdalamda')/(piepsilon_(0))`
C. `(lamdalamda')/(2piepsilon_(0))`
D. `(lamdalamda')^(piepsilon_(0))`

Explanation

The electrostatic field intensity at a point on the ring is `E=(lamda)/(2piepsilon_(0))(1)/(R)`
the force on the elementary charge dq is
`dF=dqE=(lamda'Rdtheta).(lamda)/(2piepsilon_(0))(1)/(R)`
The sine component of dF will get cancelled and cosine component will get added. Net force on the ring
`F=2int_(0)^(+pi//2)dFcostheta=2int_(0)^(+pi//2)(lamda.lamda')/(2piepsilon_(0))dtheta.costheta=(lamdalamda')/(piepsilon_(0))` ans `(lamdalamda')/(piepsilon_(0))`