A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field `vecE` at the centre O is

Options

A. `q/(4pi^2epsilon_0r^2)hatJ`
B. `-q/(4pi^2epsilon_0r^2)hatJ`
C. `-q/(2pi^2epsilon_0r^2)hatJ`
D. `q/(2pi^2epsilon_0r^2)hatJ`

Answer

C

Explanation

Linear charge density
`lamda =(2q)/(pi R), E= int DE sin theta (- hat j)= int_0 pi int (k.dq)/r^2 sin theta (- hat j)`
`E=k/r^2 int_0^pi (qr)/(pi r) d theta sin theta (- hat j) =k/r^2 q/ (pi) int_0^(pi) sin theta d theta (- hat j) =1/(4 pi e_0) q/( pi r^2) [- cos theta]_0^(pi) (-hat j)`