The electric field at the centre of a uniformly charged ring is zero. What is the electric field and potential at the centre of a half ring if the charge on it be `Q` and its radius be `R`?

Options

A. `(1)/(4 pi epsilon_(0)) (Q)/(pi R^(2)) ,(1)/(4 pi epsilon_(0)) (Q)/(pi R)`
B. `(1)/(4 pi epsilon_(0)) (Q)/(R^(2)) ,(1)/(4 pi epsilon_(0)) (2Q)/(pi R)`
C. `(1)/(4 pi epsilon_(0)) (2Q)/(pi R^(2)) ,(1)/(4 pi epsilon_(0)) (Q)/(R)`
D. `(1)/(4 pi epsilon_(0)) (2Q)/( R^(2)) ,(1)/(4 pi epsilon_(0)) (3Q)/(pi R)`

Answer

3

Explanation

`dq = (Q)/(pi R) . R d theta = (Q)/(pi) d theta`
`dE = (1)/(4 pi in_(0)) (dq)/(R^(2))`
`E_(y) = 0` (due to symmetry )
`E_(o) = E_(x) = int dE_(x) = int dE cos theta`
`= (Q)/( 4 pi^(2) in_(0) R^(2)) int_(-pi//2)^(pi//2) cos theta d theta`
`= (Q)/( 2 pi^(2) in_(0) R^(2)) = (1)/(4 pi in_(0)) .(2Q)/(pi R^(2))`
`V_(o) = (1)/( 4 pi in_(0)) .(Q)/(R )`