Electric field at centre of a uniforly charged semicirlce of radius a is

Options

A. ` lambda / (2 pi epsilon_0a^3)`
B. `lambda/(2pi epsilon_0a^2)`
C. `(lambda_2)/(2pi epsilon_0a)`
D. `(lambda)/(2pi epsilon_0a)`

Answer

D

Explanation

Electtric field intensity at O due to small elemental length dl of charged ring .
`dE = (1)/(4 pi epsi_(0)) * (lambda dl)/(a^(2)) = (1)/(4 pi epsi_(0)) * (lambda ad theta)/(a^(2))`
`dE = (1)/(4 pi epsi_(0)) * (lambda)/(a)"" d theta`
`therefore ` Net electric field at centre O is
E = `int d E sin theta = overset(pi)underset(0)(int) (1)/(4 pi epsi_(0)) (lambda)/(a)"" sin theta d theta`
`= (lambda)/(4 pi epsi_(0) a) [ - cos theta]_(0)^(pi)`
`therefore E = (lambda)/(2 pi epsi_(0) a)`.