The dipole moment of a system of charge `+q` distributed uniformly on an arc of radius T subtending an angle `pi//2` at its centre where another charge `-q` is placed is

Options

A. `(2sqrt2qr)/(pi)`
B. `(sqrt2qr)/(pi)`
C. `(qr)/(pi)`
D. `(2qr)/(pi)`

Answer

A

Explanation

Linear charge density
`lamda =(2q)/(pi R)`
Charge on element ` d_q= lamda Rd theta`
`P_x= int dp costheta= int_0 ^(pi//2) lamda R^2 cos theta d theta`
`P_x= lamda R^2 |sin theta|_0^(pi//2) =lamda R^2`
Similarly `P_y= int_0^(pi//2) lamda R^2 sin theta d theta =lamda R^2|- cos theta|_0^(pi//2) =lamda R^2`
`P=sqrt(P_X^2+ P_Y^2)= lamda R^2 sqrt2 =(2q)/(pi R). R^2 sqrt2`