For the hemisphere of radius `R` shown in the figure the surface charge density `sigma` varies with `theta` (shown in figure) as `sigma=sigma_(0) costheta` the total charge on the sphere is

Options

A. `2sigma_(0)piR^(2)`
B. `(sigma_(0)piR^(2))/(2)`
C. `(sigma_(0)piR^(2))/(3)`
D. `sigma_(0)piR^(2)`

Answer

D

Explanation

`dA=2pirRd theta =2piR^(2)sintheta(d theta)`
`dQ=2sigma_(0)piR^(2)sintheta costheta d theta =pisigma R^(2)sin2thetad theta`
`Q=pisigma_(0)R^(2) int_(0)^(pi//2)sin2thetad theta =(-pisigma_(0)R^(2))/(2)(cos2theta )_(0)^(pi//2)`
`rArr q=pisigma_(0)R^(2)`