V = , E =

Sol.

Electric potential at center:

If we assume an imaginary identical hemisphere of same charge distribution to complete the sphere, then potential at center = . So, due to symmetry, potential at centre due to left half is equal to Right half. So, potential due to a hemisphere at center = =

= =

or V =

Electric Field

To calculate the electric field strength at center, we take a ring element which makes angle θ on the center and having width of R dθ. Due to this ring, electric field strength at center:

⇒ =

{here dq = charge on Ring; r = radius of ring, x = distance b/w center of ring and hemisphere}

By figure, x = R cos θ and dq = σ.2π R sinθ (Rdθ)

⇒ =

= πkσ [sin2θ dθ]

= =

= [– cos π + cos 0 ]

= . =