A point charge q is placed on the top of a cone of semi vertex angle `theta`. Show that the electric flux through the base of cone is `(q(1-costheta))/(2epsilon_0)`

Explanation

Method 1 : For point charge, the Gaussian surface should
be spherical. Consider a Gaussian sphere with its center at the
apex and radius the slant length of the cone. The flux through
the whole sphere is `q//epsilon_0`. Therefore, the flux through the base
of the cone is `phi_E = (A//A_0)q//epsilon_0`. Here, `A_0` is the area of the
whole sphere `(4piR^2)`, and A is the area of the sphere below the
base of the cone. Consider a differential ring of radius r and
thickness dr, then
`dA = (2pir)Rdalpha = (2piRsinalpha) Rd alpha (as r = R sin alpha)`
`A = int_(0)^(theta) (2piR^2) sin alpha d alpha = 2piR^2 (1-costheta)`
The desired flux is
`phi_E = (A/A_0)1/(epsilon_0) = ((2piR^2)(1-costheta))/(4piR^2) (q/epsilon_0)`
`= ((1-costheta)q)/(2epsilon_0)`
Method 2: The total solid angle around a point in space is `4pi`
steradian. The solid angle subtended by the base of the cone
at the apex of the cone is `(2pi(1-costheta)).` As the flux associated
with solid angle `4pi` is `q//epsilon_0` , the flux associated with the solid
angle `2pi = (1-cos theta)` is
`phi = q/epsilon_0 (2pi(1-costheta))/(4pi) = (q(1-costheta))/(2epsilon_0)`.