# A point charge q is placed on the top of a cone of serni-vertex angle $\;\theta\;$ Find the electric flux through the base of the cone

$(a)\;\large\frac{q(1-cos \theta)}{2 \in_{0}}\qquad(b)\;\large\frac{q sin \theta}{2\in_{0}} \qquad(c)\;\large\frac{q(1-cos \theta)}{ \in_{0}}\qquad(d)\;\large\frac{q sin \theta}{2 \in_{0}}$

Answer : (a) $\;\large\frac{q(1-cos \theta)}{2 \in_{0}}$
Explanation :
Consider a Gaussian surface , a sphere with its centre at the top and radius the slant length of the cone . The flux through the whole sphere is $\;\large\frac{q}{\in_{0}}\;.$ Therefore the flux through the base of the cone is
$\phi_{e}=(\large\frac{S}{S_{0}})\;\large\frac{q}{\in_0{}}$
$S_{0}=\;$ area of whole sphere
S = area of sphere below the base of the cone .
Because all those field lines which pass through the base of the cone will pass through the cap of sphere
Let R= radius of Gaussion sphere
$S_{0}= \;$ area of whole sphere =$\; 4 \pi R^2$
S = area of sphere below the base of the cone which can be found by integration .
$dS = (2 \pi r )\;R\;d \alpha$
$dS=2 \pi R^2 sin \alpha \;d \alpha$
$S=\int_{0}^{\theta}\;dS =\int_{0}^{\theta}\;2 \pi R^2 sin \alpha \;d \alpha$
$=2 \pi R^2\;(1-cos \theta)$
The desired flux $\;\phi_{e} = \large\frac{S}{S_{0}}\;\large\frac{q}{\in_{0}}$
$=\large\frac{2 \pi R^2 (1-cos \theta ) }{4 \pi R^2}\;\large\frac{q}{\in_{0}}$
$\phi_{e} = \large\frac{q}{2 \in_{0}}\;(1-cos \theta)$

edited Aug 16, 2014