Question

Calculate the total electric flux and flux related to each vertices for a cube of side a when charge $q$ is placed at
(i) centre of cube
(ii) an end of the cube
(iii)a plane of cube.

Answer 1

(i) As in cube , there are $6$ faces of equal area .

$\therefore$ The leaving out flux through each face of cube

$\varphi =\frac{1}{6}{\varphi }_E=\frac{1}{6}\frac{q}{6{ϵ}_0}$

As each face of cube is in symmetry.

$\therefore$ Total flux $\varphi =6\times {\varphi }_s=\frac{q}{{ϵ}_0}$ because ${\varphi }_s=\frac{q}{6{ϵ}_0}$

(ii) As in cube are four corners on each surface. therefore total flux bond through cube = $\frac{q}{4{ϵ}_0}$

$\therefore$ The leaving out flux through any one face

$=\frac{1}{4}\left(\frac{q}{4{ϵ}_0}\right)=\frac{q}{16{ϵ}_0}$

(iii) As in cube each surface has two part

$\therefore$ flux bonding to cube = $\frac{q}{2{ϵ}_0}$

$\therefore$ the leaving out flux through any one face of the cube

$=\frac{1}{5}\left(\frac{q}{2{ϵ}_0}\right)=\frac{q}{10{ϵ}_0}$