Question

What will be the total flux through the faces of the cube (figure) with side of length a, if a charge $q$ is placed at



(i) A: a corner of the cube.
(ii) B: mid point of the edge of the cube.
(iii) C: centre of a face of the cube
(iv) D: midpoint of B and C

Answer 1

(i) According to the Gauss’s law net flux through any surface is given by,

$\Phi =\frac{q_{\text{e}nclosed}}{{ϵ}_0}$

Here, $q_{\text{e}nclosed}$ is the charge enclosed by the surface.
For the Gaussian surface the charged particle is placed at the centre of the cube of side 2a. the charge is being shared by the 8 cubes.

Therefore, total flux through the faces of the cube = $\frac{q}{8{ϵ}_0}$

Final Answer: $\frac{q}{8{ϵ}_0}$

(ii) According to the Gauss’s law net flux through any surface is given by,

$\Phi =\frac{q_{\text{e}nclosed}}{{ϵ}_0}$

Here, $q_{\text{e}nclosed}$ is the charge enclosed by the surface.
For the Gaussian surface the charged particle (at middle point of an edge of cube) is at centre of four cubes. So this charge is being shared by the 4 cubes.


Therefore, total flux through the faces of the cube = $\frac{q}{4{ϵ}_0}$

Final Answer: $\frac{q}{4{ϵ}_0}$


(iii) According to the Gauss’s law net flux through any surface is given by,

$\Phi =\frac{q_{\text{e}nclosed}}{{ϵ}_0}$

Here, $q_{\text{e}nclosed}$ is the charge enclosed by the surface.
For the Gaussian surface the charged particle (at centre of face of cube) is at centre of two cubes. So this charge is being shared by the 2 cubes.



Therefore, total flux through the faces of the cube = $\frac{q}{2{ϵ}_0}$

Final Answer: $\frac{q}{2{ϵ}_0}$

(iv) According to the Gauss’s law net flux through any surface is given by,

$\Phi =\frac{q_{\text{e}nclosed}}{{ϵ}_0}$

Here, $q_{\text{e}nclosed}$ is the charge enclosed by the surface.
For the Gaussian surface the charged particle (at midpoint of B and C.) is at centre of two cubes. So this charge is being shared by the 2 cubes.


Therefore, total flux through the faces of the cube = $\frac{q}{2{ϵ}_0}$

Final Answer: $\frac{q}{2{ϵ}_0}$