Question

A charge $Q$ is kept at the corner of a cube. Electric flux passing through one of the faces not touching that charge is
(A) $\dfrac{Q}{{24{\varepsilon _0}}}$
(B) $\dfrac{Q}{{3{\varepsilon _0}}}$
(C) $\dfrac{Q}{{8{\varepsilon _0}}}$
(D) $\dfrac{Q}{{6{\varepsilon _0}}}$

Answer 1

The charge, being at the corner of one cube can be assumed to be placed at the centre of 8 cubes. The sides of one cube not touching the charge are three, hence, the sides not touching the charge of the 8 cube are 24 (being 8 times 3)

Formula used: In this solution we will be using the following formulae;
$\phi = \dfrac{Q}{{{\varepsilon _0}}}$ where $\phi$ is the electric flux passing through a closed surface, $Q$ is the charge enclosed by the surface, and ${\varepsilon _0}$ is the permittivity of free space.

Complete Step-by-Step solution
To answer the question, we shall recall gauss law which can be given mathematically, in its constant form, as
$\phi = \dfrac{Q}{{{\varepsilon _0}}}$ where $\phi$ is the electric flux passing through a closed surface, $Q$ is the charge enclosed by the surface, and ${\varepsilon _0}$ is the permittivity of free space.
Now, the charge is actually placed at the corner of the cube, hence, the charge is not entirely enclosed. To enclose the charge, we must add seven other cubes around the charge to enclose it.
Hence, the flux passing through the entire cube is $\dfrac{Q}{{{\varepsilon _0}}}$ .
But three sides of each cube are on the inner part and hence touch the cube, the other sides not touching the cube by 3 times 8, which is 24. Hence the flux passing through one of the surface would be
$\phi = \dfrac{1}{{24}}\dfrac{Q}{{{\varepsilon _0}}} = \dfrac{Q}{{24{\varepsilon _0}}}$
Hence, the correct answer is A.

Note
For clarity, although it may be quite confusing why the number of sides of the cube is 24 instead of 48 sides. This is because on each of those boxes, 3 of its sides are on the inner surface, hence can be cut neglected as they do not make up the surface which actually encloses the entire charge. From outside for example, we would only see the 24 sides of the tiny cubes making six sides of the bigger cube.