Question

A point charge $q$ is placed on the line AH and just outside the cube (of side $d$) at a distance of ${}^{\prime }{r}^{\prime }(r<<d)$. Total flux of electric field through the surface ABCD $\left({\varphi }_{ABCD}\right)$ is $-\frac{nq}{8{\epsilon }_0}$. Find the value of n.

Answer 1

Construct an imaginary cube of side $2d$ with charge $q$ at the centre.

Using Gauss' law, total electric flux through all the faces of the cube should be $\frac{q}{{ϵ}_0}$

Thus,
${\varphi }_{BCGF}={\varphi }_{CDHG}={\varphi }_{EFGH}=\frac{q}{24{\epsilon }_0}$
(since they are all outer faces)

${\varphi }_{EHAD}=0$

Let ${\varphi }_{ABFE}={\varphi }_{ABCD}=x$

Using Gauss' law, total electric flux through all the faces of the cube ABCDEFGH should be zero (because charge enclosed is zero)

$2\times x+0+\frac{q}{24{\epsilon }_0}\times 3=0$
$\Rightarrow x=-\frac{q}{16{\epsilon }_0}$