Question

a charge q is placed at each corner of the cube of a side a the potential at the centre of the cube is​

Answer 1

Given that,

Charge = q

Side = a

Suppose, a charge (-q) is placed at center of the cube and find the potential energy at the center of the cube.

The same charge distributes at the each corner of the cube.

So the total charge will be 8q.

The length between the charge and charge of center of the cube is l.

We need to calculate the length of diagonal

Using formula of length of diagonal

$\text{length of diagonal}= \sqrt{3}\times side$

Put the value in to the formula

$\text{length of diagonal}=\sqrt{3}\times a$

We need to calculate the length between the charge and charge of center

Using formula of length

$l=\dfrac{\text{length of diagonal}}{2}$

$l=\dfrac{\sqrt{3}\times a}{2}$

We need to calculate the potential energy at the center of the cube

Using formula of potential energy

$V=\sum \dfrac{kq(-q)}{l}$

Put the value into the formula

$V=-\dfrac{8q^2}{4\pi\epsilon_{0}\times\dfrac{\sqrt{3}a}{2}}$

$V=-\dfrac{4q^2}{\sqrt{3}\pi\epsilon_{0}a}$

Hence, The potential energy at the center of the cube is $-\dfrac{4q^2}{\sqrt{3}\pi\epsilon_{0}a}$