Question

Eight equal charges q are placed at each corner of the cube of side a. Work done in carrying a charge -q from its center to infinity will be.

Answer 1

In the above question, it is given that the point charge is placed at the center of the cube. We know, all the corners of a cube are equidistant from the center of the cube. Therefore, find the effective distance from the center to one of the corners and find the potential energy of the cube initially. The final potential energy is zero as the distance is infinity.

Formula used:
$PE=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}$

Complete step by step answer:
Let us find the effective distance first from the cadette of the cube to the corner of the cube.
The distance will be,
$x=\dfrac{\sqrt{3}a}{2}$
Now, the initial potential energy of the particle will be,
$\begin{aligned} & PE=-8\times \dfrac{1}{4\pi {{\varepsilon }_{0}}}\times \dfrac{-{{q}^{2}}}{\dfrac{\sqrt{3}a}{2}} \\\ & \Rightarrow PE=\dfrac{-4{{q}^{2}}}{\sqrt{3}a\pi {{\varepsilon }_{0}}} \\\ \end{aligned}$
When the particle is at infinity, the potential energy will eventually become zero.
Hence, the work done by the particle to move from the initial position, the center to the final position, the infinity will be equal to $\dfrac{-4{{q}^{2}}}{\sqrt{3}a\pi {{\varepsilon }_{0}}}$

Additional information:
Electric potential energy is posed by an object by the virtue of two elements, those being, the charge possessed by an object itself and the relative position of an object with respect to other electrically charged objects. The magnitude of electric potential depends on the amount of work done in moving the object from one point to another against the electric field.
When an object is moved against the electric field it gains some amount of energy which is defined as the electric potential energy. For any charge, the electric potential is obtained by dividing the potential energy by the quantity of charge.

Note:
The potential energy of the particle which is at infinity is taken as zero because the potential energy of a particle or object is inversely proportional to the distance form which it has moved or measured. As the inverse of infinity is equal to zero, the potential energy will eventually turn to zero as the particle move towards infinity.