Name: Equipotential Surfaces and Its Properties
Uploaded: 2019-07-19
Duration: 7 min 22 s
Description: The surface over which the electric potential is the same everywhere is called an equipotential surface. Let's more about it and its properties in this video.

Question

Define electric potential due to a point charge and arrive at the expression for the electric potential at a point due to a point charge.

Accepted Answer

The electric potential at a point in an electric field is defined as the amount of work done in moving a unit positive charge from infinity to that point along any path when the electrostatic forces are applied.
Suppose there is a positive charge $+ q$ and we need to find the electric potential at $P$ which is at a distance $r$ from the charge. As the force is variable so the work done is given by

$W = \int F . d x$ $- equation (1)$

where $F$ is the force at that location and $d x$ is the infinitesimal distance towards the point charge.

Let us assume that the unit charge is at a distance $' x'$ from the point charge.

Hence, we can find the force of repulsion using Coulomb $'$ s law,
$F = \frac{1}{4 {πε}_{0}} \frac{q}{x^{2}}$ $- equation (2)$

where $ε_{0}$ is the dielectric constant of space.

Substitute the value of force from equation $(2)$ to equation $(1)$

$W = \int_{\infty}^{r} \frac{1}{4 {πε}_{0}} \frac{q}{x^{2}} d x$

Since the force is repulsive in nature, work done will be positive.

$W = - {[\frac{1}{4 {πε}_{0}} \frac{q}{x}]}_{\infty}^{r}$

$\Rightarrow W = - [- \frac{1}{4 π ε_{0}} \frac{q}{r}]$

$\Rightarrow W = \frac{q}{4 π ε_{0} r}$

So, the electrostatic potential at a distance $r$ from a point charge $(+ q)$ is given by,
$U = \frac{1}{4 {πε}_{0}} \frac{q}{r}$

Both the point charge and the unit charge are positive. So, we had to work against the electrostatic force. Hence, the potential is positive. If, however, the point charge had negative value, we had to work along the force. So, in that case potential would have been negative.

Define electric potential due to a point charge and arrive at the expression for the electric potential at a point due to a point charge.

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