# Electric Potential Formula

Definition: Electric potential energy of any given charge or system of changes is defined as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration. Electric potential energy is also defined as the total potential energy a unit charge will possess if located at any point in the outer space. Electric potential energy is a scalar quantity, and possesses only magnitude and no direction. It is measured in terms of Joules and is denoted by V.

An object is said to possess electric potential energy by virtue of two elements, those being, the charge possessed by the object itself and the relative position of the object with respect to other electrically charged objects.

Formula:

The electric potential energy of a unit charge ‘q’ present in an electric field of magnitude E, at a position ‘r’ is given by the negative of work done by an external force, given by Fext in bringing the charge from infinity to the present configuration.

$U_{E}=-W_{ext}=-\int _{ref}^{r}qE.ds$

Where, E is the electrostatic field and ds is the displacement vector.

The electric potential energy of a point charge given by q1 present at a distance r from another point charge q2 is given using the following formula.

$U_{r}=\frac{kq_{1}q_{2}}{r}$

Here, Ur is the electric potential energy that is to be calculated, q1 and q2 are the two charges between which the electric potential is to be calculated and r is the distance between these two charges.

Derivation:

Let us consider a charge q1. Let us say, they are placed at a distance r from each other. The total electric potential of the charge is defined as the total work done by an external force in bringing the charge from infinity to the given point.

We can write it as,

$\int _{r_{a}}^{r_{b}}F.dr=-(U_{a}-U_{b})$

Here, we see that the point rb  is present at infinity and the point ra  is r.

Substituting the values we can write,

$\int _{r}^{} \infty F.dr\;=\;-(U_{r}-{U\infty})$

As we know that $U_{infinity}$ is equal to zero.

$\int _{r}^{\infty}F.dr=-(U_{r})$

Using Coulomb’s Law, between the two charges we can write.

$-\int _{r}^{\infty}\frac{kqq_{0}}{r^{2}}dr=-(U_{r})$

$-kqq_{0}\frac{1}{r}=U_{r}$

$U_{r}=\frac{-kqq_{0}}{r}$

Example 1: Let us say we have two charges of magnitude 1 C and 2C placed at a distance 2 metre from each other. Calculate the electric potential between these two charges. (k= 1)

Solution:

Given that, the magnitude of charges are q1 = 1C, and q2 = 2C. The distance between these two charges is r = 2m.

The electric potential between these two charges is given by,

$U_{r}=-\frac{kqq_{0}}{r}$

Substituting the values we get,

$U_{r}$ = -1 J

Example 2:

How much work is required to be done, in order to bring two charges of magnitude 3 C and 5 C from a separation of infinite distance to a separation of 0.5 m?

Solution:

∆E=E0–Eg

$=0-\frac{-9\times 10^{9}(5\times 3)}{0.5}$

$=27\times 10^{10}J$