# Electrostatic Potential

## Electric Potential Formula:

After understanding the electric field it becomes essential to understand what are the effects of the electric field on source charge. The electrostatic potential or electric potential plays a vital role in electrostatics. The electrostatic potential is the amount of work required in bringing a point charge from a reference point to a specific position against the effect of the electric field. The concept of electric potential is used to demonstrate the effect of the electric field of a source charge in terms of the position within the limits of the electric field. In this article, we will look upon the electric potential formula and detailed information on the electric potential difference equation.

### Potential Difference Formula:

In order to understand the potential difference formula, we must be aware of the concepts such as electrostatic potential energy. Before jumping into the potential difference formula let us have a look at the concept of what is potential energy or electric potential energy.

### Electric Potential Energy Formula:

Whenever an object or a particle is placed in a certain position or configuration, then the external work done on the object will be stored in the form of potential energy. Therefore, in general, potential energy is a form of stored energy. If the work done required to change the position or configuration of an object is more then the potential energy stored in the object will also be more. The magnitude of the potential energy is directly proportional to the external work done on the object.

Let us consider an example, assume that an object of mass m is placed on the ground. To displace the object from the ground to a height h we need to apply an external force which is equal to mg. Then the work done in bringing the object from ground level to height h will be equal to mgh and it is known as the gravitational potential energy. Thus, the work done on the object from one point to another will be equal to the difference in objective potential energies.

Now, what is electric potential energy? The electrostatic potential energy is almost similar to the gravitational potential energy. We discuss the electric potential energy formula with reference to the gravitational potential energy concept. When an electric charge is subjected to an external electric field, then the external work done on the electric charge will be stored in the form of electric potential energy or electrostatic potential energy. Therefore, the electric potential energy is defined as the external work done by an agent in bringing a charge or system of charges from infinity to the required position without accelerating the charge.

Consider a positive charge q placed in an external electric field, let a test charge +q0 is placed at a point A (say). Due to the electric field around the charge +q the test charge +q0 will experience an electrostatic force Fe directed away (or outward direction) from the charge. Since both the charges are of the same nature the force exerted will repulsive in nature i.e., Fext = -Fe

Let the potential energy of the charge +q0 at point A be UAand it is displaced by a distance dr towards the charge +q. The magnitude of an external force acting on the test charge will be equal to the electrostatic force.

The work done in displacing the test charge from point A to dr distance, we write:

$\Rightarrow dw = F_{ext} \cdot dr$ ……..(1)

If we want to displace the test charge from point A to point B, then the total work done is given by:

$\Rightarrow \int_{A}^{B} dw = \int_{A}^{B} F_{ext} \cdot dr$ …….(2)

We know that Fext=-Fe,therefore equation (2) changes to,

$\Rightarrow W = -\int_{A}^{B} F_{e} \cdot dr$ ………(3)

Therefore the total potential energy stored in the charge is equal to the difference in potential energies at point A and point B respectively. So, the electrostatic potential energy formula can be derived by calculating the potential difference at two points.

### What is Electrostatic Potential Difference?

Now, let us have a look at what is electrostatic potential and electric potential difference formula. The electrostatic potential is defined as the electric potential energy per unit charge. The external work done per unit charge is equal to the change in potential of a point charge. Consider an electric charge q and if we want to displace the charge from point A to point B and the external work done in bringing the charge from point A to point B is $W_{AB}$ then the electrostatic potential is given by:

$\Rightarrow \Delta V = V_{A} - V_{B} = \frac{W_{AB}}{q}$ ……..(1)

Where,

$W_{AB} -$ The external work done in bringing charge from point A to B.

Equation (1) is known as the electric potential difference equation or electrostatic potential equation. If the initial position of the charge is at infinity we know that the potential at infinity will be zero therefore we write,

$\Rightarrow \Delta V = V_{\infty} - V_{B} = \frac{W_{\infty B}}{q}$

$\Rightarrow V_{B} = \frac{W_{AB}}{q}$ ….(2)

Equation (2) is known as the electric potential equation. Therefore, the electrostatic potential is defined as the total external work done in bringing the point charge from infinity to the required position.

### Example:

1. Calculate the electrostatic potential due to a point charge placed at a distance r.

Sol:

The electric potential at a point in an electric field is defined as the amount of external work done in moving a unit positive charge from infinity to that point along any path(i.e., it is path independent) when the electrostatic forces are applied.

Suppose that a positive charge is placed at a point P in a given external electric field. The charge placed at that point will exert an external force due to the presence of an electric field. The electric potential at any point at a distance r from the positive charge +q is given by:

$\Rightarrow V \frac{Kq}{r}$ ……..(1)

Where,

K - is the coulomb constant and is equal to $\frac{1}{4 \pi \epsilon_{0}}$

r - The position vector.

When external work is done in moving a charge of 1 coulomb from infinity to a particular point due to an electric field against the electrostatic force, then it is said to be 1 volt of the electrostatic potential at a point.