# Formula of electrostatic potential energy | electric potential

In other articles, we became to know that an electric charge can produce an electric field around it. Then we leaned the electrostatic force between the charges. After that, we introduced the Gauss’s law to find the electric fields due to charged conductors. Again, in mechanics, we learned that a work is to be done to move an object by a distance in the application of an external force. This fact is equally applicable for the electric force too. In this article, we are going to discuss the formula for the work done to move a charge in an electric field. This is nothing but the formula of electrostatic potential energy.

1. Definition of electric potential
2. Formula for electric potential
3. Units of electric potential
4. Derivation of the formula for the electric potential due to a point charge
5. Electrostatic potential difference
6. Relation between electric field and electric potential
7. Electrostatic potential energy
8. Formula of electrostatic potential energy
9. Units of electric potential energy
10. Expression for the minimum velocity of a charge to cross a potential difference

## Definition of electric potential

Electrostatic potential at a point inside an electric field is defined as the work done in bringing a unit positive charge from infinity to that point. Another name of the electrostatic potential is electric potential.

Electric potential is a scalar quantity.

## Formula for electric potential

The formula or equation of electric potential at r distance from a point charge Q is $\small {\color{Blue} V=\frac{Q}{4\pi \epsilon_{0}r}}$. This formula is not same for all the conductors. Expressions for electric potential are different for different shaped conductors. The above formula is applicable for a point charge and for outside point of a spherical conductor.

## Units of electric potential

The SI unit of the electric potential is Volt and the CGS unit of the electric potential is Stat-Volt. one Stat-volt is equal to 300 Volts.

## Dimension of electric potential

The dimensional formula for the electric potential is [ML2T-3I-1].

One can determine the dimension of the electric potential from the formula of potential energy, W=qV. Where, W is the work done, q is the electric charge and V is the electric potential.

## Derivation of the formula for the electric potential due to a point charge

We consider an electric charge Q produced electric field around it. We want to find the work done in bringing a unit positive charge from infinity to a point P at r distance from the charge inside this field region. This gives the electric potential at that point.

Let, at any instant the unit charge is at x distance from Q inside the electric field. Then electric force on the unit charge at that point is, F=$\small \frac{Q}{4\pi \epsilon _{0}x^{2}}$.

Now, the work done in moving the charge for dx distance is, dW=$\small \vec{F}.d\vec{x}$.

Here, the electric force is in the outward direction and the displacement is occurred in the inward direction. So, the angle between electric force and the displacement is 180 degree.

Then, dW=F.dx.cos180°

or, dW=$\small \frac{Q}{4\pi \epsilon _{0}x^{2}}$dx

Then, total work done in bringing the unit positive charge from infinity to r distance is, W=$\small \int dW$

or, W=$\small \frac{Q}{4\pi \epsilon _{0}}$$\small \int_{\infty }^{r}\frac{dx}{x^{2}}$

or, W=$\small \frac{Q}{4\pi \epsilon _{0}r}$…………….(1)

This work done on the unit positive charge is renamed as the electric potential. Equation-(1) gives the expression for electric potential at r distance from a point charge Q.

So, electric potential, V=$\small \frac{Q}{4\pi \epsilon _{0}r}$…………(2)

## Electrostatic potential difference

The definition is clearly visible on its name. The difference of the electric potential between two points in an electric field is the electrostatic potential difference between these points. If VA and VB be the electric potentials at the points A and B respectively, then the potential difference between these two points is VAB=(VA-VB).

## Relation between electric field and electric potential

From equation-(2), we get the electric potential at r distance from a point charge Q is V=$\small \frac{Q}{4\pi \epsilon _{0}r}$. Again, the lectric field at that point is, E=$\small \frac{Q}{4\pi \epsilon _{0}r^{2}}$.

Then, V=Er in terms of the magnitude of the electric field and distance.

In vector form, V=$\small \vec{E}.\vec{r}$ ………..(3)

The dot product between the electric field and the displacement vector gives the electric potential.

Now, from the concept of the electric potential we became to know that the electric charge is displaced in opposite direction to the electric field. So, the angle between E and r is 180 degree.

So, V=-Er

In integral form, V=$\small \int \vec{E}.d\vec{r}$= – $\small \int E dr$…………(4)

In the differential form, E=-$\small \frac{\mathrm{d} V}{\mathrm{d} r}$………(5)

## Electrostatic potential energy

To move a positive charge from lower potential to higher potential region, the external agent needs to work. This work is stored as the potential energy in the electric field. This is the electrostatic potential energy.

## Formula of electrostatic potential energy

The work done by the external agent to move a charge through a potential difference is stored as the electrostatic potential energy in the electric field.

Now, electric potential V is the work done per unit charge. So, the work done in moving a charge q through a potential V is qV.

Thus, the formula for electrostatic potential energy, W=qV………..(6)

Now, If VA and VB be the electric potentials at the points A and B respectively, then the potential difference between these points is VAB=(VA-VB). Then electrostatic energy required to move q charge from point-A to point-B is,

W=qVAB

or, W=q(VA-VB) ……………(7)

Again, putting the value of the potential in equation (6) we get, the electrostatic potential energy to bring a charge q from infinity to r distance of a source charge Q is,

W=$\small {\color{Blue} \frac{qQ}{4\pi \epsilon_{0}r}}$ ………….(8)

## Units of electrostatic potential energy

Units of electric potential energy are similar to that of the energy we know. So, its SI unit is Joule (J) and CGS unit erg.

## Expression for minimum velocity of a charge q to cross a potential difference V

Let an electric charge q is to cross a potential difference V with a velocity u. Now, kinetic energy of the charge will be used to do the electrostatic work to overcome the potential difference. This satisfies the law of conservation of energy.

So, $\frac{1}{2}$mu2= qV

or, $u=\sqrt{\frac{2qV}{m}}$ …………..(9)

This is the minimum velocity that a charged particle requires to cross the potential difference. For an electron, we need to replace q by the charge of an electron e.

## Homework problems related to electric potentialand potential energy

1. Find the electric potential at 10 cm from a point charge 80 micro-Coulomb. If we place another charge of 2 micro-coulomb at that point, then what will be the electrostatic potential energy of the system of these two charges?
2. Find the minimum velocity that an electron should have to cross a potential difference of 20 volts.

## Summary

So, in this article we became to know the electric potential and electric potential difference and its formula, units and dimension. Also, we have discussed electric potential energy and its formula.

This is all from this article on the definition, units and formula of electrostatic potential energy and electric potential. If you have any doubt you can ask me in the comment section.

Thank you!

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