# Electrical Potential Energy

## What is Electrical Potential Energy?

We can define Electrical Potential Energy or Electrostatic Potential Energy as the energy that arises from the collection of charges when the charge will exert a force on any other charge. It is the total amount of work done in the bringing the charges from an infinite distance to their respective positions in the system. This is termed as electrostatic potential of charges in a system.

There are two key elements on which the electric potential energy of an object depends.

• It’s own electric charge
• And it’s relative position to other electrically charged objects.

Electrical Potential Energy is used to define potential energy in time-variant electric fields and Electrostatic potential Energy is used to define potential energy in time-invariant electric fields.

Common Symbol Used to define Electric potential Energy is $U_{E}$.

Its SI Unit is joule(J). In CGS system erg is unit of energy which is equal to $10^{-7}J$

Electrostatic Potential Energy $U_{E}$ of a point charge at position r under an influence of the Electric field E, is defined as the negative (-ve) work done by electrostatic force to bring from a position $r_{ref}$ to r position.

$U_{E}(r)=-W_{r_{ref}\rightarrow r}=-\int_{r_{ref}}^{r}qE(r’).dr'$

dr’ is the replacement vector

The electrostatic potential energy can also be defined from the electric potential as follows

$U_{E}(r)=q\Phi (r)$

In the above equation electrostatic potential energy, $U_{E}$ of a point charge q at position r in the electric potenial Φ .

### Electric Potential Energy of one Point Charge

• q is a one point charge in presence of another charge Q, charge separation is infinite between the two charges.

$U_{E}(r)=k_{e}\frac{qQ}{r}$

where $k_{e}=\frac{1}{4\Pi \epsilon _{o}}$ is the Coloumb’s constant

• q is a one point charge in the presence of several n point charges $Q_{i}$ , charge spearation between the charges is infinite.

$U_{E}(r)=k_{e}q\sum_{i=1}^{n}\frac{Q_{i}}{r_{i}}$

### Electrical Potential Energy for a system of charges

For obtaining the electrostatic potential energy of a system we have to consider two point charges for a system. When we consider two point charges q1 and q2 lying at two different points A and B, at some distance say, r1 and r2 respectively. Firstly the q1 charge is brought from the infinity to its original position A. When charge q1 is moved there is no work because no electrostatic force is required as the other charge present there opposes it. But when q2 is brought from infinity to its position then work has to be done as q1 will have some electric field.

It is given by,

U = $\frac {1}{4~\pi~ε_0}\frac{q_1~q_1}{r_{12}}$

The potential energy of the two charges is given by the work done in bringing those charges in their respective positions.

### Electric Dipole in an Electrostatic Field

When electric dipole moment vector is parallel to the field and maximum, the potential energy of an electric dipole in an electric field is zero. An electric dipole always experiences a torque, when it is placed in a uniform electric field whereas it experiences a net force as well as torque when placed in a non-uniform field.

“The work done in rotating an electric dipole against the torque is stored in the dipole due to the electric field in the form of its potential energy.”

The torque acting on the dipole is given by:

τ = p E sinθ

Where the electric dipole has dipole moment which is placed along a direction making an angle θ in the direction of an external uniform electric field E.

The potential energy stored in electric dipole is given by:

U = $– \overrightarrow{-p}.\overrightarrow{E}$