Potential energy in an External Field

The potential energy of a single charge is the energy contained by an object. This energy is produced due to the relative position of the object and depends upon its mass. It also depends on the distance of an object from the centre of mass of another object. This type of energy is called gravitational potential energy.

Forces that are derived from potential are called conservative forces. The work done by a conservative force is given by,

W = -△U

where,

△U is called the change in potential energy, which is related to the force. 

The negative sign indicates that the work done against a force increases potential energy. 

Potential energy of a single charge

Let (q) be the charge of magnitude that is placed in an electric field. 

The magnitude of this electric field is (E). 

Due to this condition, (q) is a very small charge. The potential energy of (q) in the electric field is equal to the work done in obtaining that charge from infinity. 

The external electric field and the potential energy of a single charge vary from point to point. The potential at infinity is taken as zero. Hence, the work done in obtaining that charge from infinity to the point is given by: qV

The potential energy of a single charge (q) at a distance r from the origin is given by: qV(r), where V(r) is the external potential at a particular point. 

Electric potential due to a point charge

The electrostatic potential is equal to the work done by an exterior force to hold a positive unit charge from infinity to a particular location. Now we will derive the formula for electric potential due to a point charge. 

Let (Q) be the charge of magnitude that is placed in an electric field. 

The magnitude of this electric field is (E), and (q) is a very small charge. 

The potential energy of (Q) in the electric field is equal to the work done in obtaining that charge from infinity. 

The electrostatic force on a single positive charge (Q) at some point (p) is equal to:

F = Q q/ 4π𝜀or2 

Hence work done is equal to,

W = Q / 4π𝜀or

The potential at P due to the charge Q is given by,

V(r) = Q / 4π𝜀or

Potential energy of a system of two charges

Let q1 and q2 with position vectors r1 and r2 be two charges relative to our point. 

Assume that q1 is the first charge to get transferred to infinity from its relative point. 

Since there is no external field, the work done is zero. 

The charge q1 gives a potential that can be stated as:

V1 = q1 / 4π𝜀or1

Point p is a random point in space. 

Let us apply the same conditions to the charge (q2). 

The potential is given by:

Work done on q2 is W = q1q2 / 4π𝜀or2

Thus, the potential energy of a system of two charges is given by:

U = q1q2 / 4π𝜀or12

Potential energy of a dipole

A dipole is a pair of equal and oppositely charged poles separated from each other. These poles can also be magnetised. Here is an example:

Let -q and +q be the charges in a dipole. 

A distance of 2l separates them. 

The dipole is placed in a uniform electric field (E)

The dipole experiences a torque due to the electric field. Torque is referred to as the moment of force. It is the rotational counterpart of linear force. The torque experienced by the dipole is given by:

T = p × E = pEsinθ

We apply an external torque (Text) that moves it in the plane to neutralise the torque. This happens without any angular acceleration. The total work done is held as the potential energy of the system. Hence, the potential energy of a dipole when it makes an angle (θ) with the electric field is given by:

U = pE (cosθ0 – cosθ)   

Conclusion 

The potential energy of a single charge is the energy contained by an object. This energy is produced due to the relative position of the object. One good example of potential energy is Gravitational Potential Energy. Gravitational Potential Energy depends upon the object’s mass and the distance of an object from the centre of mass of another object. 

The potential energy of a single charge (q) at a distance r from the origin is given by qV(r), where V(r) is the external potential at a particular point. The electrostatic potential is equal to the work done by a force to hold a positive charge from infinity to a location.