Electric Potential Energy Formula

If W is the work done in transferring a unit positive charge q from infinity to a particular point in the electric field, this work done energy will be stored in form of the electric potential energy or electrostatic potential energy.

Let’s derive the expression for electric potential energy,

Consider the electrostatic field E that exists as a result of a charge arrangement. Consider the electric field E caused by a charge Q placed at the origin for simplicity.

Consider moving a test charge q from a point R to a point P while resisting the charge Q’s repulsive force. If Q and q are both positive or both negative, this will happen with reference. Let’s use Q as an example, with q > 0,

Assume that the test charge q is so little that it has no effect on the original configuration, specifically the charge Q at the origin (or that Q is held fixed at the origin by some unknown force). Second, apply an external force F_ext exactly enough to counter the repulsive electric force F_E (i.e. F_ext= –F_E ) as the charge q move from R to P.

This means that when the charge q is transported from R to P, it experiences no net force or acceleration, implying that it is transported at an infinitesimally slow constant speed. In this case, the work done by the external force is minus the work done by the electric force, and the potential energy of the charge q is fully stored.

If the external force is withdrawn when the charge reaches P, the electric force will pull the charge away from Q – the stored energy (potential energy) at P is used to provide kinetic energy to the charge q, preserving the sum of the kinetic and potential energies. 

Therefore, the work done by external forces in moving a charge q from R to P can be written as,

[Tex]W_{RP}=\int_{R}^{P} F_{ext}\cdot{dr}[/Tex]

Since, F_ext= –F_E, then we can write, 

[Tex]W_{RP}=-\int_{R}^{P} F_{E}\cdot{dr}[/Tex]

The above expression is the work done against electrostatic opposing force and gets stored as potential energy. A particle with charge q has a definite electrostatic potential energy at every location in the electric field.

The work done raises its potential energy by an amount equal to the potential energy difference between points R and P. Therefore, the potential energy difference can be expressed as,

∆U = U_P – U_R = W_RP

Note that this displacement is in the inverse direction of the electric force, hence the work done by the electric field is negative, i.e., –W_RP.

As a result, the work required by an external force to move (without accelerating) charge q from one location to another for an electric field of any arbitrary charge configuration can be defined as the electric potential energy difference between two points. At this point, two key points should be made,

1. The work done by an electrostatic field in transferring a charge from one location to another is solely reliant on the initial and final points and is unaffected by the path used to get there. This is a conservative force’s defining attribute.
2. The above expression defines the difference in potential energy in terms of a physically meaningful quantity of work. Within an additive constant, potential energy is clearly uncertain.
3. This indicates that the actual value of potential energy has no physical significance; only the change in potential energy is essential. We can always add an arbitrary constant to potential energy at any time since the potential energy difference will not change,

(U_P – β ) – (U_R – β ) = U_P  – U_R

To put it another way, the point where potential energy is zero can be chosen at will. Electrostatic potential energy 0 at infinity is a convenient choice. If we take the point R at infinity with this option,

W_∞P = U_P – U_∞ = U_P – 0 = U_P

The above expression defines the potential energy of a charge q at any moment in time.

The work done by the external force (equal and opposite to the electric force) in bringing the charge q from infinity to that location (in the presence of field due to any charge configuration) is called potential energy of charge q at a point.

Electrical potential energy is the cumulative effect of the position and configuration of a charged object and its neighboring charges. The electric potential energy of a charged object governs its motion in the local electric field.

Sometimes electrical potential energy is confused with electric potential, however, the electric potential at a specific point in an electric field is the amount of work required to transport a unit charge from a reference point to that specific point and electrical potential energy is the amount of energy required to move a charge against the electric field.

In this article, let’s understand the electrical potential energy, electric potential, their key concepts, applications, and solved problems.