Electric Potential Energy: Definition, Formula, and Problems

Consider an electric field generated by a positive point charge. The direction of the electric field is such that it is radially outwards. When a positive test charge is brought closer to the point charge, it will experience repulsion due to electrostatic or Coulomb force. Energy is needed to overcome the repulsive force and move the test charge closer to the point charge, which is a source charge. This energy is known as electric potential energy.

The electric potential energy is a scalar quantity. It is not a vector, although the electric field responsible for it is a vector. The magnitude depends upon two factors:

The magnitude of the charges
Distance between the two charges

Suppose q1and q2are the magnitudes of the two charges and r is the separation distance between them. Then, the electric potential energy U is given by

U ∝ q1q2/r

Or, U = kq1q2/r

Where k is a proportionality constant known as Coulomb constant, given by k = 1/(4πεo), whose value is 9 x 109 N m2/C2.

SI Unit: Joule (J)

Dimension: [ML2T-2]

Symbol: U

Electric Potential Energy Formula

The electric potential energy per unit charge is known as electric potential.

Electric Potential

It can be obtained by dividing the electric potential energy by the magnitude of the test charge.

V = U/q1

Or, V = kq1/r

Replacing k by 1/(4πεo) and q1 by Q, we get the formal expression of the electric potential.

V = Q/(4πεor)

Unit: Volt (V) or Joule/Coulomb (J/C)

Dimension: [ML2T-3A-1]

Symbol: V

The above equation gives the electric potential at a distance r from the source charge Q. If this charge is negative, the electric potential is negative and given by

V = -Q/(4πεor)

Electric Potential Formula

Work and Electric Potential

Suppose a unit charge is moved from point A to B such that B is closer to the source charge than A. Let rA and rB represent the distances of A and B from Q. Then, rA> rB. The potential difference between two points, A and B, can be written as

ΔV = VB – VA = (Q/4πεo)(1/rB – 1/rA)

Work is done against the electric field to move the unit charge from A to B. The work done is the negative change in electric potential.

W = – ΔV

The work done is negative because the displacement is opposite to the electric field. In other words, the charge is displaced in a direction opposite to the electric field.

The electric potential at a point is said to be one volt if one joule of work moves one Coulomb of the electric charge against the electric field.

Zero Potential

The zero potential is a reference point from which electric potential values are measured. For a point charge, it is clear from the above equation that the electric potential is zero at infinity. This point is taken as a reference point. Hence, work done is the change in electric potential when a unit charge is brought from infinity to a point under consideration. Alternatively, the electric potential at a point is the work done in moving a unit charge from infinity to that point.

The fundamental difference between electric potential energy and electric potential is that the former is the energy required to move an electric charge against an electric field. On the other hand, the latter is the work done in moving a unit charge from infinity to the point under consideration.