Electric Potential and Potential Energy

When a force is conservative, work done against that force is stored as potential energy. In electrostatics, the Coulomb force between stationary charges is conservative, so a system of charges possesses electrostatic potential energy.

Electrostatic potential energy is the energy stored in a system of charges due to their relative positions.

Core Idea: If an external force moves a test charge against the electrostatic force, the work done by the external agent is stored as electrostatic potential energy.

Consider a fixed point charge Q at the origin and a test charge q moved from point R to point P. If Q > 0 and q > 0, the electrostatic force is repulsive, so an external force must do work to bring the test charge closer.

Physical Meaning

• The electrostatic field opposes the inward motion of the positive test charge.
• The external force supplies energy to the system.
• That supplied energy appears as an increase in electrostatic potential energy.

If the test charge is moved from R to P, the work done by the external force is given by:

This work is stored as an increase in electrostatic potential energy.

Important Relation

The change in potential energy is: ΔU = U_P − U_R = W_RP

This means the change in electrostatic potential energy equals the work done by the external force in moving the charge from R to P.

Key Interpretation

• If external work is positive, electrostatic potential energy increases.
• If the electric field itself does the work, the system's potential energy decreases.
• Only the difference in potential energy between two points is physically significant.

Only the difference matters

An arbitrary constant may be added to potential energy without affecting physics, so the absolute value is not important; only the change in potential energy matters.

The zero of electrostatic potential energy can be chosen conveniently. In electrostatics, it is commonly taken to be zero at infinity.

Standard Convention

If the potential energy at infinity is taken as zero, then the potential energy at a point is the work done in bringing the charge from infinity to that point.

Remember: In electrostatics, potential energy at infinity is usually taken as zero because the force becomes negligible at a very large distance.

A conservative force is one for which work done depends only on the initial and final positions, not on the path followed. The electrostatic force satisfies this condition.

Consequences

• Work done between two points is path-independent.
• A closed path has zero net work for electrostatic force.
• Electrostatic potential energy can be defined for a charge system.

Flowchart

Analogy 1: Pushing similar poles of magnets together

Bringing two like magnetic poles closer requires effort. In a similar way, bringing like charges closer requires external work, which gets stored as energy in the system.

Analogy 2: Lifting an object upward

Just as lifting a book increases its gravitational potential energy, bringing a positive charge closer to another positive charge increases electrostatic potential energy.