Electric Potential Energy of Two Point Charges

Electric potential energy depends on charge type and separation. For same-polarity charges (both positive or negative), the repulsive force means work must be done to bring them together, resulting in positive potential energy. For opposite charges, the attractive force means the system releases energy as they approach, making potential energy negative. This sign convention reflects whether energy is stored in or released from the system.

Work done by the electric field equals the negative change in potential energy. When the field does positive work on a test charge, potential energy decreases. Conversely, when negative work is done, potential energy increases. This relationship holds because the work integral depends only on the endpoints, not the path taken by the charge through the field.

At infinite separation, two charges have negligible interaction because the electrostatic force approaches zero. Defining potential energy as zero at infinity provides a convenient reference point for calculating potential energy at any finite distance. This choice simplifies calculations and allows potential energy to be expressed as a single-valued function of separation distance.

The total electric potential energy is the algebraic sum of potential energies due to each pair of charges. Each pairwise interaction contributes independently to the total. This superposition principle allows you to calculate the potential energy of a test charge in a field produced by several point charges by summing the individual contributions from each static charge.

No, the work done by the electric field depends only on the initial and final positions, not the path taken. Whether the test charge moves radially or in an arbitrary direction, the work is determined solely by the endpoints. This path-independence property is fundamental to conservative forces and allows potential energy to be defined as a function of position alone.

When two like charges (both positive or both negative) are brought closer, positive work must be done on the system against the repulsive force. This increases the system's potential energy. Since potential energy is inversely proportional to separation, decreasing distance between like charges makes the potential energy more positive, storing more energy in the system.

When opposite charges are brought closer, the attractive force does negative work on the system, meaning energy is released. This decreases the system's potential energy. Since potential energy is already negative for opposite charge pairs, bringing them closer makes the potential energy more negative, representing a net decrease in stored energy as the system moves toward a lower-energy configuration.