24.3
Electric Potential Energy of Two Point Charges
The electric potential energy of a test charge in a uniform eclectic field can be generalized to any electric field produced by static charge distribu…
The electric potential energy of a test charge in a uniform electric field can be generalized for any electric field caused by static charge distribution.
Consider a static charge at point O and a test charge at point B. If the test charge displaces away from the static charge, then the work done by the electric field on the test charge is the integral of the dot product of force and displacement. Here the force is repulsive, resulting in positive work.
This work can be expressed as the negative difference of electric potential energy possessed by the test charge at the final and initial positions. Since the work done is positive, the system's potential energy decreases.
The electric potential energy is defined as zero at infinity. It is always positive for charges of the same polarity; for opposite charges, the potential energy is always negative.
Suppose the test charge is in an electric field caused by several point charges. The electric potential energy is an algebraic sum of electric potential energies due to all static charges.
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Q1: Why is electric potential energy positive for same-polarity charges and negative for opposite 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.
Q2: How does the work done by an electric field relate to changes in potential energy?
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.
Q3: Why is electric potential energy defined as zero at infinite separation?
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.
Q4: How do you calculate electric potential energy when multiple point charges are present?
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.
Q5: Does the path taken by a test charge affect the work done by the electric field?
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.
Q6: What happens to potential energy when two like charges are brought closer together?
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.
Q7: How does potential energy change when opposite charges are brought closer?
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.