Electric potential energy Knowledge-Based Study

Study notes

1. Work and Potential Energy in Electrostatics

   The electrostatic force between charged particles is a conservative force, analogous to the gravitational force. When an external force moves a charged particle in an electric field without changing its kinetic energy, the work done by the conservative electric force $W_{\text{elec}}$ is directly related to the change in electric potential energy $\Delta U$:

   $$\Delta U = -W_{\text{elec}}$$

   If the electric force acts in the direction of displacement, it does positive work, resulting in a decrease in potential energy. Conversely, if work is done against the electrostatic force, the system gains electric potential energy.

   Point Charge Formulation

   For two point charges $q_1$ and $q_2$ separated by a distance $r$, the reference point is conventionally set at infinite separation, where $U(\infty) = 0$. The electric potential energy of the two-charge system is expressed as:

   $$U = \frac{k q_1 q_2}{r}$$

   where $k$ is Coulomb's constant (approximately $8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$). Key physical implications of this relation include:

   • Like Charges ($q_1 q_2 > 0$): The potential energy is positive. Bringing like charges closer together increases the potential energy because external work must be supplied against electrostatic repulsion.
   • Opposite Charges ($q_1 q_2 < 0$): The potential energy is negative. External work must be performed to separate the attracting charges to infinity.

   The Principle of Superposition

   Because electric potential energy is a scalar quantity, the total potential energy of an assembly of point charges is the algebraic sum of the potential energies of all distinct pairs:

   $$U_{\text{total}} = \sum_{i < j} \frac{k q_i q_j}{r_{ij}}$$

   For a three-charge configuration $(q_1, q_2, q_3)$, the total electrostatic potential energy is:

   $$U_{\text{total}} = \frac{k q_1 q_2}{r_{12}} + \frac{k q_1 q_3}{r_{13}} + \frac{k q_2 q_3}{r_{23}}$$

   This scalar sum represents the total external work required to assemble the charges from infinite separation to their given positions without acceleration.

   Uniform Electric Fields

   In a region where the electric field $E$ is constant in magnitude and direction, such as between two oppositely charged parallel conducting plates, the change in potential energy when a charge $q$ undergoes a displacement $d$ parallel to the field lines is:

   $$\Delta U = -q E d$$

   When a positive charge moves in the direction of the field, its potential energy decreases as it converts into kinetic energy. When a negative charge moves along the field lines, its potential energy increases.