Electric Potential Energy & Electric Potential – Summary Sheet

Electric Potential Energy

• When two point charges $q<em>1$ and $q</em>2$ separated by a distance $r$ interact through the Coulomb force, they possess an electric potential energy
  • $U = k<em>e \dfrac{q</em>1 q<em>2}{r}$, where ke = 8.99 \times 10^9\, \text{N·m}^2/\text{C}^2.
• For a system of three or more point charges, potential energy is the sum over every unique pair:
  • $U = \displaystyle \sum<em>{i<j} k</em>e \dfrac{q<em>i q</em>j}{r<em>{ij}}$, with $r</em>{ij}$ the separation between charges $i$ and $j$.
• Key idea: Potential energy is a property of the configuration as a whole, not of a single charge in isolation.

Electric Potential (Scalar Potential)

• Describes how charge alters space independently of any specific test charge.
• Defined through its influence on a test charge $q$:
  • $V = \dfrac{U}{q} \quad \Longleftrightarrow \quad U = qV$.
• Units: $[V] = \text{Volt} = 1\, \text{J}/\text{C}$.
• Physical meaning: the energy per unit charge that a test charge would have at a given point.

Calculating Electric Potential for Different Charge Configurations

• General recipe (principle of superposition):
  1. Break the total charge $Q$ into infinitesimal pieces $dQ$ that each act like point charges.
  2. Compute the potential contribution $dV$ from each $dQ$ at the observation point $P$.
  3. Add (integrate) all $dV$ to get the net potential $V$.
  • Because $V$ is a scalar, no vector components are required during summation.

Point Charge

• Formula: $V(\mathbf r) = k_e \dfrac{q}{r}$ (positive for positive $q$).
  • Shows the familiar $1/r$ spatial dependence.

Multiple Discrete Point Charges

• Simply add individual contributions:
  • $V(\mathbf r) = \displaystyle \sum<em>{i} k</em>e \dfrac{q<em>i}{r</em>i}$.

Continuous Charge Distributions

• Replace the sum with an integral:
  • $V(\mathbf r) = k_e \int \dfrac{1}{|\mathbf r - \mathbf r'|}\, dQ(\mathbf r')$.
  • Integration variable $\mathbf r'$ sweeps through the charge distribution.

Relationship Between Electric Field and Potential

• Mathematical link mirrors that between force and potential energy:
  • $\mathbf E = -\nabla V$ (electric field is the negative gradient of potential).
  • $\mathbf F = -\nabla U$ (force is the negative gradient of potential energy).
• Conceptual contrast:
  • Force/Electric-field: acts locally on a particle.
  • Potential Energy/Potential: defined everywhere in space, independent of a specific particle’s presence.

Mechanical Energy Conservation

• In an isolated, non-dissipative system (no friction, radiation, etc.) total mechanical energy remains constant:
  • $K<em>i + U</em>i = K<em>f + U</em>f$.
• The form of $U$ used depends on the interaction (electric, gravitational, elastic, …).
  • For electric problems use $U$ expressions given above.

Potential Inside a Parallel-Plate Capacitor (Uniform Field Example)

• Parallel plates: left plate negative, right plate positive.
• Potential difference between plates of separation $s$ and uniform field $E$:
  • $\Delta V = V<em>+ - V</em>- = -\int_{-}^{+} \mathbf E \cdot d\mathbf s = Es$ (taking the magnitude and field direction).
• One convenient reference choice: set the negative plate at $0\,\text{V}$, so the positive plate is at $+\Delta V$.
• Illustrates that potential can be specified up to an arbitrary additive constant; only differences matter.

Key Takeaways & Connections

• Potential energy $U$ and potential $V$ are central scalar quantities from which vector forces and fields are derived by spatial gradients.
• Superposition simplifies calculations because both $U$ and $V$ add directly.
• Conservation of mechanical energy ties the electrostatic concepts back to broader mechanics: knowing potentials lets you track kinetic energy changes without computing forces explicitly every step.
• Capacitor example foreshadows applications in circuits, energy storage, and uniform-field approximations in many practical devices.