The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Electric Potential – Lecture 4 (University Physics II)

Overview

This lecture introduces energy methods into electrostatics, building from familiar mechanics up to two new ideas: electric potential energy and electric potential (voltage). It opens with a recap of kinetic energy, conservative forces, work, and the work–energy theorem, then derives potential-energy functions for gravity (both the near-surface form and Newton's law) by integrating the work done by the force. The same recipe is applied to Coulomb's law to obtain the electric potential energy of point charges. Finally, dividing potential energy by the test charge gives electric potential, measured in volts, which becomes the central quantity for the rest of the course. The unifying theme is that solving problems with energy conservation is often far simpler than tracking forces.

Key Concepts

Kinetic energy. The energy stored in an object's motion: $K = \tfrac{1}{2}mv^2$. It takes energy to set an object moving and energy to stop it. Units are joules (J), where $1\,\text{J} = 1\,\text{N}\cdot\text{m} = 1\,\text{kg}\cdot\text{m}^2/\text{s}^2$.

Work. The energy transferred by a force acting through a displacement. Only the component of force along the displacement counts: $W = \int_A^B \vec{F}\cdot d\vec{s}.$ Pushing on a wall that does not move does no work.

Conservative forces. Forces (e.g. gravity, the electrostatic force) with four defining properties:

• The work done is independent of the path taken from A to B.
• The process is reversible (energy put in can be recovered).
• The total work around any closed loop is zero.
• Each conservative force has an associated potential energy function $U$. Friction is the standard counterexample — none of these hold.

Work–energy theorem. The net work on an object equals its change in kinetic energy: $W = K_B - K_A = \tfrac{1}{2}mv_B^2 - \tfrac{1}{2}mv_A^2.$ It follows from $\vec{F}=m\vec{a}$ with $a = \tfrac{dv}{dt}$, and extends to three dimensions because $v_x^2 + v_y^2 + v_z^2 = v^2$.

Potential energy from work. Integrating the work done by a conservative force yields $W = U_A - U_B$ (note the order). The function $U$ that makes this true is the potential energy. Combining with the work–energy theorem gives conservation of mechanical energy when only conservative forces act: $U_A + K_A = U_B + K_B.$

Gravitational potential energy.

• Near Earth's surface (constant $g = 9.8\,\text{m/s}^2$): $U = mgy$.
• Newton's law of gravity, $F = \dfrac{Gm_1m_2}{r^2}$ (attractive): $U = -\dfrac{Gm_1m_2}{r}$. The negative sign reflects the attractive force; $U \to 0$ as $r \to \infty$.

Reference point. Only changes in potential energy are physical — adding a constant to $U$ cancels in $U_A - U_B$. We are free to choose where $U = 0$; "infinitely far away" is the convenient choice for $1/r$ potentials. Potential energy is a property of the relationship between two bodies, not of either one alone.

Electric potential energy. Coulomb's law $F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q\,q_0}{r^2}$ integrates exactly as gravity did, giving $U = \frac{1}{4\pi\varepsilon_0}\frac{q\,q_0}{r}.$ Unlike gravity it has no leading minus sign: like charges repel, so two like charges have maximum potential energy when close and $U \to 0$ as they separate. Convention: $q_0$ is the charge being moved through the field; $q_1, q_2, \dots$ generate the field. $U$ is the work needed to bring $q_0$ from infinity to its location.

Superposition. For a test charge near several source charges, the potential energy is the sum of pairwise contributions: $U = \frac{q_0}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}.$

Electric potential (voltage). Potential energy per unit charge — the "altitude" analogue: $V = \frac{U}{q_0}, \qquad V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}\ \text{for a point charge}.$ Voltage usually means a difference in potential, $V_{AB} = V_A - V_B$, measured in volts ($1\,\text{V} = 1\,\text{J/C}$). A 1.5 V battery delivers 1.5 J per coulomb moved between its terminals. Dividing $\vec F = q_0\vec E$ and $W = \int\vec F\cdot d\vec s$ by $q_0$ shows the field–potential analogue of force–energy: $V_A - V_B = \int_A^B \vec{E}\cdot d\vec{s}.$

Units and the electron-volt. The electric field can be expressed as N/C or, equivalently, V/m. One electron-volt is the energy to move one electron's charge across one volt: $1\,\text{eV} = e \times (1\,\text{V}) \approx 1.6\times10^{-19}\,\text{J}$ — the standard energy unit at atomic scales.

Worked Example

A dust particle of charge $q_0 = +2\,\text{nC}$ starts from rest at point A, midway in a line of fixed charges $q_1 = +3\,\text{nC}$ and $q_2 = -3\,\text{nC}$ spaced 1 cm apart, and moves to point B. At A it is 1 cm from $q_1$ and 2 cm from $q_2$; at B it is 2 cm from $q_1$ and 1 cm from $q_2$. Find its speed at B using energy conservation, $U_A + K_A = U_B + K_B$ with $K_A = 0$.

Compute each potential energy from the superposition formula, then solve $K_B = U_A - U_B$ for $v_B$. Because $q_2 = -q_1$, the cross-terms combine so that $U_A - U_B = \dfrac{q_0 q_1}{4\pi\varepsilon_0\,(0.01)}$.

Summary

• Kinetic energy $K = \tfrac{1}{2}mv^2$ measures energy of motion; work $W = \int \vec F\cdot d\vec s$ transfers energy, and the work–energy theorem gives $W = K_B - K_A$.
• Conservative forces are path-independent, reversible, do zero work around closed loops, and each has a potential energy with $W = U_A - U_B$; with only such forces, $U + K$ is conserved.
• Gravity gives $U = mgy$ near the surface and $U = -Gm_1m_2/r$ in general; only changes in $U$ matter, so the zero point is arbitrary.
• The same integration of Coulomb's law gives electric potential energy $U = \dfrac{q q_0}{4\pi\varepsilon_0 r}$ (positive for like charges), summed over multiple sources.
• Electric potential $V = U/q_0$ is energy per unit charge, measured in volts (J/C); voltage is a potential difference $V_A - V_B = \int_A^B \vec E\cdot d\vec s$.
• Equivalent units: $\vec E$ in N/C or V/m, and energy at atomic scale in electron-volts, $1\,\text{eV}\approx 1.6\times10^{-19}\,\text{J}$.