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

Electric Charge and Electric Fields – Lecture 2 (University Physics II)

Overview

This lecture builds the foundation of electrostatics: how charged objects exert forces on one another and how those forces can be repackaged as an electric field that fills space. It opens with Coulomb's law — the inverse-square force between two point charges — and contrasts it with Newton's law of gravity. It then introduces the electric field as force-per-unit-charge, along with the field-line picture used to visualize it. The second half applies calculus to find the field produced by continuous charge distributions: a charged ring, a finite rod, a charged disc, and finally a pair of parallel plates. A recurring theme is checking each result against limiting cases (very near, very far) to confirm it behaves sensibly.

Key Concepts

Charge basics (recap). All charge comes in integer multiples of the elementary charge $e = 1.602 \times 10^{-19}\ \text{C}$; protons carry $+e$, electrons $-e$. Charge is conserved — never created or destroyed, only separated and recombined by moving electrons. Conductors let electrons flow freely; insulators do not. One coulomb is enormous: about $6 \times 10^{18}$ electron charges. Real static charges are typically microcoulombs ($\mu\text{C}$) or nanocoulombs ($\text{nC}$).

Coulomb's law. The electrostatic force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them: $F = k\,\frac{q_1 q_2}{r^2}, \qquad k = 8.988 \times 10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2.$

• The force is radial: like charges repel, opposite charges attract.
• Doubling either charge doubles the force; doubling the separation quarters it.
• The constant is often rewritten as $k = \dfrac{1}{4\pi\varepsilon_0}$, with $\varepsilon_0 = 8.854 \times 10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2)$ — a form that simplifies later results.
• The same structure appears in Newton's gravity, $F = G\,\frac{m_1 m_2}{r^2}$, but gravity is purely attractive while the electric force can attract or repel. The electric force is vastly stronger — for two alpha particles the ratio $F_E/F_g \approx 3 \times 10^{35}$.

Electric field. A charge creates a field in the space around it; any other charge placed there feels a force. The field is the force per unit charge measured with a small test charge $q_0$: $\vec{E} = \frac{\vec{F}}{q_0}.$

• For a point charge, dividing out the test charge gives $E = \dfrac{kQ}{r^2}$, directed radially ($\hat{r}$).
• Units are newtons per coulomb (N/C). The force on any charge in the field is recovered by $\vec{F} = q\vec{E}$.
• $\vec E$ always points in the direction a positive charge would be pushed; a negative charge feels a force of equal magnitude in the opposite direction.

Field lines. A way to visualize $\vec{E}$: the field is tangent to the line at each point, pointing the way a positive charge would move. Lines emanate from positive charges and terminate on negative ones. Where lines crowd together the field is strong; where they spread out it is weak. The lines show direction and strength, not the trajectory a particle would follow.

Continuous charge distributions. When charge is spread over a ring, rod, surface, or volume, slice it into infinitesimal point-like pieces $dq$, find each piece's field, and integrate. Symmetry usually cancels all components except one (e.g. the axial $x$-component), so include a $\cos\alpha$ factor and integrate that component alone. Charge density bridges $dq$ to geometry:

• Linear density $\lambda = Q/L$, so $dq = \lambda\, ds$.
• Surface density $\sigma = Q/A$, so $dq = \sigma\, dA$.
• Volume density $\rho = Q/V$.

Key distribution results (on the axis, distance $x$):

• Ring of radius $a$: $\quad E = \dfrac{kQx}{(x^2 + a^2)^{3/2}}$. Zero at the center, point-like ($kQ/x^2$) far away.
• Finite rod of length $2a$: $\quad E = \dfrac{2akQ/(2a)}{x\sqrt{x^2+a^2}} = \dfrac{2k\lambda a}{x\sqrt{x^2+a^2}}$.
• Infinite line ($a \gg x$): $\quad E = \dfrac{2k\lambda}{x}$ — falls off only as $1/x$, not $1/x^2$.
• Disc / infinite plate ($R \gg x$): $\quad E = \dfrac{\sigma}{2\varepsilon_0}$ — uniform, independent of distance.
• Parallel plates (equal and opposite $\sigma$): the two plates' fields add between them and cancel outside, giving $E_{\text{inside}} = \dfrac{\sigma}{\varepsilon_0}$ and $E_{\text{outside}} = 0$ — the basis of the capacitor.

Worked Example

Electron accelerated between charged plates. Two large plates produce a uniform field $E = 1 \times 10^{4}\ \text{N/C}$ and sit $d = 1\ \text{cm} = 0.01\ \text{m}$ apart. An electron is released from rest at the negative plate. Find its acceleration, speed at the positive plate, and crossing time.

Start from $\vec F = q\vec E$ and Newton's second law, then the work–energy theorem $W = qEd = \tfrac{1}{2}mv^2$:

The electron reaches about 2% of the speed of light in roughly 3.4 nanoseconds.

Summary

• Coulomb's law: $F = k\,q_1 q_2 / r^2$ with $k = 8.988\times10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2 = 1/(4\pi\varepsilon_0)$; force is radial, attractive or repulsive, and inverse-square.
• The electric force shares gravity's mathematical form but is enormously stronger and can repel as well as attract.
• The electric field is force per unit charge, $\vec E = \vec F/q_0$; for a point charge $E = kQ/r^2$, and any charge feels $\vec F = q\vec E$. Field lines point the way a positive charge would move.
• For continuous distributions, integrate the field of infinitesimal pieces $dq$ using the appropriate charge density ($\lambda$, $\sigma$, $\rho$) and keep only the surviving (symmetric) component.
• Limiting cases organize the results: a ring gives zero field at its center and looks point-like far away; an infinite line falls off as $1/x$; an infinite plate gives a uniform field $\sigma/2\varepsilon_0$.
• Two oppositely charged plates produce a uniform field $\sigma/\varepsilon_0$ between them and zero field outside — the working principle of a capacitor.