Topic 3

Electromagnetism

Electric charges, magnetic fields, and light itself — Maxwell's equations unify electricity and magnetism into a single, elegant theory that underlies all of modern technology.

  1. Electric charge & Coulomb's law
  2. Electric fields & potential
  3. Circuits & currents
  4. Magnetism & the Biot-Savart law
  5. Maxwell's equations
  6. Light, waves & optics

1. Electric Charge & Coulomb's Law

Electric charge is the fundamental property that makes objects attract or repel. There are two kinds — positive and negative — and unlike charges attract while like charges repel. Charge is quantized in units of the elementary charge $e = 1.602\times10^{-19}$ C.

$$\boxed{F = k_e\frac{|q_1 q_2|}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}}$$

This is Coulomb's law: the electric force is proportional to the product of the charges and inversely proportional to the square of their separation. By contrast with gravity (always attractive), electric forces can attract or repel — and they're roughly 10³⁶ times stronger!

Conservation of charge — The total charge in an isolated system is constant. You can transfer charge, but you can't create or destroy net charge (charge density can cancel to zero, but the total is conserved).

2. Electric Fields & Potential

To avoid "action at a distance," we say a charge $q$ creates an electric field $\vec{E}$ that exerts a force $\vec{F} = q\vec{E}$ on any other charge. Field lines point away from positive charges and toward negative ones.

$$\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i(\vec{r}-\vec{r}_i)}{|\vec{r}-\vec{r}_i|^3},\qquad \vec{F} = q\vec{E}$$

Because the electric force is conservative, we can define a scalar electric potential $V$ such that $\vec{E} = -\nabla V$:

$$V = \frac{U}{q},\qquad \Delta V = -\int \vec{E}\cdot d\vec{l}$$

Voltage is potential difference. A battery maintains a steady $\Delta V$, pushing charges through a circuit.

Electric field lines from two charges. Drag the sliders to flip charge signs and change separation. Field arrows (lines) show the direction a positive test charge would move; denser lines mean stronger field.

3. Current, Resistance & Circuits

Current $I$ is the rate of charge flow. Ohm's law $V = IR$ relates voltage, current, and resistance for ohmic materials, where resistance depends on geometry and material:

$$R = \rho\frac{L}{A},\qquad V = IR,\qquad P = IV$$
Think of electricity like water. Voltage is like water pressure (height of a reservoir), current is like flow rate (liters per second), and resistance is like a narrow pipe. A battery is a pump that maintains pressure even as water flows.
$$\text{Series: } R_{\text{eq}} = \sum R_i \quad\quad \text{Parallel: } \frac{1}{R_{\text{eq}}} = \sum \frac{1}{R_i}$$
Example. Two 100 Ω resistors in parallel across a 12 V battery.
Equivalent resistance: $1/R_{\text{eq}} = 1/100 + 1/100 = 2/100$, so $R_{\text{eq}} = 50$ Ω.
Current: $I = V/R = 12/50 = 0.24$ A. Power dissipated: $P = IV = 2.88$ W.

4. Magnetism & the Biot-Savart Law

Whereas stationary electric charges create electric fields, moving charges (currents) create magnetic fields. The Biot-Savart law gives the field from a small current element:

$$\vec{B} = \frac{\mu_0}{4\pi}\int \frac{I\,d\vec{\ell}\times\hat{r}}{r^2}$$

A charge $q$ moving at velocity $\vec{v}$ in a magnetic field feels the Lorentz force $\vec{F} = q\vec{v}\times\vec{B}$, which is always perpendicular to velocity — so it bends trajectories without changing speed. This is why the aurora curves along Earth's magnetic field lines.

Faraday's law of induction — A changing magnetic flux $\Phi_B$ induces an electromotive force: $\mathcal{E} = -d\Phi_B/dt$. This is the principle behind generators, transformers, and induction cooktops.
$$\mathcal{E} = -\frac{d\Phi_B}{dt},\qquad \Phi_B = \int \vec{B}\cdot d\vec{A}$$

5. Maxwell's Equations

James Clerk Maxwell (1865) gathered the laws of electricity and magnetism into four equations. In differential form, in vacuum:

$$\begin{aligned} \nabla\cdot\vec{E} &= \frac{\rho}{\varepsilon_0} & &\text{(Gauss's law — electric charges make E-fields)}\\\\ \nabla\cdot\vec{B} &= 0 & &\text{(No magnetic monopoles)}\\\\ \nabla\times\vec{E} &= -\frac{\partial\vec{B}}{\partial t} & &\text{(Faraday — changing B makes E)}\\\\ \nabla\times\vec{B} &= \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t} & &\text{(Ampère–Maxwell)} \end{aligned}$$
The displacement current is Maxwell's genius stroke. Ampère's original law had a gap: it worked for steady currents but broke down for time-varying fields (like the charging capacitor, where current flows into a plate but not through the gap). Maxwell added the $\mu_0\varepsilon_0\,\partial\vec{E}/\partial t$ term — a current made of changing electric fields. With this fix, the equations predict self-sustaining waves that travel exactly at $c = 1/\sqrt{\mu_0\varepsilon_0} = 3\times10^8$ m/s. Light is an electromagnetic wave.
SymbolMeaningValue
$\varepsilon_0$Vacuum permittivity$8.854\times10^{-12}$ F/m
$\mu_0$Vacuum permeability$4\pi\times10^{-7}$ H/m
$c$Speed of light$2.998\times10^8$ m/s
$k_e$Coulomb constant$8.988\times10^9$ N·m²/C²

6. Light, Waves & Optics

Light is a transverse electromagnetic wave with wavelength $\lambda$ and frequency $f$ related by $c = \lambda f$. The electromagnetic spectrum spans radio waves (kilometers long) to gamma rays (picometers), with visible light a narrow 400–700 nm band in between.

$$\vec{E}(x,t) = E_0\sin(kx - \omega t),\qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = \lambda f$$

Refraction follows Snell's law, $n_1\sin\theta_1 = n_2\sin\theta_2$, where $n = c/v$ is the index of refraction. Reflection, diffraction, and interference reveal light's wave nature — confirmed decisively by Young's double-slit experiment in 1801.

interface (air/glass) incident ray refracted ray n₁sinθ₁ = n₂sinθ₂
Snell's law: light bends toward the normal when entering a higher-index medium (glass). The angle of incidence equals the angle of reflection.

Key Takeaways

〰️ Next: Waves & Oscillations →