Electromagnetism

Coulomb's law

$\mathbf{F}_{12}$

Force on charge 1 due to charge 2. A vector. By Newton's third law, the force on charge 2 due to charge 1 is equal and opposite.

$q_1, q_2$

The two charges, in coulombs. Can be positive or negative.

$r$

Distance between them, in meters.

$\hat{\mathbf{r}}$

Unit vector pointing from charge 2 to charge 1. Tells you the direction; the magnitude is encoded in the scalar part.

$1/(4\pi\epsilon_0)$

Coulomb's constant, $k_e \approx 8.99 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$. The $4\pi$ is there so that Gauss's law below looks clean; it is a bookkeeping choice.

Sign convention. If $q_1 q_2 > 0$ (same signs), the force is along $+\hat{\mathbf{r}}$, pushing them apart. If $q_1 q_2 < 0$ (opposite signs), the force is along $-\hat{\mathbf{r}}$, pulling them together. Coulomb's law looks exactly like Newton's inverse-square law for gravity, with charge replacing mass. The difference is that charges come in two flavors and can cancel, while mass only attracts.

Electric field from point charges

$\mathbf{E}(\mathbf{r})$

The electric field vector at the field point $\mathbf{r}$. Tells you the force a unit positive charge would feel if placed there.

$q_i$

The $i$-th source charge.

$\mathbf{r}_i$

Position of the $i$-th source charge.

$|\mathbf{r} - \mathbf{r}_i|$

Distance from the $i$-th source charge to the field point.

$\hat{\mathbf{r}}_i$

Unit vector from the $i$-th source charge to the field point.

$\sum_i$

Sum over all source charges — this is the principle of superposition: fields from multiple charges simply add vectorially.

Why fields, not forces. The field is a property of the region of space, independent of whether there is a test charge there to feel it. You can compute $\mathbf{E}$ once, then ask about the force on any charge just by multiplying: $\mathbf{F} = q\mathbf{E}$. It also sets up the mental picture you'll need for Maxwell's equations and for the idea that fields carry energy and momentum of their own, independent of any particles.

Gauss's law (integral form)

$\oint$

A closed-surface integral — integrate over the entire boundary of some 3D region.

$\mathbf{E}\cdot d\mathbf{A}$

The electric field dotted with an outward-pointing area element. Measures how much field is piercing through that patch of surface.

$\partial\mathcal V$

The boundary of the volume $\mathcal V$ — the Gaussian surface. You are free to choose any closed surface; symmetry tells you which choice makes the integral trivial.

$Q_\text{enc}$

The total charge enclosed inside that surface. Charges outside the surface contribute nothing to the total flux, though they contribute plenty to the local field.

$\epsilon_0$

Permittivity of free space — the same constant from Coulomb's law.

The trick. Gauss's law is always true, but it is only useful when the symmetry of the problem lets you pull $\mathbf{E}$ out of the integral. For a spherical charge distribution, a concentric spherical Gaussian surface gives $|\mathbf{E}| \cdot 4\pi r^2 = Q/\epsilon_0$, so $|\mathbf{E}| = Q/(4\pi\epsilon_0 r^2)$ — you just recovered Coulomb's law in one line. For an infinite line of charge, a coaxial cylinder gives $|\mathbf{E}| \propto 1/r$. For an infinite sheet, a pillbox gives $|\mathbf{E}| = \sigma/(2\epsilon_0)$, independent of distance. Three problems, three symmetry choices, three trivial integrals.

Field outside a uniform sphere

$Q$

Total enclosed charge.

$4\pi r^2$

Surface area of the Gaussian sphere.

$r$

Distance from the sphere's center to the field point.

The shell theorem. This looks exactly like the field of a point charge $Q$ at the sphere's center. Outside any spherically symmetric distribution, the field is identical to a point charge at the center. Newton proved the same thing for gravity. It is a happy consequence of the inverse-square law and spherical symmetry.

Field inside a uniform sphere

$Qr^3/R^3$

The enclosed charge inside a Gaussian sphere of radius $r$ — total charge times the volume fraction.

$r$

Same as before, but now $r < R$.

Linear inside, inverse-square outside. Inside, the field grows linearly with $r$ — it is zero at the center (by symmetry) and rises to $Q/(4\pi\epsilon_0 R^2)$ at the surface. Outside, it falls off as $1/r^2$. The two pieces match at $r = R$, so the field is continuous even though its functional form changes. Plot it and you see a triangle rising to a peak then decaying.

Electric potential

$V(\mathbf{r})$

Electric potential at point $\mathbf{r}$, in volts. Potential energy per unit charge.

$-\nabla V$

Negative gradient of $V$. Same relation as force from potential energy in mechanics — electric field "points downhill" on the potential landscape.

$1/|\mathbf{r}-\mathbf{r}_i|$

Notice the single inverse power, not inverse-square. Integrating $E \sim 1/r^2$ gives $V \sim 1/r$.

Why scalar fields are nice. Adding potentials from many charges is a simple scalar sum — no vector components to juggle. You compute $V$ from a point-charge distribution by adding, then take the gradient to get $\mathbf{E}$. Much easier than vector superposition of fields, especially for complicated geometries. The potential is also what voltmeters measure: a voltage reading is a difference in electric potential between two probe points.

Parallel-plate capacitor

$C$

Capacitance. How much charge you can store per volt of potential difference.

$A$

Plate area.

$d$

Plate separation.

$U$

Energy stored in the capacitor — equivalent to the energy stored in the electric field between the plates.

Where the energy goes. Filling a capacitor takes work because you have to push like charges together against their own repulsion. That work is stored in the electric field in the gap. The energy density of an electric field is $u_E = \tfrac{1}{2}\epsilon_0 |\mathbf{E}|^2$, so electric fields themselves carry energy — not just the particles in them. This is the first hint that fields are physical objects, not mathematical conveniences.

Ohm's law and power

$V$

Voltage drop (potential difference) across the resistor, in volts.

$I$

Current through the resistor, in amperes. One ampere is one coulomb per second.

$R$

Resistance, in ohms. Depends on the material, geometry, and temperature.

$P$

Power dissipated — the rate at which electrical energy is turned into heat. Units: watts.

The water analogy. Voltage is pressure, current is flow rate, and resistance is pipe narrowness. More pressure pushes more water through a pipe of fixed narrowness. A thin pipe resists more. The power $I^2 R$ dissipated is like the heat you'd generate forcing water through a small hole — the resistor's job description is "convert electrical work into heat." This is what every electric heater and incandescent bulb does deliberately, and what every CPU does by accident.

Lorentz force

$q$

The charge feeling the force.

$\mathbf{E}$

Local electric field.

$\mathbf{v}\times\mathbf{B}$

Cross product of the particle's velocity with the local magnetic field. Perpendicular to both $\mathbf{v}$ and $\mathbf{B}$.

$\mathbf{B}$

Local magnetic field.

Three things the Lorentz law tells you. First, a stationary charge feels no magnetic force — magnetic fields only act on moving charges. Second, the magnetic force is always perpendicular to the velocity, so it changes the direction of motion but not the speed — magnetic fields do no work on point charges. Third, charges moving in a uniform magnetic field travel in circles (or helixes if they have a velocity component along the field). This is the principle behind cyclotrons, mass spectrometers, and the funnels that guide electron beams in old CRT televisions.

Cyclotron motion

$r$

Radius of the circular orbit. Proportional to speed, inversely proportional to field strength. Fast particles orbit larger circles.

$\omega_c$

Angular frequency of the circular motion, independent of the particle's speed. This speed-independence is what made the cyclotron work.

$m, q$

Mass and charge of the particle.

Why mass spectrometers work. For two particles with the same charge but different masses sent through the same field at the same speed, the heavier one traces a bigger circle. Fire a beam of ionized atoms into a magnetic field and each isotope lands at a different place on the detector plate. This is how isotope abundances are measured and how trace chemicals are identified by their characteristic mass spectra.

Ampère's law (static form)

$\oint_C$

Integral around a closed loop $C$ — an Amperian loop, chosen for symmetry.

$\mathbf{B}\cdot d\boldsymbol{\ell}$

Magnetic field dotted with an infinitesimal directed segment of the loop.

$\mu_0$

Permeability of free space, $4\pi\times 10^{-7}\,\text{T}\cdot\text{m/A}$.

$I_\text{enc}$

Current piercing any surface bounded by the loop.

Same trick as Gauss. For a long straight wire, draw a circular loop around it: $|\mathbf{B}|\cdot 2\pi r = \mu_0 I$, so $|\mathbf{B}| = \mu_0 I / (2\pi r)$. For a solenoid, draw a rectangle with one side inside and one outside: $|\mathbf{B}| = \mu_0 n I$ inside, 0 outside. Two problems, two symmetry choices, two trivial integrals. Maxwell later added a correction term (the displacement current) to make Ampère's law work when fields change in time — that correction is what makes electromagnetic waves possible.

Faraday's law

$\Phi_B$

Magnetic flux through surface $S$ — integrate $\mathbf{B}\cdot d\mathbf{A}$ over the surface.

$d\Phi_B/dt$

Rate of change of that flux with time.

$\oint \mathbf{E}\cdot d\boldsymbol{\ell}$

Electromotive force (EMF) around a closed loop — the line integral of the induced electric field.

The minus sign

Lenz's law: the induced current flows in whichever direction opposes the change in flux. If the flux through a loop is increasing, the induced current tries to create a flux that cancels the increase.

Why generators work. Spin a coil of wire inside a fixed magnet (or a fixed magnet inside a stationary coil). The flux through the coil varies sinusoidally, so the induced EMF is sinusoidal, and alternating current flows in the wire. This is every power plant's output stage. The energy comes from whatever is doing the spinning — steam from nuclear or coal, water from a dam, wind from a turbine. The physics is identical. A transformer uses the same principle with two coils sharing flux through an iron core: the primary's varying current creates varying flux which induces an EMF in the secondary. Voltage ratios match turn ratios.

Gauss's laws (for E and B)

$\nabla\cdot\mathbf{E}$

Divergence of the electric field — measures how much the field "spreads out" from a point.

$\rho$

Electric charge density, in coulombs per cubic meter. Local amount of charge per unit volume.

$\nabla\cdot\mathbf{B} = 0$

The magnetic field has no divergence anywhere. In other words, magnetic monopoles do not exist — every magnetic field line that enters a closed region must also leave it. If you cut a bar magnet in half, you get two smaller bar magnets, not an isolated north and south pole.

What divergence means. Think of a fluid. The divergence of its velocity field at a point tells you whether fluid is being created or destroyed there. $\nabla\cdot\mathbf{E} = \rho/\epsilon_0$ says electric field lines are created at positive charges and destroyed at negative charges, with strength proportional to the local charge density. $\nabla\cdot\mathbf{B} = 0$ says magnetic field lines have nowhere to start or end — they always form closed loops.

Faraday's and Ampère-Maxwell laws

$\nabla\times\mathbf{E}$

Curl of the electric field — measures how much the field circulates around a point.

$-\partial\mathbf{B}/\partial t$

Negative time derivative of the magnetic field. A changing $\mathbf{B}$ drives a curling $\mathbf{E}$. This is Faraday's law in differential form.

$\mathbf{J}$

Electric current density, in amperes per square meter.

$\mu_0 \mathbf{J}$

Ampère's original source term: currents produce curling magnetic fields.

$\mu_0 \epsilon_0\,\partial\mathbf{E}/\partial t$

Maxwell's displacement current — a changing electric field also produces a curling magnetic field, even in empty space with no actual current. Without this term, the equations would be mathematically inconsistent and electromagnetic waves would not exist.

What Maxwell noticed. Ampère's original law said only real currents produced magnetic fields. Maxwell realized that if you apply this to a capacitor being charged — current flows into the plates but not through the gap — you get a contradiction at the capacitor gap. His fix: the changing electric field between the plates acts like a current for the purpose of magnetic generation. Add this term and the four equations together become self-consistent and predict that ripples in the field propagate as transverse waves at speed $1/\sqrt{\mu_0\epsilon_0}$. Maxwell computed this number and got $3\times 10^8$ m/s, matching measurements of the speed of light. "We can scarcely avoid the inference," he wrote, "that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Unification.

Electromagnetic wave equation

$\nabla^2$

The Laplacian — sum of second partial derivatives, $\partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2$.

$\partial^2/\partial t^2$

Second partial derivative with respect to time.

$c$

The speed of light in vacuum, $2.998 \times 10^8$ m/s. Comes out of the equations as $1/\sqrt{\mu_0\epsilon_0}$, with no reference to light.

Why this is a wave equation. The standard form $\nabla^2 f = (1/c^2) \partial^2 f/\partial t^2$ describes any quantity that propagates as a wave at speed $c$. Plane-wave solutions like $\mathbf{E} = \mathbf{E}_0 \cos(kx - \omega t)$ with $\omega/k = c$ satisfy it immediately. The $\mathbf{E}$ and $\mathbf{B}$ fields are perpendicular to each other and to the direction of propagation, with amplitudes related by $|\mathbf{B}| = |\mathbf{E}|/c$. They oscillate in step, shedding energy as they go.