Electromagnetism
Charges, fields, light. Every formula below opens into a live, hands-on simulation.
Coulomb's Law
Electric force between two charges falls off as the square of the distance between them.
Electric Field of a Point Charge
Each charge creates a field that tells other charges how much force they would feel.
Gauss's Law
The total electric flux through a closed surface equals the enclosed charge divided by ε₀.
Electric Potential (Point Charge)
Potential is the energy per unit charge — it falls off as 1/r, not 1/r².
Parallel Plate Capacitor
Bigger plates and smaller gaps store more charge per volt.
Ohm's Law
Voltage is the push, resistance is the friction, current is how much flows.
Biot–Savart Law
Each bit of current creates a magnetic field perpendicular to both the current and distance.
Ampère's Law (with Maxwell's correction)
Magnetic field loops around currents; total circulation equals enclosed current times μ₀.
Faraday's Law of Induction
A changing magnetic flux through a loop induces a voltage that opposes the change.
Lorentz Force Law
Electric fields push charges; magnetic fields deflect moving charges sideways.
Gauss's Law for Magnetism
Magnetic field lines always close on themselves — no isolated north or south poles.
Energy Stored in a Capacitor
To charge a capacitor you must push charge against a growing voltage — the work done is stored as electric field energy between the plates.
RC Circuit Charging
Current is largest at t = 0 when the capacitor is empty; as voltage builds up, current decays, asymptotically filling the capacitor to V₀.
Magnetic Force on a Current-Carrying Wire
Each moving charge in the wire feels a Lorentz force; summed over all of them, the wire experiences a net push perpendicular to both the current and the field.
Magnetic Field Inside a Solenoid
Each turn contributes a circular field; many tightly packed turns add up inside the tube and cancel outside — leaving a near-uniform interior field.
Self-Inductance of a Solenoid
When current changes, flux through every loop changes, inducing an EMF that opposes the change. The geometric constant tying flux to current is the inductance.
Energy Stored in an Inductor
To establish a current in an inductor you must do work against the back-EMF. That work survives as magnetic field energy stored in the coil's interior.
LC Oscillation Frequency
Energy sloshes between the capacitor's electric field and the inductor's magnetic field — a pure electromagnetic version of a mass-on-a-spring.
Poynting Vector
Wherever both E and B exist, energy flows perpendicular to both. The Poynting vector tells you how much energy crosses a unit area per second, and in what direction.
Motional EMF
Free charges inside the moving rod feel a magnetic force qv×B that pushes them along the rod — positive charge piles up at one end until the resulting electric field balances the push. That charge separation is a battery made of motion.
Cyclotron Frequency
The magnetic force qvB always points to the center, bending the path into a circle. Faster particles ride bigger circles, but the extra path length exactly cancels the extra speed — every lap takes the same time.
RL Circuit Current Growth
An inductor hates changes in current: its back-EMF L·dI/dt initially cancels the battery entirely, so current starts at zero and creeps up. As the current settles, the back-EMF dies away and the resistor takes over.
Ideal Transformer Equation
Both coils share the same changing flux through the iron core, so each turn of wire sees the same induced voltage. Voltage per coil is just volts-per-turn × turns — the ratio of turns is the ratio of voltages.
Speed of Electromagnetic Waves
A changing electric field makes a magnetic field (Ampère–Maxwell); a changing magnetic field makes an electric field (Faraday). The two regenerate each other in a self-sustaining ripple whose speed is fixed by how strongly each effect couples — ε₀ and μ₀.
Ampère–Maxwell Law
Maxwell noticed Ampère's law contradicts itself at a charging capacitor: a loop around the wire sees current, but the same loop with its surface bulged through the gap sees none. His fix: a changing electric field acts exactly like a current — the displacement current — and creates magnetic field just the same.
Larmor Formula
A charge at rest has radial field lines. Accelerate it, and the news of its new motion spreads outward at c — the field lines develop a traveling kink. That kink is a transverse field pulse carrying energy: radiation. No acceleration, no kink, no radiation.
Drift Velocity
Electrons in a wire already zip around at ~10⁶ m/s thermally, but in random directions that cancel. An applied field adds a tiny net drift on top — a slow river current under a churning sea. The signal is fast (a field change at near light speed) but the carriers themselves barely move.
Temperature Dependence of Resistivity
Resistance comes from electrons scattering off the lattice. Heat makes the ions vibrate harder, so electrons collide more often and drift less freely — resistivity climbs nearly linearly over ordinary temperature ranges.
Magnetic Dipole Moment & Torque
A current loop is a tiny bar magnet: its moment μ points along the loop's axis. A uniform field can't push it (forces cancel) but it twists it, trying to align μ with B — exactly how a compass needle swings north.
Maxwell's Equations (Unified)
Two of the four equations say charges make diverging E-fields and that there are no magnetic charges; the other two say a changing B makes a curling E and a changing E (or a current) makes a curling B. That mutual feedback lets the fields regenerate each other and sail off as light — no charges required.
Why Birds on Power Lines Don't Get Shocked
Current follows the path of least resistance. The bird's body (tens of kΩ) sits in parallel with a few centimeters of thick wire (micro-ohms). The wire is millions of times more conductive, so essentially all the current stays in the metal and the bird carries a vanishing trickle. Shock needs a voltage difference across you — and across two nearby points on one wire that difference is microscopic.
Hall Effect
Moving carriers in a perpendicular field feel a sideways Lorentz push and pile up on one edge. The charge buildup creates a transverse electric field until it exactly cancels the magnetic deflection. The leftover voltage across the strip reveals both the sign and the density of the carriers — a direct count of charge carriers.
Why a Faraday Cage Shields You
A conductor's free charges move until they kill any field inside the metal. Put a cavity inside and those charges arrange on the outer surface to cancel the external field throughout the hollow — leaving the interior field-free. For oscillating waves a mesh works too, as long as the holes are far smaller than the wavelength, because the induced surface currents re-radiate a canceling field.
How MRI Imaging Works (Larmor)
Every proton is a tiny spinning magnet. In a strong field B₀ it doesn't just align — it precesses like a wobbling top, at a frequency set only by the field. Hit it with a radio pulse at exactly that frequency and it absorbs energy, tips over, then relaxes back while broadcasting a faint radio signal. A field gradient makes the frequency a position label — turning the echo into an image.
Skin Effect in Conductors
An alternating current sets up a changing magnetic field inside the conductor, which by Faraday's law drives eddy currents that oppose the flow in the core and reinforce it near the surface. The faster the oscillation, the more the current is squeezed into a thin surface layer of thickness δ — the skin depth.
Physics of the Aurora
A charged particle spirals around a magnetic field line. As it moves toward a pole the field gets stronger and tighter, and conservation of the magnetic moment (the first adiabatic invariant) forces its spiral to widen in pitch until it stops and reflects — a magnetic mirror. Particles with small enough pitch angle slip through the mirror and slam into the upper atmosphere, exciting oxygen and nitrogen that glow green and red.
Waveguide Cutoff Frequency
A guided wave can be pictured as a plane wave zig-zagging between the conducting walls. To satisfy the boundary conditions, exactly a half-wavelength (for TE₁₀) must fit across the width a. If the free-space wavelength is too long — frequency too low — it can't fit at any bounce angle, so the wave can't propagate and instead decays exponentially: it's evanescent.
Dirac Magnetic Monopole Quantization
Wrap an electron's quantum wavefunction around a monopole and its phase must come back to itself — single-valuedness. The phase picked up is set by the magnetic flux from the monopole; demanding it be a multiple of 2π forces the product of electric and magnetic charge to be quantized. Turn it around: even one monopole makes every electric charge a multiple of a basic unit.
Kirchhoff's Voltage Law
Voltage is electrical height. Go around a closed loop and every rise (battery) must be exactly cancelled by the drops (resistors) — you can't gain energy by walking in a circle. It is energy conservation wearing a circuit hat.
Self-Inductance
A coil stores energy in its magnetic field and hates sudden change. Push current in and it pushes back; cut current and it strains to keep it flowing — electrical inertia, the magnetic analogue of mass.
RLC Series Resonance
An inductor and capacitor swap energy back and forth like a pendulum swapping speed and height. At one special frequency their opposing reactances cancel, the circuit looks purely resistive, and the current rings loudest.
Inductive Reactance
The faster AC tries to change, the harder a coil's back-EMF pushes against it. Reactance is that opposition — zero for DC, growing straight-line with frequency, like air resistance that gets worse the faster you go.
Capacitance
Capacitance is how much charge a conductor pair will swallow for each volt you push — a bucket size for electric charge. Bigger plates, closer together, with the right filling, hold more per volt.
RC Time Constant
Resistance throttles the current that fills (or drains) the capacitor, so their product sets one natural tick. In one τ the gap to the final value shrinks by 63%; after ~5τ it's essentially done.
LC Resonant Frequency
The capacitor stores energy in its electric field, the inductor in its magnetic field, and they trade it endlessly. Their sizes set the period exactly as mass and spring set a pendulum's swing.
Thévenin's Theorem
Any linear two-terminal network, no matter how complex, behaves at its terminals exactly like a single voltage source V_th in series with a single resistor R_th. Replace the whole box with two numbers.
Wheatstone Bridge
Two voltage dividers fed from the same source; when their midpoints sit at the same voltage, no current crosses the bridge. Balance turns an unknown resistor into a ratio of three known ones.
Solenoid Inductance
Each turn both makes flux and catches it. Double the turns and you double the field AND double the linkage — hence the square. Pack them on a long, fat coil for the most inductance.
How Electric Motors Spin
A current loop is a magnet; a field twists it to align, just like a compass needle. Keep flipping the current (commutator) so it never settles, and the loop spins forever — that's a motor.
How Generators Make Electricity
As a coil rotates, the flux through it rises and falls sinusoidally; Faraday's law says the EMF is the rate of change of that flux — so the output is a sine wave whose peak grows with how fast you spin.
Why Transformers Hum
The core's iron physically stretches in a magnetic field (magnetostriction), and the strain depends on B². Since B² peaks twice per AC cycle — once each polarity — the mechanical vibration, and the sound, comes out at double the line frequency.
Transmission-Line Characteristic Impedance
A long line is a ladder of tiny inductors and capacitors. A wave entering it sees a constant ratio of voltage to current — √(L/C) — set by the per-metre L and C. Match a load to it and there's no reflection.