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38 formulas

Electromagnetism

Charges, fields, light. Every formula below opens into a live, hands-on simulation.

electrostatics
F=ke∣q1q2∣r2F = k_e \frac{|q_1 q_2|}{r^2}

Coulomb's Law

Electric force between two charges falls off as the square of the distance between them.

electrostatics
E⃗=14πε0qr2r^\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}

Electric Field of a Point Charge

Each charge creates a field that tells other charges how much force they would feel.

electrostatics
∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

Gauss's Law

The total electric flux through a closed surface equals the enclosed charge divided by ε₀.

electrostatics
V=14πε0qrV = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}

Electric Potential (Point Charge)

Potential is the energy per unit charge — it falls off as 1/r, not 1/r².

electrostatics
C=ε0AdC = \varepsilon_0 \frac{A}{d}

Parallel Plate Capacitor

Bigger plates and smaller gaps store more charge per volt.

circuits
V=IRV = IR

Ohm's Law

Voltage is the push, resistance is the friction, current is how much flows.

magnetostatics
dB⃗=μ04πI dl⃗×r^r2d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}

Biot–Savart Law

Each bit of current creates a magnetic field perpendicular to both the current and distance.

magnetostatics
∮B⃗⋅dl⃗=μ0(Ienc+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Ampère's Law (with Maxwell's correction)

Magnetic field loops around currents; total circulation equals enclosed current times μ₀.

electromagnetic induction
E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}

Faraday's Law of Induction

A changing magnetic flux through a loop induces a voltage that opposes the change.

electromagnetic dynamics
F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Lorentz Force Law

Electric fields push charges; magnetic fields deflect moving charges sideways.

magnetostatics
∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0

Gauss's Law for Magnetism

Magnetic field lines always close on themselves — no isolated north or south poles.

electrostatics
U=12CV2U = \tfrac{1}{2} C V^2

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.

circuits
V(t)=V0(1−e−t/τ),τ=RCV(t) = V_0 \left(1 - e^{-t/\tau}\right), \quad \tau = RC

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.

magnetostatics
F⃗=IL⃗×B⃗\vec{F} = I \vec{L} \times \vec{B}

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.

magnetostatics
B=μ0nIB = \mu_0 n I

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.

magnetostatics
L=μ0n2V=μ0N2AℓL = \mu_0 n^2 V = \mu_0 \frac{N^2 A}{\ell}

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.

magnetostatics
U=12LI2U = \tfrac{1}{2} L I^2

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.

circuits
f=12πLCf = \frac{1}{2\pi \sqrt{LC}}

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.

electromagnetic waves
S⃗=1μ0E⃗×B⃗\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}

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.

electromagnetic induction
ε=BLv\varepsilon = B L v

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.

electromagnetic dynamics
fc=qB2πmf_c = \frac{q B}{2\pi m}

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.

circuits
I(t)=V0R(1−e−t/τ),τ=LRI(t) = \frac{V_0}{R}\left(1 - e^{-t/\tau}\right), \quad \tau = \frac{L}{R}

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.

electromagnetic induction
VsVp=NsNp\frac{V_s}{V_p} = \frac{N_s}{N_p}

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.

electromagnetic waves
c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

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.

electromagnetic dynamics
∮B⃗⋅dl⃗=μ0(I+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

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.

electromagnetic waves
P=q2a26πε0c3P = \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3}

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.

circuits
vd=InAqv_d = \frac{I}{n A q}

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.

circuits
ρ=ρ0 [1+α(T−T0)]\rho = \rho_0\,[1 + \alpha (T - T_0)]

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.

magnetostatics
τ=μBsin⁡θ,μ=NIA\tau = \mu B \sin\theta,\quad \mu = N I A

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.

electromagnetic waves
∇ ⁣⋅ ⁣E⃗=ρε0, ∇ ⁣⋅ ⁣B⃗=0, ∇ ⁣× ⁣E⃗=−∂tB⃗, ∇ ⁣× ⁣B⃗=μ0J⃗+μ0ε0∂tE⃗\nabla\!\cdot\!\vec{E}=\tfrac{\rho}{\varepsilon_0},\ \nabla\!\cdot\!\vec{B}=0,\ \nabla\!\times\!\vec{E}=-\partial_t\vec{B},\ \nabla\!\times\!\vec{B}=\mu_0\vec{J}+\mu_0\varepsilon_0\partial_t\vec{E}

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.

circuits
ΔV=I Rseg=I ρ LA\Delta V = I\,R_{seg} = I\,\rho\,\frac{L}{A}

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.

magnetostatics
VH=IBnqtV_H = \frac{I B}{n q t}

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.

electrostatics
E⃗inside=0\vec{E}_{inside} = 0

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.

applications
f=γB02πf = \frac{\gamma B_0}{2\pi}

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.

ac circuits
δ=2ωμσ=1πfμσ\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}}

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.

plasma physics
sin⁡2α1B1=sin⁡2α2B2\frac{\sin^2\alpha_1}{B_1} = \frac{\sin^2\alpha_2}{B_2}

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.

electromagnetic waves
fc=c2a(TE10)f_c = \frac{c}{2a}\quad(\text{TE}_{10})

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.

theoretical
e g=2πnℏ=nh  ⇒  g=nhee\,g = 2\pi n \hbar = n h \;\Rightarrow\; g = \frac{n h}{e}

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.