A field guide

Electromagnetism.

Suppose you place a charge in empty space. The space around it is no longer neutral bookkeeping. It has an electric field. Move charge, and magnetic field joins the story. Change either field in time, and the other one can appear.

The subject is the grammar of those fields: where they start, where they loop, how they store energy, and how they travel as light.

The drawing is illustrative: charge makes electric field lines, current makes magnetic loops, and changing fields can propagate as waves.

II · Field means local force rule

A charge distribution can be complicated, but the field at one point has a simple job. It tells a small positive test charge which way it would be pushed. For point charges, the calculation is just vector addition.

Move the charges apart and change their signs. The widget computes the Coulomb field on a grid in arbitrary units, with arrows scaled by field magnitude.

Electric field map

Computes superposed Coulomb fields from two point charges in normalized units.

sample field0
patterndipole-like
null cluebetween charges

III · A closed surface counts enclosed charge

Flux is field passing through an area. Gauss's law says that total electric flux through a closed surface depends only on net charge inside. Outside charges can bend the arrows, but their field enters and leaves.

The figure uses a circular Gaussian surface in a two-dimensional drawing as an illustration of the three-dimensional law. The readout reports the enclosed charge and the corresponding closed-surface flux scale.

Gauss surface

Counts enclosed point charges and reports the Gauss-law flux scale Q/epsilon0.

enclosed charge0
flux scale0
outside effectbends field

IV · Voltage is energy per charge

In electrostatics, electric field points downhill in potential: $\mathbf{E}=-\nabla V$. A capacitor stores separated charge and field energy. The plate formula is simple because the geometry is simple.

Change area, gap, voltage, and dielectric constant. The widget computes $C=\epsilon_0\epsilon_r A/d$ and $U=\frac12CV^2$ for an ideal parallel-plate capacitor.

Parallel plates

Computes capacitance, field strength, stored energy, and energy density for ideal plates.

capacitance0
field0
stored energy0
energy density0

V · Circuits are compressed field problems

A circuit diagram is a low-frequency reduction of Maxwell's equations. It keeps currents and voltages while hiding the detailed fields in and around the wires. The reduction works when components are small compared with the relevant wavelength.

The RC model below computes capacitor voltage and resistor current during charging. The curve is exponential because the remaining voltage across the resistor shrinks as charge accumulates.

RC charging

Computes ideal RC capacitor voltage and current over five time constants.

time constant0
capacitor voltage0
current0

VI · Magnetism turns motion sideways

The Lorentz force is $q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$. A static magnetic field changes direction, not speed, for a point charge. That is why a uniform field makes circular or helical motion.

This widget computes a normalized circular orbit in a uniform field and a straight-wire magnetic-field scale from Ampere's law.

Lorentz orbit

Computes cyclotron radius and frequency in normalized units, plus B around a long wire.

orbit radius0
cyclotron omega0
wire field at 5 cm0

VII · Changing flux makes circulation

Faraday's law says $\mathcal{E}=-d\Phi_B/dt$. The minus sign is not decoration; it is the Lenz-law direction that prevents the induced current from creating energy for free.

Move the magnet speed slider. The widget uses a toy flux curve through a loop and computes the induced emf from its time derivative.

Magnet and loop

Computes a toy magnetic flux through a loop and the induced emf from -dPhi/dt.

flux0
emf0
current directionclockwise

VIII · Field energy can oscillate

A capacitor stores electric energy. An inductor stores magnetic energy. Connect them and the energy trades places, just as a mass-spring system trades kinetic and potential energy.

The ideal LC readout computes $\omega_0=1/\sqrt{LC}$ and the energy split at the selected phase. The damping slider is an illustrative envelope, not a full driven RLC solver.

LC oscillator

Computes ideal LC frequency and the electric/magnetic energy split over phase.

frequency0
electric energy0
magnetic energy0

IX · The math asks what starts, loops, and flows

The same law can be written locally or globally. Divergence becomes flux through a closed surface. Curl becomes circulation around a closed loop. Potentials give fields by derivatives, but they include gauge freedom: different potentials can describe the same physical $\mathbf{E}$ and $\mathbf{B}$.

Select a statement. The drawing shows the geometric object being tested: a pillbox for flux, a loop for circulation, an interface for boundary conditions, or an energy-flow arrow for Poynting's theorem.

Divergence theorem$\int_V\nabla\cdot\mathbf{F}\,dV=\oint_{\partial V}\mathbf{F}\cdot d\mathbf{A}$
Stokes' theorem$\int_S(\nabla\times\mathbf{F})\cdot d\mathbf{A}=\oint_{\partial S}\mathbf{F}\cdot d\boldsymbol{\ell}$
Potentials$\mathbf{B}=\nabla\times\mathbf{A}$ and $\mathbf{E}=-\nabla\phi-\partial\mathbf{A}/\partial t$
Gauge move$\mathbf{A}'=\mathbf{A}+\nabla\chi$, $\phi'=\phi-\partial\chi/\partial t$

Operator translator

Translates key E&M statements between local form, integral form, geometry, and caveat.

operatordivergence
geometryclosed surface
checkenclosed charge

X · Maxwell closes the loop

Maxwell's equations say what fields can start on, loop around, and carry away. The displacement-current term makes changing electric field a magnetic source, completing the symmetry needed for waves in vacuum.

The wave panel computes a plane-wave sketch in normalized units. The electric and magnetic fields are perpendicular to each other and to the propagation direction.

Gauss electric$\nabla\cdot\mathbf{E}=\rho/\epsilon_0$
Gauss magnetic$\nabla\cdot\mathbf{B}=0$
Faraday$\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$
Ampere-Maxwell$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\partial\mathbf{E}/\partial t$

Electromagnetic wave

Computes a normalized plane wave and the vacuum speed relation c=1/sqrt(mu0 epsilon0).

frequency0
E/B relationE = cB
energy flow+x

XI · Matter answers the field

Materials are not passive backgrounds. Electric fields polarize dielectrics. Magnetic fields align or oppose microscopic moments. Conductors move charge. These responses become $\epsilon$, $\mu$, and $\sigma$ only in simple ranges.

The widget compares four schematic material responses. The curves are teaching curves, not design data.

Material response

Draws schematic polarization, conduction, and magnetization responses for common material classes.

response0
memorylow
warninglinear range only

XII · Radiation carries the field away

Accelerating charge radiates. An antenna is a shaped current distribution that couples a circuit to traveling electromagnetic modes. Its length, feed, environment, and frequency decide how power leaves.

Choose a frequency. The widget computes wavelength in vacuum, a rough half-wave dipole length, and places the frequency on a simplified spectrum.

Spectrum and dipole

Computes wavelength and a rough half-wave dipole length from frequency in vacuum.

frequency0
wavelength0
half-wave length0
bandradio

XIII · Workshop: make the geometry visible

The workshop builds a Lorentz-force particle step, breaks it, repairs it, adds crossed-field drift, and then lets electric and magnetic fields update each other as a one-dimensional wave.

Fragment I

Euler heats a magnetic orbit

Step 1 of 4

XIV · Anatomy plates

Five plates, one field theory. Field lines show sources and sinks. Maxwell's equations show divergence and curl bookkeeping. The capacitor, induction, and wave plates show energy storage, conversion, and transport.

Plate I · charge and field lines
Plate II · Maxwell operator map
Plate III · capacitor energy
Plate IV · induction loop
Plate V · plane wave triad