Foundations

Current measures charge crossing a chosen surface per unit time.

An electric field assigns force per unit test charge.

Magnetic force depends on charge motion relative to the field.

A linear current-voltage relation requires suitable conditions.

Four field equations connect electric and magnetic fields with charge and current.

The elementary charge has an exact SI numerical value.

Scalar and vector potentials encode electric and magnetic fields.

A linear two-conductor arrangement stores separated charge.

Changing current changes linked magnetic flux and induced emf.

The rate of field-energy change balances outward energy flow and power transferred to charges.

Lumped circuit models balance node currents and loop voltages.

A simple resistor–capacitor circuit has an exponential timescale.

Sinusoidal response combines resistance and net reactance.

In vacuum electrostatics, a point charge q at r′ produces E(r)=q(r−r′)/(4πε₀|r−r′|³).

Electrostatic potential obeys ∇²φ=−ρ/ε₀ in vacuum; charge-free regions obey Laplace's equation.

For a charge q at r₀ outside a grounded conducting sphere of radius a centered at the origin, an image q′=−qa/|r₀| at r′=a²r₀/|r₀|² gives the exterior potential q/(4πε₀|r−r₀|)+q′/(4πε₀|r−r′|).

Outside a localized charge distribution, the potential expands in powers of inverse distance.

A localized steady current distribution in vacuum produces B(r)=μ₀/(4π)∫J(r′)×(r−r′)/|r−r′|³d³r′.

Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary.

Magnetization M contributes bound volume current J_b=∇×M and surface current K_b=M×n.

For a stationary interface with unit normal n from medium 1 to 2, n·(D₂−D₁)=σ_free, n·(B₂−B₁)=0, n×(E₂−E₁)=0 and n×(H₂−H₁)=K_free.

In a homogeneous nonmagnetic isotropic dielectric, a monochromatic plane wave has k²=ε_r(ω)ω²/c² and B₀=k×E₀/ω.

A damped bound-electron oscillator gives ε_r(ω)=1+Ne²/[ε₀m(ω₀²−ω²−iγω)] for the exp(−iωt) convention.

At normal incidence on a lossless nonmagnetic dielectric interface, the electric-amplitude reflection coefficient is r=(n₁−n₂)/(n₁+n₂).

A nonrelativistic accelerated point charge radiates total power P=q²a²/(6πε₀c³).

At an interface between lossless isotropic nonmagnetic dielectrics, normal energy-flux ratios are R_s=[(n₁cosθ_i−n₂cosθ_t)/(n₁cosθ_i+n₂cosθ_t)]² and R_p=[(n₂cosθ_i−n₁cosθ_t)/(n₂cosθ_i+n₁cosθ_t)]², with T_s,p=1−R_s,p.

Vacuum fields E=−∇φ−∂_tA and B=∇×A are unchanged by A′=A+∇χ, φ′=φ−∂_tχ.

In Coulomb gauge ∇·A=0, the vacuum scalar potential satisfies ∇²φ=−ρ/ε₀ while the vector potential obeys (∇²−c⁻²∂_t²)A=−μ₀J_T, J_T=J−ε₀∇∂_tφ and ∇·J_T=0.

The source-generated vacuum Lorenz potentials are φ(r,t)=(1/(4πε₀))∫ρ(r′,t−R/c)d³r′/R and A(r,t)=(μ₀/(4π))∫J(r′,t−R/c)d³r′/R, where R=|r−r′|.

Differentiating the causal potentials gives E=(1/(4πε₀))∫([ρ]n/R²+[ρ̇]n/(cR)−[J̇]/(c²R))d³r′ and B=(μ₀/(4π))∫([J]/R²+[J̇]/(cR))×n d³r′.