Classical Electrodynamics — formula sheet
Every boxed result from the 12 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Laplacian of 1/r as a Delta Function
The Newtonian kernel \( 1/r \) is harmonic everywhere except at the origin, where it hides a point source of strength \( -4\pi \). Applied to \( -\frac{1}{4\pi r} \), the Laplacian returns exactly \( \delta^3(\mathbf{r}) \), which is why \( G(\mathbf{r}) = -\frac{1}{4\pi r} \) is the Green's function of \( \nabla^2 \). Physically, the flux of the Coulomb field through any closed surface enclosing the charge is the same — Gauss's law in miniature.
Electrostatic Potential and Irrotational E
A static electric field is irrotational, so it is the gradient of a single scalar potential. Feeding that potential into Gauss's law yields Poisson's equation, a single linear second-order PDE for \( \phi \). Given \( \phi \) or its normal derivative on the boundary, the solution is unique: electrostatics is completely determined by charge in the volume plus data on the surface.
Multipole Expansion of the Potential
Far from any bounded static source the potential is an ordered sum: a Coulomb tail set by the net charge \(Q\); if \(Q=0\), a dipole field set by \(\vec p=\int\vec r'\rho\,d^3r'\); if \(\vec p=0\) too, a quadrupole field set by the tensor \(Q_{ij}\); and so on. The leading non-vanishing moment controls the far field and, uniquely, is independent of where the origin is placed.
Ampère's Circuital Law from Biot–Savart
The magnetostatic field circulates around current: its curl at a point is set entirely by the local current density there, and the net circulation around any loop counts only the current piercing that loop, weighted by \(\mu_0\). Regions with no current carry a curl-free (but not necessarily zero) field.
Vector Potential and Bound Currents
A magnetized body produces exactly the field of a current \(\nabla\times\mathbf{M}\) threading its interior plus a sheet current \(\mathbf{M}\times\hat{\mathbf n}\) wrapping its surface — no more, no less. The field \(\mathbf{H}\) is engineered so that only the currents you drive through wires appear as its source; the material's own response is hidden inside \(\mathbf{H}\)'s definition through \(\mathbf{M}\).
Faraday's Law and the Induced Field
The circulation density of the electric field at a point equals minus the rate at which the magnetic field there is changing. A magnetic field growing out of the page ( \(\partial\vec{B}/\partial t\) out of the page) drives \(\vec{E}\) to circulate clockwise, so the induced electric field opposes the change that created it, this is Lenz's law written locally. The field is genuinely non-conservative: \(\oint\vec{E}\cdot d\vec{\ell}\neq 0\).
Maxwell Stress Tensor and Field Momentum
Momentum is conserved locally in electrodynamics, but only if the field is credited with momentum of density \(\mathbf{g} = \mathbf{S}/c^2\). The rate of change of (mechanical + field) momentum inside any closed surface equals the stress-tensor flux through that surface: \(T_{ij}\,da_j\) is the \(i\)-th component of force transmitted across the area element \(d\mathbf{a}\). Equivalently, \(-T_{ij}\) is the flux density of the \(i\)-component of momentum in the \(j\)-direction. The diagonal elements are pressures/tensions: along a field line the field pulls with tension \(\tfrac{1}{2}\varepsilon_0E^2 + \tfrac{1}{2}B^2/\mu_0\) per unit area; transverse to the lines it pushes with the same magnitude of pressure.
Electromagnetic Wave Equation in Vacuum
Each Cartesian component of E and of B obeys the same three-dimensional wave equation. A disturbance in the fields propagates without a medium at the fixed speed c, determined solely by the two vacuum constants. The two fields are not independent: they are locked together through the original first-order equations (which force E ⊥ B ⊥ propagation direction and E = cB in magnitude), but each separately satisfies the second-order wave equation.
Retarded Potentials in Lorenz Gauge
The potential here-and-now is the Coulomb/Biot–Savart-like sum over all source elements, but each element speaks to us from the moment its light-cone reaches us. Information about a source point a distance \(R\) away is always old by \(R/c\). The functional form is identical to the static case except that "now" is replaced by "then." When the sources change slowly (or not at all), \(t_r\to t\) and the static expressions are recovered.
Larmor Formula for Radiated Power
An accelerating charge sheds energy as light at a rate fixed by the square of its acceleration. Doubling the acceleration quadruples the radiated power; the sign of the charge and the direction of the acceleration are irrelevant to the total. The emission is beamed sideways — a \( \sin^2\theta \) doughnut with its dead axis along \( \vec{a} \), maximal in the plane perpendicular to the acceleration.
Covariant Formulation of Electrodynamics
The first equation is Gauss's law (\(\nu=0\)) and the Ampère–Maxwell law (\(\nu=1,2,3\)) fused into one four-component tensor equation; the source is the four-current. The second is the homogeneous pair (\(\nabla\cdot\vec{B}=0\) and Faraday's law), which holds automatically because the field is derived from a potential. Every symbol carries spacetime indices, so both equations keep their form under any Lorentz transformation — covariance is manifest, not something to be verified frame by frame. \(\vec{E}\) and \(\vec{B}\) are not independent fields but components of the single object \(F^{\mu\nu}\).
Transformation of E and B under Boosts
Components of the field along the boost survive untouched; components perpendicular to the boost are amplified by \(\gamma\) and, crucially, mixed — the perpendicular electric field in \(S'\) picks up a \(\vec v\times\vec B\) contribution and the perpendicular magnetic field picks up a \(-\vec v\times\vec E/c^2\) contribution. There is no frame-independent way to say "this is an electric field": \(\vec E\) and \(\vec B\) are the time-space and space-space parts of the single antisymmetric tensor \(F^{\mu\nu}\), and a boost simply re-slices that tensor. The combinations \(\vec E\cdot\vec B\) and \(B^2-E^2/c^2\) are the two quantities the slicing leaves fixed.