Formula sheet · PU-204

Electromagnetism II — formula sheet

Every boxed result from the 19 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.

D-160

Faraday's Law and the Flux Rule

∇ × E = −∂B/∂t ⇔ ∮_C E·dl = −∫_S ∂B/∂t ·dA ⇒ ℰ = −dΦ_B/dt

The local law says a magnetic field changing in time is inseparable from an electric field that curls around it; the electric field lines close on themselves rather than terminating on charges. The global flux rule then states that the EMF around any loop is minus the total rate of change of the magnetic flux threading it — whether that change comes from the field varying in time (transformer action) or from the loop moving and reshaping in a static field (motional action). The minus sign is Lenz's law: the induced EMF opposes the change that produced it, guaranteeing energy conservation.

D-161

The Displacement Current from Charge Conservation

∇×B = μ₀J + μ₀ε₀ ∂E/∂t, with displacement current density J_d = ε₀ ∂E/∂t

A time-varying electric field sources a magnetic field exactly as a real current does. The added term is the unique local correction that makes the total current density J + ε₀ ∂E/∂t divergence-free, so that the field equation never contradicts charge conservation. In a charging capacitor, the conduction current in the wire is handed off to the displacement current in the gap without interruption.

D-162

Assembly of Maxwell's Equations

∇·E = ρ/ε₀ ∇·B = 0 ∇×E = −∂B/∂t ∇×B = μ₀J + μ₀ε₀ ∂E/∂t

Two scalar (divergence) equations fix the sources of the fields: electric field lines begin and end on charge; magnetic field lines never end. Two vector (curl) equations govern the dynamics: a changing magnetic field circulates the electric field, and currents plus a changing electric field circulate the magnetic field. The divergence pair are constraints on initial data; the curl pair are evolution equations. Together with the Lorentz force F = q(E + v×B) they constitute all of classical electromagnetism.

D-163

Potentials and Gauge Freedom

B = ∇×A, E = −∇φ − ∂A/∂t

D-164

Electromagnetic Wave Equation in Vacuum

∇²E = μ₀ε₀ ∂²E/∂t² ∇²B = μ₀ε₀ ∂²B/∂t² c = 1/√(μ₀ε₀)

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.

D-165

Structure of Monochromatic Plane Waves

k·E = k·B = 0, B = (k̂×E)/c, |E| = c|B|, E₀ = a_xx̂ + a_ye^iδŷ

A vacuum plane wave is transverse: E and B both lie in the wavefront, perpendicular to the propagation direction k̂. They are mutually perpendicular, oscillate in step (same phase, peaking and vanishing together), and (E, B, k̂) forms a right-handed triad, so E×B points along k̂ — the direction of energy flow. Their magnitudes are locked at |E| = c|B|. All remaining freedom is the polarization, carried by the single complex phase δ and amplitude ratio.

D-166

Poynting's Theorem and Field Energy

\[ \frac{\partial u}{\partial t} + \nabla\cdot\vec{S} = -\,\vec{J}\cdot\vec{E}, \qquad u = \frac{\varepsilon_0 E^2}{2} + \frac{B^2}{2\mu_0}, \qquad \vec{S} = \frac{\vec{E}\times\vec{B}}{\mu_0} \]

At every point of space, the electromagnetic energy density \(u\) can change in exactly two ways: energy can flow away as the flux \( \vec{S} \) (energy per unit area per unit time, directed along \( \vec{E}\times\vec{B} \)), or it can be handed to matter at the rate \( \vec{J}\cdot\vec{E} \) per unit volume (Joule heating when positive, generator action when negative). With no charges present, \( \partial_t u + \nabla\cdot\vec{S} = 0 \) is a strict continuity equation: field energy is locally conserved and transported.

D-167

Maxwell Stress Tensor and Field Momentum

\[ \frac{d}{dt}\left(\mathbf{p}_{\text{mech}} + \mathbf{p}_{\text{field}}\right)_i = \oint_{\mathcal{S}} T_{ij}\,da_j, \qquad T_{ij} = \varepsilon_0\!\left(E_iE_j - \tfrac{1}{2}\delta_{ij}E^2\right) + \tfrac{1}{\mu_0}\!\left(B_iB_j - \tfrac{1}{2}\delta_{ij}B^2\right), \qquad \mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \frac{\mathbf{S}}{c^2} \]

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.

D-168

Retarded Potentials and Jefimenko's Equations

\[ \varphi(\mathbf{r},t) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\mathbf{r}',t_r)}{R}\, d^3r', \qquad \mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}',t_r)}{R}\, d^3r' \] \[ \mathbf{E}(\mathbf{r},t) = \frac{1}{4\pi\varepsilon_0} \int \left[ \frac{\rho(\mathbf{r}',t_r)}{R^2}\,\hat{\mathbf{R}} + \frac{\dot{\rho}(\mathbf{r}',t_r)}{cR}\,\hat{\mathbf{R}} - \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{c^2 R} \right] d^3r' \] \[ \mathbf{B}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \left[ \frac{\mathbf{J}(\mathbf{r}',t_r)}{R^2} + \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{cR} \right] \times \hat{\mathbf{R}}\; d^3r', \qquad t_r = t - \frac{R}{c} \]

The potentials are the static Coulomb and vector-potential integrals with one amendment: each source element speaks from its own past, delayed by the light-travel time \( R/c \). The fields (Jefimenko's equations) say that \( \mathbf{E} \) is built from the retarded charge density (Coulomb-like, \( 1/R^2 \)), the retarded rate of change of charge density (\( 1/R \)), and the retarded acceleration of currents (\( 1/R \)); \( \mathbf{B} \) is retarded Biot–Savart plus a \( \dot{\mathbf{J}} \) radiation term. Only the \( 1/R \) terms carry energy to infinity, since the Poynting flux \( \sim E B R^2 \) then survives as \( R \to \infty \). Everything is causal: no term references the present or future of the source.

D-169

Fields of a Moving Point Charge

\[ V = \frac{1}{4\pi\varepsilon_0}\frac{qc}{Rc - \mathbf{R}\cdot\mathbf{v}}, \quad \mathbf{A} = \frac{\mathbf{v}}{c^2}V, \qquad \mathbf{E} = \frac{q}{4\pi\varepsilon_0}\frac{R}{(\mathbf{R}\cdot\mathbf{u})^3}\left[(c^2\!-\!v^2)\,\mathbf{u} + \mathbf{R}\times(\mathbf{u}\times\mathbf{a})\right], \quad \mathbf{B} = \frac{1}{c}\hat{\mathbf{n}}\times\mathbf{E} \]

All source quantities \( \mathbf{R}, R, \hat{\mathbf{n}}, \mathbf{v}, \mathbf{a} \) are evaluated at the retarded time \( t_r \) fixed by \( R = c(t-t_r) \), and \( \mathbf{u} = c\hat{\mathbf{n}} - \mathbf{v} \). The field at any point is the sum of two physically distinct pieces. The velocity field \( \propto (c^2-v^2)\mathbf{u}/R^2 \) is the charge's deformed Coulomb field: it points along \( \mathbf{u} \), which is exactly the direction from the charge's linearly extrapolated present position, and it stays attached to the charge. The acceleration field \( \propto \mathbf{R}\times(\mathbf{u}\times\mathbf{a})/R \) exists only while the charge accelerates, is transverse (\( \perp \hat{\mathbf{n}} \) in the far zone), and, falling only as \( 1/R \), carries a Poynting flux \( \sim E^2R^2 \) that survives to infinity: this piece is electromagnetic radiation. The factor \( (\mathbf{R}\cdot\mathbf{u})^{-3} = (Rc\kappa)^{-3} \) with \( \kappa = 1 - \hat{\mathbf{n}}\cdot\mathbf{v}/c \) produces violent forward beaming as \( v \to c \).

D-170

Larmor Formula for Radiated Power

\[ P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3} = \frac{\mu_0 q^2 a^2}{6\pi c} \]

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.

D-171

Electric Dipole Radiation

\[ \vec E = -\frac{\mu_0 p_0 \omega^2}{4\pi}\frac{\sin\theta}{r}\cos\!\left[\omega\!\left(t-\tfrac{r}{c}\right)\right]\hat{\boldsymbol\theta}, \qquad \vec B = \frac{\hat{\mathbf r}\times\vec E}{c}, \] \[ \frac{d\langle P\rangle}{d\Omega} = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\sin^2\theta, \qquad \langle P\rangle = \frac{\mu_0 p_0^2 \omega^4}{12\pi c} = \frac{p_0^2 \omega^4}{12\pi \varepsilon_0 c^3} \]

An oscillating dipole launches an outgoing spherical wave whose electric field lies along \( \hat{\boldsymbol\theta} \) (in the plane containing the dipole axis and the line of sight), with magnitude falling as \( 1/r \) and modulated by \( \sin\theta \): nothing is radiated along the dipole axis, and emission peaks in the equatorial plane. The radiated power grows as the fourth power of the frequency and the square of the dipole amplitude, and is the same through every distant sphere — energy genuinely leaves the source and never comes back.

D-172

Reflection, Refraction and Fresnel Equations

\[ n_1 \sin\theta_i = n_2 \sin\theta_t, \qquad r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}, \quad r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}, \] \[ t_s = \frac{2 n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}, \qquad t_p = \frac{2 n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t} \]

Phase matching along the interface fixes where the reflected and refracted rays go (law of reflection and Snell's law); amplitude matching fixes how much field goes each way. The reflection coefficients are, in essence, normalized impedance mismatches: whenever the two media present the same effective impedance to the wave at that angle and polarization, \( r \) vanishes. For p-polarization this happens at a real angle — Brewster's angle \( \tan\theta_B = n_2/n_1 \) — while for s-polarization between ordinary dielectrics it never does. A negative \( r \) means the reflected field is phase-shifted by \( \pi \) (reflection off a denser medium, for s-polarization).

D-173

Waves in Conductors and Skin Depth

\[ \tilde{k}^2 = \mu\varepsilon\omega^2 + i\,\mu\sigma\omega, \qquad \vec{E} = E_0\, e^{-z/\delta} \cos\!\left(\frac{z}{\delta} - \omega t\right)\hat{\mathbf{x}}, \qquad \delta = \sqrt{\frac{2}{\mu\sigma\omega}} \;\;(\sigma \gg \varepsilon\omega) \]

Inside a conductor an electromagnetic wave is strangled as it propagates: its amplitude falls by a factor \( e \) every skin depth \( \delta \), and in a good conductor the wavelength inside the metal is only \( 2\pi\delta \) — the wave barely completes one oscillation before it is extinguished. Higher frequency, higher conductivity, or higher permeability all thin the skin, each under a square root. The magnetic field is enormously enhanced relative to \( E \) (by \( \sqrt{\sigma/\varepsilon\omega} \) compared with a vacuum wave) and lags it by \( 45^\circ \): energy in a conductor is stored overwhelmingly in the magnetic field.

D-174

Dispersion from the Lorentz Oscillator Model

\[ \varepsilon_r(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad \omega_p^2=\frac{Ne^2}{\varepsilon_0 m},\qquad \tilde{n}=n+i\kappa=\sqrt{\varepsilon_r},\qquad n-1\approx\frac{\omega_p^2}{2}\frac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2},\quad \kappa\approx\frac{\omega_p^2}{2}\frac{\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \]

Below resonance the electrons oscillate in phase with the field, their re-radiated waves retard the transmitted phase, and \(n>1\), growing as \(\omega\) approaches \(\omega_0\) — this is the normal dispersion that makes prisms work. Crossing the resonance the electron's phase flips by \(\pi\); within one linewidth of \(\omega_0\) the index plunges with frequency (anomalous dispersion) while absorption peaks, and above resonance \(n<1\): the electrons now respond as if free, opposing the applied field. Absorption is confined to a Lorentzian of width \(\gamma\); everywhere else the medium is transparent and normally dispersive.

D-175

The Electromagnetic Field Tensor

\[ F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu, \qquad \partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \]

All of electromagnetism is two tensor statements about one antisymmetric object. The first equation couples the field to its sources and contains Gauss's law (\( \nu = 0 \)) and Ampère–Maxwell (\( \nu = 1,2,3 \)). The second is an identity — automatically true once fields come from potentials — and contains \( \nabla\cdot\vec{B} = 0 \) and Faraday's law. Because every index is a spacetime index contracted correctly, both equations hold in identical form in every inertial frame: covariance is visible by inspection, and charge conservation \( \partial_\nu J^\nu = 0 \) is forced by antisymmetry.

D-176

Relativistic Transformation of E and B

\[ \begin{aligned} E'_x &= E_x & E'_y &= \gamma\left(E_y - vB_z\right) & E'_z &= \gamma\left(E_z + vB_y\right) \\ B'_x &= B_x & B'_y &= \gamma\left(B_y + \tfrac{v}{c^{2}}E_z\right) & B'_z &= \gamma\left(B_z - \tfrac{v}{c^{2}}E_y\right) \end{aligned} \]

Field components along the boost pass through unchanged; components perpendicular to the boost are enhanced by \( \gamma \) and mixed: the moving observer sees part of the old \( \vec{B} \) as new \( \vec{E} \) (the \( v\times B \) terms) and part of the old \( \vec{E} \) as new \( \vec{B} \) (the \( vE/c^{2} \) terms). Electric and magnetic fields are one tensor viewed from two frames — a purely electric field in the rest frame of a charge is a magnetic field to anyone watching the charge move.

D-177

Field Lagrangian and Gauge Invariance

\[ \partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0, \qquad \partial_\mu J^\mu = 0 \]

All four Maxwell equations emerge from one scalar Lagrangian: the sourced pair (Gauss, Ampère–Maxwell) as genuine equations of motion, the sourceless pair (no monopoles, Faraday) as an identity of the potential description. The third relation is not optional: demanding that the action be blind to the local redefinition \( A_\mu \to A_\mu + \partial_\mu \chi \) forces the current to be conserved, and equivalently the antisymmetry of \( F^{\mu\nu} \) makes the field equation inconsistent for any non-conserved source.

D-429

AC Circuits: Phasors, Impedance and Series RLC Resonance

\[ I_0 = \frac{V_0}{\sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}}, \quad \tan\phi = \frac{\omega L - \frac{1}{\omega C}}{R}, \quad \omega_0 = \frac{1}{\sqrt{LC}} \]

The current amplitude is maximized at the resonant frequency \(\omega_0\). The phase \(\phi\) indicates whether the circuit is inductive (\(\phi>0\), current lags voltage) or capacitive (\(\phi<0\), current leads voltage). At resonance, \(\phi=0\) and the circuit behaves as a pure resistor.

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