3.5 Electromagnetics

Maxwell's equations at a conceptual level

The four Maxwell's equations unify electricity and magnetism, and the FE tests them conceptually rather than by surface/line-integral derivation:

• Gauss's law (electric): the electric flux out of a closed surface equals the enclosed charge divided by epsilon. Charges are the sources of E fields.
• Gauss's law (magnetic): the net magnetic flux through any closed surface is zero (no magnetic monopoles); B field lines always close on themselves.
• Faraday's law: a time-changing magnetic field induces an electromotive force (EMF): EMF = -d(flux)/dt. This underlies generators, transformers, and inductors.
• Ampere's law (with Maxwell's displacement-current correction): a current and/or a changing electric field produces a magnetic field.

Together they predict self-propagating electromagnetic waves, in which changing E and B fields continuously regenerate each other and travel outward at the speed of light. In an electrostatics-only problem (no time variation), only the two Gauss-type relations matter; in magnetostatics, Ampere's law I = integral of H around a loop and the Biot-Savart law govern the B field of steady currents.

The key conceptual takeaways the FE tests: only the electric Gauss law has a non-zero right-hand side (charge), because there are no magnetic monopoles; a changing magnetic flux is what induces voltage (Faraday), so a static field induces nothing; and Maxwell's displacement-current term is what lets fields propagate through a vacuum where no conduction current exists.

Fields, forces, capacitance, and inductance

The electric field from a point charge is E = Q/(4piepsilonr^2); the force on a charge in a field is F = qE (Coulomb's law gives F = Q_1 Q_2/(4piepsilonr^2) between two charges). The magnetic force on a moving charge is the Lorentz force F = q*(v x B), perpendicular to both velocity and field.

Energy-storage elements derive from field geometry:

• Parallel-plate capacitor: C = epsilonA/d, where epsilon = epsilon_repsilon_0 and epsilon_0 = 8.854x10^-12 F/m. Stored energy = (1/2)CV^2.
• Solenoid/coil inductance: L = muN^2A/length, where mu = mu_rmu_0 and mu_0 = 4pi*10^-7 H/m. Stored energy = (1/2)LI^2.

Increasing plate area A or relative permittivity epsilon_r raises capacitance; decreasing plate spacing d also raises it. Because N appears squared, doubling a coil's turns quadruples its inductance — a frequently tested sensitivity. Inserting a high-permeability magnetic core (large mu_r) likewise multiplies inductance, just as a high-permittivity dielectric multiplies capacitance.

For electric and magnetic energy density in a field region, the FE Handbook gives w_E = (1/2)epsilonE^2 and w_B = (1/2)*B^2/mu (joules per cubic meter); integrating these over the field volume recovers the (1/2)CV^2 and (1/2)LI^2 lumped-element results, tying the field picture to the circuit picture.

Plane waves and the 377-ohm free-space impedance

In free space, electromagnetic waves propagate at the speed of light:

c = 1/sqrt(mu_0*epsilon_0) = 3x10^8 m/s, with wavelength lambda = c/f.

The ratio of the electric- to magnetic-field magnitude of a plane wave is the intrinsic (characteristic) impedance of the medium: eta = sqrt(mu/epsilon). For free space this evaluates to eta_0 = sqrt(mu_0/epsilon_0) = 120*pi, approximately 377 ohms — worth memorizing exactly because it appears throughout antenna, radiation, and wave-impedance problems.

Worked example: a 1 GHz wave in free space has lambda = c/f = (3x10^8)/(1x10^9) = 0.3 m. A plane wave with E = 7.54 V/m has H = E/eta_0 = 7.54/377 = 0.02 A/m.

Transmission-line basics

At high frequency, a line's distributed inductance and capacitance set its characteristic impedance Z_0 = sqrt(L/C) for a lossless line — a real number in ohms (e.g., 50 ohm coax). When a line of impedance Z_0 is terminated in a load Z_L, the reflection coefficient at the load is:

Gamma = (Z_L - Z_0)/(Z_L + Z_0).

Key cases:

• Matched (Z_L = Z_0): Gamma = 0, no reflection, all power delivered — the design goal.
• Open circuit (Z_L = infinity): Gamma = +1, full reflection.
• Short circuit (Z_L = 0): Gamma = -1, full reflection with phase inversion.

The standing wave ratio is SWR = (1 + |Gamma|)/(1 - |Gamma|); SWR = 1 means a perfectly matched line. A signal advances one wavelength when the line length equals lambda, so quarter-wave (lambda/4) sections act as impedance transformers, with input impedance Z_in = Z_0^2 / Z_L. Worked example: a 75 ohm line feeding a 300 ohm load has Gamma = (300-75)/(300+75) = 225/375 = 0.6, so SWR = (1+0.6)/(1-0.6) = 1.6/0.4 = 4. The magnitude of Gamma ranges from 0 (matched) to 1 (total reflection); SWR ranges from 1 (matched) to infinity (open/short).

A quarter-wave transformer matches a real load R_L to a line Z_0 by choosing an intermediate section of impedance Z_0' = sqrt(Z_0 * R_L). For the 75-to-300 ohm case, a lambda/4 section of sqrt(75*300) = sqrt(22,500) = 150 ohm would give a perfect match. Transmission-line effects matter when the line length is a non-trivial fraction of a wavelength (roughly > lambda/10); at low frequency or short lengths a wire is just a node and reflections are negligible. The signal's propagation velocity on the line is v = 1/sqrt(L'C') where L' and C' are per-unit-length values, always at or below c.