3.5 Electromagnetics

Maxwell's equations at a conceptual level

The four Maxwell's equations unify electricity and magnetism:

• Gauss's law (electric): electric flux out of a closed surface equals enclosed charge / epsilon. Charges are the sources of E fields.
• Gauss's law (magnetic): net magnetic flux through any closed surface is zero (no magnetic monopoles).
• Faraday's law: a time-changing magnetic field induces an electromotive force (EMF): EMF = -d(flux)/dt. This underlies generators and transformers.
• Ampere's law (with Maxwell's correction): current and a changing electric field both produce a magnetic field.

Together they predict self-propagating electromagnetic waves, where changing E and B fields regenerate each other.

Fields, forces, capacitance, and inductance

The electric field from a point charge is E = Q/(4piepsilonr^2); the force on a charge is F = qE. The magnetic force on a moving charge is F = qv x B (the Lorentz force).

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 or relative permittivity raises capacitance; increasing turns (N appears squared) sharply raises inductance.

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, and 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. This constant appears in antenna, radiation, and wave-impedance problems and is worth memorizing exactly.

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.

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. This is the design goal.
• Open circuit (Z_L = infinity): Gamma = +1, full reflection.
• Short circuit (Z_L = 0): Gamma = -1, full reflection with inversion.

The standing wave ratio is SWR = (1 + |Gamma|)/(1 - |Gamma|); SWR = 1 means a perfectly matched line. A signal travels one wavelength when the line length equals lambda, so quarter-wave (lambda/4) sections are used as impedance transformers.