Electromagnetic Fields and Waves
What this domain tests
This domain tests Maxwell's equations, boundary conditions, plane-wave propagation, transmission-line behaviour, and loss mechanisms such as skin effect.
Core rules
- Maxwell equations in time form: curl E = -dB/dt, curl H = J + dD/dt, div D = rho, div B = 0. In a source-free region, fields propagate as waves.
- Constitutive relations: D = epsilon E, B = mu H, J = sigma E. Intrinsic impedance eta = sqrt(mu/epsilon); in free space eta is about 377 ohm.
- Wave velocity v = 1/sqrt(mu epsilon) = c/sqrt(epsilon_r mu_r); wavelength lambda = v/f.
- Boundary conditions: tangential E and tangential H are continuous without surface current or charge; normal B and normal D are continuous without surface sources.
- Skin depth delta = sqrt(2/(omega mu sigma)); current crowds toward the surface as frequency rises.
- Transmission line: reflection coefficient Gamma = (Z_L - Z_0)/(Z_L + Z_0); VSWR = (1+|Gamma|)/(1-|Gamma|).
Worked example
A 100 MHz wave travels in a dielectric with epsilon_r = 4 and mu_r = 1. Find the wavelength. v = c/sqrt(4) = 1.5 x 10^8 m/s. lambda = v/f = 1.5 x 10^8 / 100 x 10^6 = 1.5 m. In free space the same frequency would give 3 m, so the dielectric halves the wavelength.
Traps
- Mixing angular frequency omega with ordinary frequency f; omega = 2 pi f.
- Reversing tangential and normal boundary conditions.
- Using free-space eta when the medium is a dielectric or conductor.
- Forgetting that a matched line has Gamma = 0 and VSWR = 1, not zero.