Electrodynamics in Matter & Wave Guides
Electrodynamics in Matter & Wave Guides
When electromagnetic waves propagate in matter, free charges and bound dipoles respond to the fields. Waveguides confine waves in two dimensions, permitting only discrete modes — technology at the heart of radar, microwave communications, and accelerator cavities.
Definition
In a linear dielectric: \(\mathbf{D} = \epsilon\mathbf{E}\), \(\mathbf{H} = \mathbf{B}/\mu\). The wave speed is \(v = c/n\) with refractive index \(n = \sqrt{\epsilon_r\mu_r}\).
Key Result
A rectangular waveguide (width \(a\)) supports TE\(_{mn}\) and TM\(_{mn}\) modes. The cutoff frequency for the dominant TE\(_{10}\) mode is \(f_c = c/(2a)\), below which the wave is evanescent.
Example 1
For TE\(_{10}\) in a waveguide of width \(a = 2\) cm: \(f_c = c/(2a) = 7.5\) GHz. Signals at 10 GHz propagate; signals at 5 GHz are cut off.
Example 2
The quality factor of a microwave cavity resonator: \(Q = 2\pi\times\)(energy stored)/(energy lost per cycle). High-Q cavities (\(Q>10^4\)) are used in particle accelerators.
Loading waveguide-modes...
Practice
- Derive the dispersion relation for TE modes in a rectangular waveguide.
- What is the physical meaning of a cutoff frequency?
- Compare TEM, TE, and TM modes in a waveguide.
- How does impedance matching prevent reflections in a transmission line?
Show Answer Key
1. For a rectangular waveguide ($a \times b$, $a>b$), TE modes: $k_z^2 = \omega^2\mu\epsilon - (m\pi/a)^2 - (n\pi/b)^2$. Propagation requires $k_z^2 > 0$: $\omega > \omega_{mn} = \pi\sqrt{(m/a)^2+(n/b)^2}/\sqrt{\mu\epsilon}$. Dispersion: $k_z = \sqrt{\omega^2\mu\epsilon - \omega_{mn}^2\mu\epsilon}$, so $v_p = \omega/k_z > c$ and $v_g = k_z/(\omega\mu\epsilon) < c$ with $v_p v_g = 1/(\mu\epsilon)$.
2. Below the cutoff frequency $f_c = \omega_{mn}/(2\pi)$, $k_z$ becomes imaginary — the mode is evanescent ($e^{-|k_z|z}$), decaying exponentially. No energy propagates. Only modes with $f > f_c$ carry power. The dominant mode (lowest $f_c$) is TE₁₀ with $f_c = c/(2a)$.
3. TEM: no cutoff, $\mathbf{E}$ and $\mathbf{B}$ both transverse (requires two conductors, e.g., coax). TE: $E_z=0$, $H_z\neq0$ (has cutoff). TM: $H_z=0$, $E_z\neq0$ (has cutoff). Hollow waveguides support only TE and TM modes (no TEM — would require $\nabla^2_t\Phi=0$ with single-boundary, forcing $\Phi=\text{const}$).
4. When a transmission line's characteristic impedance $Z_0$ doesn't match the load $Z_L$, a fraction $\Gamma = (Z_L-Z_0)/(Z_L+Z_0)$ of the wave reflects. $|\Gamma|=0$ when $Z_L=Z_0$ (matched): all power is absorbed, no standing waves. Matching networks (quarter-wave transformers, stubs) transform $Z_L$ to $Z_0$. VSWR $= (1+|\Gamma|)/(1-|\Gamma|) = 1$ at perfect match.