For a conducting medium, $\mathbf{J} = \sigma \mathbf{E}$ (Ohm's Law), $\rho_v$ is often assumed to be zero for time-harmonic fields.

Maxwell's equations lead to modified wave equations: $$ \nabla^2 \mathbf{E} - \mu \sigma \frac{\partial \mathbf{E}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$

For time-harmonic fields ($e^{j\omega t}$ dependence), the wave equation for $\mathbf{E}$ becomes: $$ \nabla^2 \mathbf{E} + \omega^2 \mu \epsilon \left(1 - j \frac{\sigma}{\omega \epsilon}\right) \mathbf{E} = 0 $$ $$ \nabla^2 \mathbf{E} + \gamma^2 \mathbf{E} = 0 $$ where $\gamma = \sqrt{j\omega\mu(\sigma+j\omega\epsilon)}$ is the complex propagation constant.

$\gamma = \alpha + j\beta$, where $\alpha$ is the attenuation constant and $\beta$ is the phase constant.

The wave is attenuated as it propagates through the conductor. $$ \mathbf{E}(z,t) = E_0 e^{-\alpha z} \cos(\omega t - \beta z) \hat{a}_x $$

Skin Depth ($\delta$): The depth at which the field amplitude reduces to $1/e$ (approx. 37%) of its surface value. $$ \delta = \frac{1}{\alpha} $$

For good conductors ($\sigma \gg \omega \epsilon$): $$ \alpha \approx \sqrt{\frac{\omega \mu \sigma}{2}} = \beta $$ $$ \delta \approx \sqrt{\frac{2}{\omega \mu \sigma}} $$

High frequency currents in good conductors tend to flow near the surface (skin effect).

Electromagnetic Energy Density

The total energy stored in an electromagnetic field per unit volume. $$ w = w_e + w_m = \frac{1}{2} \mathbf{D} \cdot \mathbf{E} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} $$ For linear, isotropic media: $$ w = \frac{1}{2} \epsilon |\mathbf{E}|^2 + \frac{1}{2} \mu |\mathbf{H}|^2 $$
Units: Joules per cubic meter (J/m$^3$).

Poynting Vector and Poynting Theorem

Poynting Vector ($\mathbf{P}$): Represents the directional energy flux density of an electromagnetic field. $$ \mathbf{P} = \mathbf{E} \times \mathbf{H} $$
Units: Watts per square meter (W/m$^2$). Represents power flow per unit area.
For a plane wave, $\mathbf{P}$ points in the direction of wave propagation.
Poynting Theorem: Expresses the conservation of energy in electromagnetic fields. $$ \oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \int_V \left(\frac{1}{2}\mathbf{D} \cdot \mathbf{E} + \frac{1}{2}\mathbf{B} \cdot \mathbf{H}\right) dV - \int_V \mathbf{E} \cdot \mathbf{J} dV $$ $$ \oint_S \mathbf{P} \cdot d\mathbf{S} = -\frac{\partial W}{\partial t} - P_{diss} $$
The net power flowing out of a closed surface $S$ (LHS) is equal to the rate of decrease of energy stored in the EM fields within volume $V$ plus the power dissipated as heat (Joule heating, $P_{diss} = \int_V \mathbf{E} \cdot \mathbf{J} dV$).

Wave Propagation in Bounded System (Waveguide)

Waveguides are structures that guide EM waves, typically hollow metal tubes of specific cross-sections (rectangular, circular).
Unlike free space, waves in waveguides cannot propagate at arbitrary frequencies or modes.
Modes: Specific field configurations that can propagate.
- Transverse Electric (TE) Modes: $\mathbf{E}$ field is entirely transverse to the direction of propagation ($E_z = 0$).
- Transverse Magnetic (TM) Modes: $\mathbf{H}$ field is entirely transverse to the direction of propagation ($H_z = 0$).
- Transverse ElectroMagnetic (TEM) Modes: Both $\mathbf{E}$ and $\mathbf{H}$ are entirely transverse to the direction of propagation ($E_z = 0, H_z = 0$). TEM modes can only exist in waveguides with at least two conduc