Electromagnetics

Classical Electromagnetics

In classical electromagnetism, the Maxwell Equations sum up the laws concerning moving electric and magnetic charges. From them, the wave equation is derived as below (no arrows for convenience):

$\ \nabla \times \nabla \times E = -\nabla \times \frac{\partial B}{\partial t}$

$\ \nabla(\nabla \cdot E) - \nabla^2 E = -\mu_{0} \frac{\partial}{\partial t}\left(\frac{\partial D}{\partial t} + J \right).$

This is almost the wave equation. In a sourceless, simple" medium like vacuum, we have the more familiar,

$\ \nabla^2 E = \mu \epsilon \frac{\partial^2 E}{\partial t^2} = \frac{\mu_r \epsilon_r}{c^2} \frac{\partial^2 E}{\partial t^2}.$

The derivation for the magnetic field is analogous and results in,

$\ \nabla^2 H = \mu \epsilon \frac{\partial^2 H}{\partial t^2}.$

It is interesting that the signs in Maxwell's equations are just right to give us a positive $\frac{1}{c^2}$ factor for both equations. Combined with appropriate boundary conditions, they describe propagation in waveguides. This factor is the squared reciprocal of the velocity of propagation of the electric field wave and the magnetic field wave in the sourceless, simple medium.

Gain Media

For a sourceless media without current, but that might cause optical attenuation or gain, we define the susceptibility, $\ \chi(\vec{r},t)$, as,

$\ \vec{P}(t) = \epsilon_0 \int_{- \infty}^{t} \chi(t - \tau) \vec{E}(t) d\tau,$

$\ \hat{P}(\omega)= \hat{\chi}(\omega) \hat{E}(\omega).$

Still assuming a sourceless medium, we have the wave equation (similar derivation to above),

$\nabla^2 \hat{E} = \frac{-\omega^2}{c^2} (1+\hat{\chi})\hat{E}.$

If the susceptibility were written as $\ \hat{\chi} = \chi_R - \dot{\imath}\chi_I$, the real part is the gain/ attenuation of light through a medium and the imaginary part is the phase shift, which gives the refractive index, thus,

refractive index - $n = 1 + \frac{1}{2} \chi_R(\omega)$

gain - $\gamma = \frac{- \omega}{c} \chi_I(\omega).$

Telegrapher's Equations

On a transmission line, the unit cell is modelled by,

.

Kirchoff's junction and voltage laws allow us to arrive at the telegrapher's equations at the limit $\Delta z \to 0$:

$\ \frac{\partial I}{\partial z} = -GV - C \frac{\partial V}{\partial t}$

$\ \frac{\partial V}{\partial z} = -RI - L \frac{\partial I}{\partial t},$

where $\ z$,$\ t$ are the spatial and temporal coordinates respectivey; $\ V(z,t)$, $\ I(z,t)$, are the voltage and current along the transmission line respectively; $\ G$, $\ C$, $\ R$, $\ L$ are respectively, the conductance between the cables, capacitance between the cables, resistance along the line and inductance along the line, all per unit length, along the transmission line.

From the telegrapher's equations, we can find that,

$\ \frac{\partial^2 V}{\partial z^2} = -R \frac{\partial I}{\partial z} - L \frac{\partial^2 I }{\partial z \partial t}$

$\ \frac{\partial^2 V}{\partial z^2} = -L \frac{\partial}{\partial t}\left( -GV - C \frac{\partial V}{\partial t} \right) - R \left( GV - C\frac{\partial V}{\partial t} \right)$

$\ \frac{\partial^2 V}{\partial z^2} = RGV+ (GL+RC) \frac{\partial V}{\partial t}+ LC \frac{\partial^2 V}{\partial t^2}.$

In a lossless transmission line, both the resistance along the line and the conductance between the cables are zero, and we have the modelling equations,

$\ \frac{\partial^2 V}{\partial z^2} = LC \frac{\partial^2 V}{\partial t^2}$ and,

$\ \frac{\partial^2 I}{\partial z^2} = LC \frac{\partial^2 V}{\partial t^2},$

which are wave equations. The factor $\ LC$ is the squared reciprocal of the velocity of propagation of a signal down a transmission line.

Reference: ECE357 Electromagnetic Fields - notes from class, 2010. Instructor: Andrea Luttgen.

Remarks

It was assumed that the field functions and the current and voltage functions were continuous and so by Clairut's theorem, the order of derivatives were inter-changable.