On this page
Essential Electromagnetics Derivations
The goal here is not to reproduce a textbook. These are derivations worth remembering because each reveals a reusable physical structure.
1. Wave equation from Maxwell’s equations
In a homogeneous, source-free, linear medium,
\[\nabla\times\mathbf E=-\mu\frac{\partial\mathbf H}{\partial t}, \qquad \nabla\times\mathbf H=\epsilon\frac{\partial\mathbf E}{\partial t}.\]Take the curl of Faraday’s law:
\[\nabla\times(\nabla\times\mathbf E) =-\mu\frac{\partial}{\partial t}(\nabla\times\mathbf H).\]Use
\[\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E.\]With no free charge in a homogeneous medium, $\nabla\cdot\mathbf E=0$. Substitute Ampère–Maxwell:
\[-\nabla^2\mathbf E=-\mu\epsilon\frac{\partial^2\mathbf E}{\partial t^2}.\]Therefore
\[\boxed{\nabla^2\mathbf E-\mu\epsilon\frac{\partial^2\mathbf E}{\partial t^2}=0}\]with speed
\[\boxed{v=\frac1{\sqrt{\mu\epsilon}}}.\]2. Poynting theorem
Start with
\[\nabla\times\mathbf H=\mathbf J+\frac{\partial\mathbf D}{\partial t}, \qquad \nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.\]Use the vector identity
\[\nabla\cdot(\mathbf E\times\mathbf H) =\mathbf H\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf H).\]Substitution gives
\[\nabla\cdot(\mathbf E\times\mathbf H) =-\mathbf H\cdot\frac{\partial\mathbf B}{\partial t} -\mathbf E\cdot\mathbf J -\mathbf E\cdot\frac{\partial\mathbf D}{\partial t}.\]For linear nondispersive media identify field-energy density
\[u=\frac12\mathbf E\cdot\mathbf D+\frac12\mathbf B\cdot\mathbf H.\]Then
\[\boxed{ \nabla\cdot\mathbf S+ \frac{\partial u}{\partial t}+ \mathbf J\cdot\mathbf E=0, \qquad \mathbf S=\mathbf E\times\mathbf H. }\]This is local electromagnetic energy conservation.
3. Reflection coefficient on a transmission line
At the load,
\[V=V^++V^-, \qquad I=\frac{V^+}{Z_0}-\frac{V^-}{Z_0}.\]Enforce $Z_L=V/I$ and define $\Gamma=V^-/V^+$. Solving gives
\[\boxed{\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}}.\]Special cases follow instantly:
- $Z_L=Z_0\Rightarrow\Gamma=0$;
- open circuit $Z_L\rightarrow\infty\Rightarrow\Gamma\rightarrow+1$;
- short circuit $Z_L=0\Rightarrow\Gamma=-1$.
4. Skin depth
In a good conductor, $\sigma\gg\omega\epsilon$. The propagation constant becomes approximately
\[\gamma\approx(1+j)\sqrt{\frac{\omega\mu\sigma}{2}}.\]Thus attenuation constant
\[\alpha=\sqrt{\frac{\omega\mu\sigma}{2}}.\]Define skin depth as the distance for amplitude to fall by $1/e$:
\[\boxed{\delta=\frac1\alpha=\sqrt{\frac{2}{\omega\mu\sigma}}}.\]The key result is $\delta\propto f^{-1/2}$.
5. Friis scaling from power density and effective aperture
A transmitter with gain $G_t$ produces far-field power density
\[S=\frac{P_tG_t}{4\pi R^2}.\]The receiving antenna captures
\[P_r=SA_e.\]Using
\[A_e=\frac{G_r\lambda^2}{4\pi}\]gives
\[\boxed{P_r=P_tG_tG_r\left(\frac{\lambda}{4\pi R}\right)^2}.\]6. Radar range resolution
For a signal bandwidth $B$, the characteristic compressed pulse/time resolution is roughly
\[\Delta t\sim\frac1B.\]Radar round-trip delay is
\[\Delta t=\frac{2\Delta R}{c}.\]Therefore
\[\boxed{\Delta R\approx\frac{c}{2B}}.\]This makes clear why range resolution is fundamentally a bandwidth problem.
7. Larmor precession
A magnetic moment experiences torque
\[\boldsymbol\tau=\boldsymbol\mu\times\mathbf B.\]For angular momentum $\mathbf F$ with $\boldsymbol\mu=\gamma\mathbf F$,
\[\frac{d\mathbf F}{dt}=\gamma\mathbf F\times\mathbf B.\]The derivative is perpendicular to $\mathbf F$, so the magnitude stays approximately constant while the vector precesses. The angular frequency is
\[\boxed{\omega_L=|\gamma|B}.\]8. Rabi frequency from the dipole Hamiltonian
For a classical oscillating electric field
\[\mathbf E(t)=\mathbf E_0\cos\omega t\]and electric-dipole interaction
\[H_I=-\mathbf d\cdot\mathbf E(t),\]| the matrix element between $ | g\rangle$ and $ | e\rangle$ is proportional to |
Under the usual resonant rotating-wave convention,
\[\boxed{\Omega=\frac{\mathbf d_{eg}\cdot\mathbf E_0}{\hbar}}\]up to amplitude/convention definitions. Polarization enters through the projected vector matrix element.
9. AC Stark shift from dressed-state expansion
For a two-level system in the rotating frame, a common Hamiltonian is
\[H=\frac{\hbar}{2} \begin{pmatrix} 0 & \Omega\\ \Omega & -2\Delta \end{pmatrix}.\]The eigenvalue separation involves
\[\sqrt{\Delta^2+\Omega^2}.\]| For $ | \Omega/\Delta | \ll1$, |
The leading correction therefore scales as
\[\boxed{\delta\omega_{AC}\sim\frac{|\Omega|^2}{4\Delta}}\]with sign and exact factor following the state/detuning convention.
10. Shockley–Ramo current from weighting potential
Define weighting field
\[\mathbf E_w=-\nabla\phi_w.\]For charge $q$ moving with velocity $\mathbf v$,
\[\frac{d\phi_w}{dt}=\nabla\phi_w\cdot\mathbf v=-\mathbf E_w\cdot\mathbf v.\]Thus, subject to electrode-current sign convention,
\[\boxed{i=q\mathbf v\cdot\mathbf E_w=-q\frac{d\phi_w}{dt}}.\]This derivation makes the central point explicit: signal depends on motion through weighting potential, not only on charge collection at the electrode.
Derivation practice
For each derivation, try to identify:
- the conservation law or symmetry used;
- the approximation that simplifies it;
- the dimensionless parameter controlling that approximation;
- the experimentally measurable quantity at the end.
That is more valuable than memorizing the algebra alone.