Advanced Electrodynamics | Physics

The scalar potential of a localised charge distribution at large distances ($r \gg d$Where $d$ is the size of the distribution):

$\phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\left[\frac{Q}{r} + \frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2} + \frac{1}{2}\sum_{ij}Q_{ij}\frac{\hat{r}_i\hat{r}_j}{r^3} + \cdots\right]$

Monopole term: $Q = \int \rho\, dV$ (total charge).

Dipole term: $\mathbf{p} = \int \mathbf{r}"\,\rho(\mathbf{r}')\,dV'$ (electric dipole moment).

Quadrupole term: $Q_{ij} = \int (3r'_ir'_j - r'^2\delta_{ij})\,\rho(\mathbf{r}')\,dV'$ (traceless quadrupole tensor).

The quadrupole term is important for nuclei with spin $I \geq 1$ and for non-spherical charge distributions. The quadrupole moment $Q = Q_{zz}$ (in the principal axis frame) characterises the deviation from spherical symmetry.

The scalar and vector potentials are not unique. The gauge transformation:

$\mathbf{A}' = \mathbf{A} + \nabla\chi, \quad \phi' = \phi - \frac{\partial\chi}{\partial t}$

Leaves $\mathbf{E}$ and $\mathbf{B}$ unchanged for any scalar function $\chi(\mathbf{r}, t)$.

Common gauges:

In the Lorenz gauge, both $\mathbf{A}$ and $\phi$ satisfy wave equations with sources:

$\Box^2\mathbf{A} = -\mu_0\mathbf{J}, \quad \Box^2\phi = -\frac{\rho}{\varepsilon_0}$

Where $\Box^2 = \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}$ is the d’Alembertian.

The electromagnetic stress-energy tensor $T^{\mu\nu}$ encodes the energy density, momentum density, and stress:

$T^{00} = \frac{1}{2}\left(\varepsilon_0 E^2 + \frac{B^2}{\mu_0}\right) \quad \text{(energy density)}$

$T^{0i} = \frac{1}{c}(\mathbf{E} \times \mathbf{B})_i = \frac{S_i}{c} \quad \text{(momentum density)}$

$T^{ij} = -\varepsilon_0 E_i E_j - \frac{1}{\mu_0}B_i B_j + \frac{1}{2}\delta_{ij}\left(\varepsilon_0 E^2 + \frac{B^2}{\mu_0}\right) \quad \text{(Maxwell stress tensor)}$

Conservation law: $\partial_\mu T^{\mu\nu} = -f^\nu$ where $f^\nu$ is the Lorentz force density on charges.

Radiation pressure: For a normally incident plane wave with intensity $I$:

$P_{\text{rad} = \frac{I}{c} = \frac{1}{2}\varepsilon_0 E_0^2}$

For a perfect reflector, the radiation pressure is $2I/c$ (momentum transfer is doubled).

1. $v_p > c$ does not violate relativity: The phase velocity in a waveguide exceeds $c$ But no information or energy travels faster than $c$. The group velocity (signal velocity) is always $v_g < c$. Similarly, the refracted phase front in a prism can appear to move faster than $c$ But the actual signal does not.

2. Gauge choice matters for potentials, not fields: Different gauges give different $\mathbf{A}$ and $\phi$ for the same $\mathbf{E}$ and $\mathbf{B}$. In quantum mechanics, the Hamiltonian depends on the gauge, but all physical observables are gauge-invariant. The Aharonov—Bohm effect shows that even in regions where $\mathbf{E} = \mathbf{B} = 0$The vector potential $\mathbf{A}$ has measurable physical effects.

3. Multipole expansion convergence: The multipole expansion converges only outside a sphere that encloses all charges. Inside the charge distribution, the expansion diverges and must not be used. The expansion parameter is $d/r$ where $d$ is the source size and $r$ is the observation distance.

4. Radiation fields vs. Near fields: At distances $r \ll \lambda$ (near field), the fields are dominated by $1/r^2$ (induction) and $1/r^3$ (electrostatic/magnetostatic) terms. The radiation fields ($\propto 1/r$) dominate only in the far field ($r \gg \lambda$). Do not apply the Larmor formula or radiation resistance in the near field.

5. Poynting vector is not unique: The Poynting vector $\mathbf{S} = \mathbf{E} \times \mathbf{H}$ is gauge-dependent and can be nonzero even in static situations (e.g., a charged capacitor in a constant magnetic field). Only the surface integral $\oint \mathbf{S} \cdot d\mathbf{a}$ (total power flow) is physically meaningful.

• Electromagnetic Waves — The wave equation, Poynting vector, and energy conservation provide the foundation for the multipole expansion and radiation analysis.

• Special Relativity and Electromagnetism — The covariant formulation of electrodynamics extends gauge invariance and the stress-energy tensor to relativistic settings.

• Radiation from Accelerating Charges — The Larmor formula and radiation resistance connect the multipole expansion to practical antenna calculations.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

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    A[11_Advanced Electrodynamics] --> B[Key Concepts]
    A --> C[Core Principles]
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    C --> F[Design patterns]
    D --> G[Real-world usage]

Antenna theory describes how oscillating currents generate electromagnetic radiation. A half-wave dipole has a current distribution shaped like a half-sine wave along its length, producing a radiation pattern that is doughnut-shaped with maximum intensity broadside to the antenna. The radiation resistance quantifies how efficiently the antenna converts input power into radiated power. The key physical picture is that the accelerating charges in the antenna create time-varying electric and magnetic fields that detach from the antenna and propagate outward as a self-sustaining electromagnetic wave. The impedance matching between the antenna and the transmission line determines how much power is reflected versus radiated.