Electrodynamic Potentials, Gauge Freedom & Radiation

Definition of Potentials in Time-Dependent Electrodynamics

In static electromagnetism, $\nabla \times \mathbf{E} = 0$ allows us to express the electric field as the gradient of a scalar potential, $\mathbf{E} = -\nabla V$. In dynamic electrodynamics, however, Faraday's law states:

Because $\nabla \times \mathbf{E} \neq 0$, $\mathbf{E}$ cannot be written as the gradient of a scalar alone.

However, Gauss's law for magnetism remains strictly valid:

Since the divergence of any curl vanishes identically ($\nabla \cdot (\nabla \times \mathbf{A}) \equiv 0$), we can always define the magnetic vector potential $\mathbf{A}$:

Substitute this definition into Faraday's law:

Because the quantity in parentheses has zero curl, it can now be expressed as the negative gradient of a dynamic electric scalar potential $V$:

These two equations represent the fundamental relations expressing physical fields $(\mathbf{E}, \mathbf{B})$ in terms of electromagnetic potentials $(V, \mathbf{A})$:

Gauge Transformations

Potentials are not directly observable physical quantities; only fields $\mathbf{E}$ and $\mathbf{B}$ exert measurable Lorentz forces. If we modify $\mathbf{A}$ and $V$ through a transformation involving an arbitrary scalar gauge function $\lambda(\mathbf{r}, t)$:

Let us compute the new fields:

The physical fields are completely invariant under this gauge transformation. This mathematical freedom (gauge freedom) allows us to impose convenient constraints on the divergence of $\mathbf{A}$.

1. The Coulomb Gauge (Radiation Gauge)

Set the constraint:

Substituting into Gauss's law $\nabla \cdot \mathbf{E} = \rho / \epsilon_0$:

Since $\nabla \cdot \mathbf{A} = 0$:

This is the standard Poisson equation. The scalar potential in Coulomb gauge is:

While computationally convenient in quantum electrodynamics and atomic physics, $V$ in Coulomb gauge appears to propagate instantaneously, disguising relativistic causality.

2. The Lorentz Gauge Condition

To preserve manifest relativistic covariance and causality, Ludvig Lorenz (and later Hendrik Lorentz) introduced the condition:

Substitute $(V, \mathbf{A})$ into the inhomogeneous Maxwell equations (Gauss's law and Ampère-Maxwell law). Both coupled equations decouple completely into symmetric 4D inhomogeneous d'Alembertian wave equations:

where $\Box \equiv \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}$ is the d'Alembertian operator.

The Retardation Principle

Because electromagnetic signals propagate through space at the finite speed of light $c$, the potential at an observation point $\mathbf{r}$ at time $t$ cannot depend on what source charges and currents are doing at that exact moment. Instead, it depends on their behavior at an earlier time $t_r$ (the retarded time), accounting for the travel time of light:

where $\boldsymbol{\imath} = \mathbf{r} - \mathbf{r}'$ is the separation vector and $\imath = |\mathbf{r} - \mathbf{r}'|$.

Using the retarded Green's function of the d'Alembertian operator:

The exact solutions to the decoupled wave equations in Lorentz gauge are the Retarded Potentials:

These equations encapsulate relativistic causality: an event at the source point $\mathbf{r}'$ can only affect the field point $\mathbf{r}$ after the light-cone delay $\Delta t = \imath/c$ has elapsed.

Potentials of a Point Charge on an Arbitrary Trajectory

Consider a point charge $q$ moving along an arbitrary trajectory $\mathbf{w}(t)$ with velocity $\mathbf{v}(t) = \dot{\mathbf{w}}(t)$. Because the charge is moving, different parts of the charge distribution during an integration volume emit signals that arrive at observation point $\mathbf{r}$ at the same time $t$. This geometric elongation introduces a Doppler-like Jacobian volume factor:

where $\boldsymbol{\beta} = \frac{\mathbf{v}}{c}$ and $\hat{\boldsymbol{\imath}} = \frac{\mathbf{r} - \mathbf{w}(t_r)}{|\mathbf{r} - \mathbf{w}(t_r)|}$.

Carrying out the 4D delta-function integral yields the celebrated Liénard-Wiechert Potentials (Alfred-Marie Liénard 1898, Emil Wiechert 1900):

where all source quantities $(\mathbf{w}, \mathbf{v}, \boldsymbol{\beta}, \imath)$ are evaluated at the retarded time $t_r$ satisfying $c(t - t_r) = |\mathbf{r} - \mathbf{w}(t_r)|$.

Computing the Fields from Liénard-Wiechert Potentials

Differentiating the potentials $\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}$ and $\mathbf{B} = \nabla \times \mathbf{A}$ requires taking gradients and time derivatives through the implicit dependence on retarded time $\nabla t_r = -\frac{\hat{\boldsymbol{\imath}}}{c(1 - \hat{\boldsymbol{\imath}} \cdot \boldsymbol{\beta})}$.

The resulting electric field separates into two distinct physical terms:

where $\mathbf{u} \equiv c \hat{\boldsymbol{\imath}} - \mathbf{v}$ and $\mathbf{a} = \dot{\mathbf{v}}(t_r)$ is the charge acceleration evaluated at retarded time.

The magnetic induction field is strictly orthogonal and transverse:

Decomposition and Physical Significance

1. The Velocity Field (Generalized Coulomb Field):

Decays as $1/\imath^2$ with distance. It is carried along with the moving charge and represents bound field energy that cannot escape to infinity.

2. The Acceleration Field (Radiation Field):

Decays as $1/\imath$ with distance! The associated Poynting energy flux scales as:

When integrated over a giant sphere of radius $\imath \to \infty$, the surface area $4\pi \imath^2$ cancels the $1/\imath^2$ decay:

Fundamental Theorem of Classical Electrodynamics: A charge moving with uniform velocity ($a = 0$) does NOT radiate. Only an accelerated charge ($a \neq 0$) radiates electromagnetic energy to infinity.

The Oscillating Electric Dipole Model

Consider two tiny conducting spheres separated by a distance $d$ along the $z$-axis, connected by a thin filament carrying an alternating current. The electric dipole moment oscillates harmonically:

where $p_0 = q_0 d$.

We evaluate the fields under the standard radiation zone approximations:

1. Short dipole compared to wavelength: $d \ll \lambda$ (dipole approximation)
2. Far radiation field: $r \gg \lambda \gg d$ where $\lambda = 2\pi c/\omega$

Derivation of the Radiation Zone Fields

The retarded vector potential in spherical coordinates $(r, \theta, \phi)$ in the far zone is:

Converting $\hat{\mathbf{z}} = \cos\theta \hat{\mathbf{r}} - \sin\theta \hat{\boldsymbol{\theta}}$:

Computing $\mathbf{B} = \nabla \times \mathbf{A}$ keeping only terms scaling as $1/r$:

From $\mathbf{E} = c(\mathbf{B} \times \hat{\mathbf{r}})$:

Key Physical Characteristics

1. Transverse Wave: $\mathbf{E}$ points along $\hat{\boldsymbol{\theta}}$, $\mathbf{B}$ points along $\hat{\boldsymbol{\phi}}$, and propagation is radially outward along $\hat{\mathbf{r}}$.

2. Frequency Dependence: Field amplitudes are proportional to $\omega^2$ (or $1/\lambda^2$). High frequencies radiate vastly more efficiently than low frequencies.

3. Angular Profile: Fields vanish along the dipole axis ($\theta = 0, \pi$) and peak in the equatorial plane ($\theta = \pi/2$).

Instantaneous and Time-Averaged Poynting Vector

The Poynting vector in the radiation zone is:

Taking the time average over an oscillation cycle ($\langle \cos^2 \rangle = 1/2$):

The Toroidal Doughnut Radiation Pattern

The radiated intensity depends on angle as $\sin^2\theta$:

• Broadside ($\theta = 90^\circ$): Maximum radiation flux perpendicular to the dipole axis.
• Endfire ($\theta = 0^\circ, 180^\circ$): Zero radiation along the axis of the antenna wire.

Total Radiated Power: Larmor's Formula for Dipoles

Integrating $\langle \mathbf{S} \rangle$ over a closed sphere of radius $r$:

The standard integral is $\int_0^\pi \sin^3\theta \, d\theta = \frac{4}{3}$:

In terms of the speed of light in vacuum ($c = 1/\sqrt{\epsilon_0 \mu_0}$):

This is the celebrated Larmor Formula for an Oscillating Dipole. Notice the profound $\omega^4$ frequency dependence: doubling the frequency increases the radiated power by a factor of $2^4 = 16$.

Radiation Resistance of a Short Dipole Antenna

The current feeding the dipole is $I(t) = \dot{q}(t) = -q_0 \omega \sin(\omega t)$, with amplitude $I_0 = q_0 \omega$. Thus $p_0 = q_0 d = \frac{I_0 d}{\omega}$. Substituting into the total power formula:

Equating to the equivalent dissipated circuit power $P = \frac{1}{2} I_0^2 R_{\text{rad}}$:

For a short dipole where $d \ll \lambda$ (e.g., $d = 0.05\lambda$):

Because radiation resistance is tiny, short antennas match poorly to standard $50\,\Omega$ RF lines, radiating inefficiently.

Magnetic Dipole Radiation

An oscillating circular loop of radius $b$ carrying alternating current $I(t) = I_0 \cos(\omega t)$ constitutes an oscillating magnetic dipole $\mathbf{m}(t) = m_0 \cos(\omega t) \hat{\mathbf{z}}$, where $m_0 = \pi b^2 I_0$.

The radiation zone fields are:

The total radiated power is:

Comparison Between Electric and Magnetic Dipoles

For comparable geometric dimensions ($p_0 \sim q d, m_0 \sim I d^2 \sim q \omega d^2$):

Magnetic dipole radiation is suppressed by a factor of $(d/\lambda)^2$ relative to electric dipole radiation. Electric dipole transitions (E1) are vastly stronger than magnetic dipole (M1) atomic transitions.

The Center-Fed Half-Wave Antenna ($L = \lambda/2$)

Practical radio communications employ resonant antennas of length $L = \lambda/2$. The current distribution is a standing wave that vanishes at the endpoints $z = \pm L/2$:

where $k = \frac{2\pi}{\lambda} = \frac{\pi}{L}$.

Integrating across the antenna length yields the far-field radiation pattern:

Integrating over a sphere gives the total power radiated:

Solving for the radiation resistance of a resonant half-wave antenna:

Engineering Significance: A radiation resistance of $73\,\Omega$ provides an outstanding impedance match to standard coaxial transmission cables ($50\,\Omega$ or $75\,\Omega$), making half-wave dipoles the universal cornerstone of modern telecommunications.