Electrodynamics §

The idea, step by step.

1. Flux is the “amount of field” through a surface. The magnetic flux $\Phi_B = \int \mathbf{B}\cdot d\mathbf{A}$ counts how much $\mathbf{B}$ threads a surface. If the field strength, the area, or the tilt changes, the flux changes.

2. A changing flux creates a circulating electric field. Faraday’s law says the line integral of $\mathbf{E}$ around any closed loop equals the negative rate of change of flux through that loop. Unlike the electrostatic field (which is conservative, $\nabla\times\mathbf{E}=0$), this induced $\mathbf{E}$ is non-conservative: it forms closed loops with no source charge. This is the key new object in electrodynamics.

3. Lenz’s law fixes the direction. The induced current’s own magnetic field opposes the change in flux. If flux into the page is increasing, the induced current circulates to make flux out of the page. This is energy conservation in disguise: you must do work pushing the magnet in, and that work becomes the electrical energy delivered.

4. Worked example: a magnet entering a coil. Push a bar magnet into a coil of $N$ turns. The flux through the coil rises, so an EMF $\mathcal{E} = -N\,d\Phi_B/dt$ appears and a current flows while the magnet moves. Pull it out and the current reverses. Hold it still and the current stops — only the change matters.

5. Authoritative source. Faraday’s law is the second of the four Maxwell equations in its modern form; see Maxwell (1865).

The idea, step by step.

1. The original Ampère law has a hole. The steady-current form $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ works for steady currents, where $\nabla\cdot\mathbf{J}=0$. Take the divergence of both sides: $\nabla\cdot(\nabla\times\mathbf{B})=0$ (always), so the law requires $\nabla\cdot\mathbf{J}=0$. But that fails for time-varying currents — charge piles up where current changes, so $\nabla\cdot\mathbf{J} = -\partial\rho/\partial t \neq 0$.

2. Charge conservation forces the fix. The continuity equation $\nabla\cdot\mathbf{J} = -\partial\rho/\partial t$ is charge conservation. Combine it with Gauss’s law $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ to get $\nabla\cdot\mathbf{J} + \varepsilon_0\,\partial(\nabla\cdot\mathbf{E})/\partial t = 0$. Adding $\varepsilon_0\,\partial\mathbf{E}/\partial t$ to the current makes the divergence of the right-hand side vanish, restoring consistency. That added term is the displacement current.

3. Concrete picture: a charging capacitor. Conduction current $I$ flows in the wire and stops at the plate — no charge crosses the gap. Yet the current must be continuous around the loop. The growing electric field between the plates carries a displacement current $I_d = \varepsilon_0\,d\Phi_E/dt = I$ that exactly matches the conduction current, so $\mathbf{B}$ is continuous whether you thread the loop through the wire or through the gap.

4. Why it matters beyond bookkeeping. The displacement term is not a bookkeeping trick: it is the term that, together with Faraday’s law, makes electromagnetic waves possible (Section 3). Without it, a changing field could not sustain itself in empty space.

5. Authoritative source. Maxwell introduced the displacement current in A Dynamical Theory of the Electromagnetic Field (1865).

The idea, step by step.

1. Each law feeds the other. Faraday’s law: a changing $\mathbf{B}$ makes a curling $\mathbf{E}$. The Maxwell–Ampère law: a changing $\mathbf{E}$ makes a curling $\mathbf{B}$. Neither field needs a charge or current to keep going once it is moving — each regenerates the other.

2. Chase the algebra. Take the curl of Faraday’s law: $\nabla\times(\nabla\times\mathbf{E}) = -\partial(\nabla\times\mathbf{B})/\partial t$. Use the vector identity $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}$; in a source-free region $\nabla\cdot\mathbf{E}=0$, so the left side is $-\nabla^2\mathbf{E}$. Substitute the Ampère law for $\nabla\times\mathbf{B}$ and the wave equation falls out.

3. The speed is a surprise. The wave speed $c = 1/\sqrt{\mu_0\varepsilon_0}$ depends only on the vacuum permittivity and permeability — the same constants that govern static electricity and magnetism. Maxwell recognized this equals the measured speed of light, unifying optics with electromagnetism.

4. Transverse, in phase, perpendicular. A plane wave traveling in direction $\hat{\mathbf{k}}$ has $\mathbf{E}$ and $\mathbf{B}$ perpendicular to each other and to the direction of travel, in phase, with $B = E/c$. The field energy flows in the direction $\mathbf{E}\times\mathbf{B}$ (Section 6).

5. Authoritative source. The wave prediction is the climax of Maxwell (1865); the propagating-wave treatment continues in Electromagnetic Radiation.

The idea, step by step.

1. Transformer: coupling two coils through shared flux. A primary coil driven by AC creates a time-varying flux in a common core; that flux links the secondary coil and induces an EMF there. The turns ratio sets the voltage ratio: $V_2/V_1 = N_2/N_1$. Power is (ideally) conserved, so current scales inversely: $I_2/I_1 = N_1/N_2$.

2. Generator: Faraday in reverse of a motor. Rotate a coil in a magnetic field and the flux through it varies sinusoidally, $\Phi_B = \Phi_0\cos(\omega t)$, giving $\mathcal{E} = \Phi_0\omega\sin(\omega t)$ — an AC voltage. This is how most of the world’s electricity is made.

3. Eddy currents: induction in bulk metal. A changing field induces circulating currents throughout a conductor, not just in a wire. These currents create their own opposing field (Lenz’s law), producing magnetic braking, damping, and losses.

4. Induction heating: eddy currents as a heater. A high-frequency field drives strong eddy currents in a metal workpiece; the resistance of the metal converts that current to heat. Because the field is applied without contact, it heats the metal directly and rapidly — used in induction cooktops, melting, and hardening.

5. Authoritative source. The transformer and generator relations follow directly from Faraday’s law in Maxwell (1865).

The idea, step by step.

1. Why lumping is an approximation. The full field solution has a phase that varies across the circuit by $\sim 2\pi L/\lambda$. When $L \ll \lambda$, that phase variation is negligible, so the voltage and current are the same at every point of a given element — the premise of lumped circuit theory. This is the same $L \ll \lambda$ criterion from the Scope section, and it is why a $50\,\text{MHz}$ circuit ($\lambda \approx 6\,\text{m}$) can be lumped on a $10\,\text{cm}$ board but a $3\,\text{GHz}$ circuit ($\lambda \approx 10\,\text{cm}$) cannot.

2. Inductor and capacitor are the two storage modes. An inductor’s voltage is proportional to the rate of change of current ($V = L\,dI/dt$): it resists changes in current and stores energy $U = \tfrac{1}{2}LI^2$ in its magnetic field. A capacitor’s current is proportional to the rate of change of voltage ($I = C\,dV/dt$): it resists changes in voltage and stores energy $U = \tfrac{1}{2}CV^2$ in its electric field. These are the circuit shadows of Faraday’s law and the displacement current.

3. RLC as an oscillator. In a series RLC circuit, energy sloshes between the inductor’s magnetic field and the capacitor’s electric field, exactly like a mass–spring (see Harmonic oscillator). The undamped natural frequency is $\omega_0 = 1/\sqrt{LC}$; the resistor damps the oscillation with time constant $\tau = 2L/R$.

4. Authoritative source. The lumped-element model and its $L \ll \lambda$ validity are standard; see Griffiths, Introduction to Electrodynamics, and any circuits text.

The idea, step by step.

1. Energy lives in the field, not the wire. The density $u$ has two equal-and-symmetric parts: electric ($\tfrac{1}{2}\varepsilon_0 E^2$), stored in capacitors, and magnetic ($B^2/(2\mu_0)$), stored in inductors. The field is the real carrier of energy.

2. The Poynting vector is the energy current. $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ has units of W/m² and points where the energy is going. In a plane wave it points along the direction of propagation, with magnitude $S = E^2/(\mu_0 c)$.

3. The surprising direction: energy flows through space. In a charging capacitor, the energy does not travel up the wire and into the plates. The Poynting vector points inward from the surrounding space into the gap between the plates — the field energy flows through the space around the wires and into the region where it is being stored. This is one of the most counterintuitive and important results in electrodynamics.

4. Authoritative source. Poynting’s theorem follows from the Maxwell equations; see Maxwell (1865) and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. No instantaneous action at a distance. In the static limit the potentials look instantaneous, but the full solution uses the source at the retarded time $t - R/c$: the field at $\mathbf{r}$ today depends on what the source did $R/c$ ago. This is causality built into Maxwell’s equations — information and energy never travel faster than $c$.

2. The retarded time is the light-travel time. A source $R$ meters away influences you only after $R/c$ seconds. For a source $1\,\text{m}$ away that delay is $\sim 3\,\text{ns}$; for the Sun ($1.5\times10^{11}\,\text{m}$) it is $\sim 500\,\text{s}$. You see the Sun as it was eight minutes ago.

3. Near field vs. far field from the same solution. Expanding the retarded potentials for a small source: the terms that fall off as $1/r^2$ and $1/r^3$ are the near (reactive) field — they store energy locally and belong to electrodynamics. The term that falls off as $1/r$ is the radiation (far) field — it carries energy away and is the subject of Electromagnetic Radiation. The boundary between them is around $r \sim \lambda/(2\pi)$.

4. Authoritative source. The retarded-potential solution of the wave equation is standard; see Jackson, Classical Electrodynamics, and Griffiths, Introduction to Electrodynamics.

Problem. A single loop of wire (radius $r = 10\,\text{cm}$) lies in a uniform magnetic field perpendicular to its plane. The field is switched on and rises linearly to $B = 0.5\,\text{T}$ over $0.1\,\text{s}$.

1. What is the magnitude of the induced EMF?
2. If the loop has resistance $R = 0.5\,\Omega$, what is the induced current, and which way does it flow (viewed from the side the field is increasing toward)?
3. Explain, using energy conservation, why the current flows that way.