= Solution
\b[<Displacement current> makes <Ampère's law> compatible with <charge conservation>.] The macroscopic <Maxwell equations> are
$$
\nabla\cdot\mathbf D=\rho_f,\qquad\nabla\cdot\mathbf B=0,\qquad\nabla\times\mathbf E=-\partial_t\mathbf B,\qquad\boxed{\nabla\times\mathbf H=\mathbf J_f+\partial_t\mathbf D.}
$$
Taking the divergence of the last equation and using <Gauss's law> gives $\nabla\cdot\mathbf J_f+\partial_t\rho_f=0$. Omitting $\partial_t\mathbf D$ would wrongly require the conduction current to be divergence free during charging. The displacement-current density $\partial_t\mathbf D$ has current-density units, but need not represent free charges crossing a region.
The <surface independence of Ampère circulation during capacitor charging> illustrates the need directly. A surface spanning the circuit can cut the wire or pass between the plates. For ideal parallel plates, $Q=\epsilon_0AE$ and
$$
I=\dot Q=\epsilon_0 A\dot E=\int_{\rm gap}\partial_t\mathbf D\cdot d\mathbf A.
$$
The corrected circulation of $\mathbf H$ is independent of that surface choice. Free conduction current stops at a plate; <displacement current> supplies the gap contribution.
\b[In vacuum the two time-dependent curl equations support waves.] Taking a curl of Faraday's law in a source-free region yields
$$
\nabla^2\mathbf E-\mu_0\epsilon_0\partial_t^2\mathbf E=0,\qquad c=(\mu_0\epsilon_0)^{-1/2}.
$$
The corresponding magnetic equation is identical. For a <plane wave>, $\mathbf k\cdot\mathbf E=\mathbf k\cdot\mathbf B=0$ and $\mathbf B=\mathbf k\times\mathbf E/\omega$. The fields carry energy through the Poynting flux. Dropping <displacement current> everywhere would remove these freely propagating solutions.<Electromagnetism by David Tong>[Sections 4.2–4.3: Maxwell's equations, charge conservation and electromagnetic waves]{ref}
\b[In a metal, conduction and displacement compete.] For a uniform ohmic material with $\mathbf J_f=\sigma\mathbf E$, constant $\epsilon,\mu$, and transverse waves,
$$
\nabla^2\mathbf E=\mu\epsilon\partial_t^2\mathbf E+\mu\sigma\partial_t\mathbf E.
$$
Use $e^{i\mathbf k\cdot\mathbf r-i\omega t}$. The <dispersion relation> and <conduction-to-displacement current ratio> are
$$
\boxed{k^2=\mu\epsilon\omega^2+i\mu\sigma\omega,\qquad\frac{|J_{\rm cond}|}{|J_{\rm disp}|}=\frac\sigma{\omega\epsilon}.}
$$
When $\sigma\gg\omega\epsilon$, $k\simeq(1+i)/\delta$ with $\delta=\sqrt{2/(\mu\sigma\omega)}$: fields are screened within a <skin depth> and energy is dissipated by <Joule heating>. This quasistatic limit neglects <displacement current> locally compared with conduction; it does not invalidate the full Maxwell law. At sufficiently high frequency, conductivity and permittivity are frequency dependent, so a constant-$\sigma$ approximation must be replaced by a material model.
\b[In an unmagnetized cold plasma, electron inertia gives a cutoff.] With immobile ions and negligible collisions, $m_e\dot{\mathbf v}=-e\mathbf E$ and $\mathbf J=-n_e e\mathbf v$ give $\sigma(\omega)=in_ee^2/(m_e\omega)$. For the <transverse wave cutoff in a cold unmagnetized plasma>, combining this carrier current with vacuum <displacement current> gives
$$
\boxed{\epsilon_{\rm eff}=\epsilon_0\left(1-\frac{\omega_p^2}{\omega^2}\right),\quad\omega_p^2=\frac{n_ee^2}{m_e\epsilon_0},\quad\omega^2=\omega_p^2+c^2k^2.}
$$
Transverse waves propagate above $\omega_p$ and are evanescent below it. The <group velocity> is $c\sqrt{1-\omega_p^2/\omega^2}<c$ even though the <phase velocity> exceeds $c$. The cold model also has longitudinal charge-density oscillations at $\omega_p$, distinct from transverse light waves. Collisions add <damping>, and <pressure> or magnetic fields modify these simple dispersions. To avoid <carrier-current double counting in effective permittivity>, once the carrier response is included in $\epsilon_{\rm eff}$ it must not be counted a second time as a separate conductivity. <Displacement current> therefore connects circuit continuity, vacuum radiation, conductor screening and plasma propagation within one consistent field theory.<Electromagnetism by David Tong>[Sections 7.6.2–7.6.3: conductors, plasma frequency and longitudinal plasma oscillations]{ref}
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2015/paper-2-plasma-and-skin.png]
{title=Electron inertia gives transverse plasma waves a cutoff, while a good conductor attenuates fields over a skin depth}
{source=paper-2-plasma-and-skin.py}
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