Past exam of the physics B course of the University of Cambridge / 2010 / Paper 1 / Section B / B6

displacement current completes Ampère's law:In vacuum $\mathbf{D}={ϵ}_0\mathbf{E}$. Taking the divergence and using $\nabla \cdot \mathbf{D}={\rho }_{\mathrm{free}}$ gives $\nabla \cdot {\mathbf{J}}_{\mathrm{free}}+{\partial }_t{\rho }_{\mathrm{free}}=0$, exactly the continuity equation. Without the added term a time-dependent charge distribution would contradict the curl equation.

A charging parallel-plate capacitor illustrates the physics. For plate area $A$, $Q={ϵ}_{0}AE$, so the gap displacement current is $I_d={ϵ}_{0}A\dot{E}=\dot{Q}$, equal to the conduction current in its leads. An Ampère contour therefore gives the same magnetic circulation whether its spanning surface crosses the wire or the gap. No electrons have to cross an ideal vacuum gap. In matter, ${\partial }_t\mathbf{D}$ includes the polarization response as well as the vacuum term. Combined with Faraday induction, the displacement term permits source-free electromagnetic waves.

A thin pillbox applied to $\nabla \cdot \mathbf{B}=0$ and a thin rectangular contour applied to the magnetostatic Ampère law giveThe normal points from medium1 to medium2, and ${\mathbf{K}}_{\mathrm{free}}$ is a free sheet current. Without such a current, tangential $H$ is continuous; normal $B$ is continuous independently. The remaining components need not be continuous because $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})$ contains the magnetization.

For linear isotropic media with no free sheet current, $B_{1n}=B_{2n}$ and $B_{1t}/{\mu }_1=B_{2t}/{\mu }_2$. If ${\theta }_i$ measures a field-line angle to the normal, $\tan {\theta }_2/\tan {\theta }_1={\mu }_2/{\mu }_1$. This magnetic field-line refraction follows from the two different component conditions, not from continuity of the complete vector field. Bound surface currents are already represented in the constitutive relation and must not be added again as free currents.

In paramagnetism, a material develops magnetization along a weak applied field, with positive susceptibility; in diamagnetism, it develops an induced moment opposing it, with negative susceptibility. In a simple localized-moment paramagnet, Zeeman alignment competes with thermal disorder. For independent two-state moments $\pm \mu$,when the field is weak and demagnetizing corrections are negligible. Stronger fields saturate the moment; interactions and conduction-electron Pauli paramagnetism can depart from this Curie law.

Ordinary diamagnetism comes from a field-induced change of orbital motion, including in closed-shell matter without permanent atomic moments. Its weak susceptibility is often nearly temperature independent away from a transition, and it contributes a background even when another magnetic response dominates. A nonuniform field tends to attract a paramagnetic specimen toward stronger field and repel a diamagnetic specimen in the weak linear regime. Calibrate the holder contribution and field gradient and measure a temperature/field series to distinguish induced response from retained magnetization. Ferromagnetic hysteresis and superconducting flux expulsion require different collective explanations.

Since $\nabla \cdot \mathbf{B}=0$, a magnetic field can locally be represented by a magnetic vector potential,It is not unique: $\mathbf{A}\to \mathbf{A}+\nabla \chi$ leaves its curl unchanged. In magnetostatics, choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Ampère's law then gives ${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$ in vacuum. For a localized steady current distribution with the usual decay condition,Its curl gives the Biot–Savart field. For a uniform field, one convenient choice is $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$. Stokes's theorem relates ${\oint }_C\mathbf{A}\cdot d\mathbf{l}$ to magnetic flux through a spanning surface. In time-dependent electrodynamics, $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$; changing $\mathbf{A}$ by $\nabla \chi$ requires $\varphi \to \varphi -{\partial }_t\chi$ to preserve the electric field. Gauge freedom simplifies calculation without changing the electromagnetic observables.