Past exam of the physics B course of the University of Cambridge / 2011 / Paper 1 / Section B / B6

Gauss's law connects electric flux through a closed surface to its enclosed charge:The differential form follows from the divergence theorem. The law is exact; extracting a field magnitude by simple division additionally needs symmetry. For spherical symmetry, $E_r(r)=Q(r)/(4\pi {ϵ}_0{r}^2)$; a uniformly charged ball therefore has a field linear in $r$ inside and an inverse-square field outside. An infinite line of linear charge density $\Lambda$ gives $E_r=\Lambda /(2\pi {ϵ}_{0}r)$ by a cylindrical surface. A uniformly charged infinite sheet gives $E=\sigma /(2{ϵ}_0)$ on either side by a pillbox.

An electrostatic conductor has zero interior field and constant potential; excess charge occupies its surface. A pillbox gives $E_{n,\mathrm{out}}=\sigma /{ϵ}_0$ when the inside is conducting. More generally, $\hat{\mathbf{n}}\cdot ({\mathbf{E}}_2-{\mathbf{E}}_1)=\sigma /{ϵ}_0$. For dielectric matter, $\nabla \cdot \mathbf{D}={\rho }_{\mathrm{free}}$ separates bound polarization charge from free charge. Gauss's law alone does not establish a uniform field for an arbitrary finite plate or recover a detailed nonsymmetric charge distribution from one flux measurement.

An electric dipole has equal and opposite charges separated by a small vector from negative to positive, with electric dipole moment $\mathbf{p}=q\mathbf{d}$. More generally, a localized neutral distribution has $\mathbf{p}=\int {\mathbf{r}}^{\prime }\rho ({\mathbf{r}}^{\prime })d^3{r}^{\prime }$, independent of origin. At distances large compared with its size, the leading nonzero term in the multipole expansion isThe field points along $\mathbf{p}$ on either axial side and opposite $\mathbf{p}$ in the equatorial plane; its axial magnitude is twice the equatorial magnitude at equal radius. Near the charges the finite-charge expression must replace the far-field approximation.

A permanent dipole in an externally imposed slowly varying electrostatic field has interaction energy $-\mathbf{p}\cdot \mathbf{E}$, torque $\mathbf{\tau }=\mathbf{p}\times \mathbf{E}$ and force $\nabla (\mathbf{p}\cdot \mathbf{E})$ for fixed $\mathbf{p}$. A uniform field aligns it but gives no net force. For a linearly induced dipole $\mathbf{p}=\alpha \mathbf{E}$, the formation-inclusive energy is $-\alpha E^2/2$, not $-\alpha E^2$: the moment itself grows while the field is applied. Dipole density gives electric polarization, whose bound charges are ${\rho }_b=-\nabla \cdot \mathbf{P}$ and ${\sigma }_b=\mathbf{P}\cdot \hat{\mathbf{n}}$.

For a fixed circuit, Faraday's law of induction readsSurface orientation and contour direction obey the right-hand rule. The negative sign is Lenz's law: a passive induced current acts against the flux change, consistently with energy conservation. The induced electric field generally has nonzero curl and cannot be represented by one single-valued electrostatic potential.

For a material circuit moving with local velocity $\mathbf{u}$, the measured electromotive force includes the magnetic Lorentz force on its carriers:This distinguishes transformer induction in a stationary coil from motional induction in a changing circuit. A straight sliding rod of length $l$ moving perpendicular to itself and to uniform $B$ has $|\mathcal{E}|=Blv$. With circuit resistance $R$, the electrical power $(Blv{)}^2/R$ equals the external mechanical power needed to oppose magnetic drag. An ideal transformer gives $V_2/V_1=N_2/N_1$ in magnitude from common flux linkage; real leakage, resistance and core losses limit that simple result. pickup coils measure flux change rather than an absolute static magnetic field, so integration and a reference state are needed to recover a field history.

In charge- and current-free vacuum, Maxwell equations reduce toTaking a curl and using $\nabla \times \nabla \times \mathbf{E}=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$ givesand the same wave equation for $\mathbf{B}$. For a plane wave $\mathbf{E}=\mathrm{Re}[{\mathbf{E}}_0{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}]$, $\omega =ck$, $\mathbf{k}\cdot {\mathbf{E}}_0=\mathbf{k}\cdot {\mathbf{B}}_0=0$ and ${\mathbf{B}}_0=\hat{\mathbf{k}}\times {\mathbf{E}}_0/c$. Thus the fields are transverse, mutually perpendicular and in phase; polarization describes the electric-field trajectory at a fixed point.

The energy density and Poynting vector areFor a linearly polarized sinusoid with peak electric amplitude $E_0$, the mean intensity is ${ϵ}_{0}c{E}_0^2/2$. Electromagnetic momentum density is $\mathbf{S}/c^2$; normal absorption produces radiation pressure $⟨S⟩/c$, while ideal retroreflection doubles it. Linear superposition permits interference and standing waves, whose local electric and magnetic fields need not have the traveling-plane-wave equality point by point.