Past exam of the optics and electrodynamics course of the University of Cambridge / 2012 / Paper 1 / 4

Take $x^{\mu }=(ct,x,y,z)$ and metric signature $(+,-,-,-)$. The four-current, four-potential and four-gradient areThe first two transform as contravariant four-vectors under an inertial-frame Lorentz transformation, within a consistently transformed electromagnetic gauge. The lowered-index gradient is a covector. Their contraction is a Lorentz scalar:Indeed, $J^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{J}^{\nu }$ and ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\alpha }{}_{\mu }{\partial }_{\alpha }$, so the two matrices cancel in the contraction. Local charge conservation in one inertial frame implies local charge conservation in every inertial frame. Integrating over a region gives the charge change equal to minus its outward current flux; global conserved charge requires no flux through the system boundary.

Let $\hat{\mathbf{r}}$ point transversely away from the line and define $\hat{\mathbf{\phi }}=\hat{\mathbf{x}}\times \hat{\mathbf{r}}$. Cylindrical Gauss's law in the line's rest frame givesThe transverse distance is unchanged by the axial boost: $r=r^{\prime }$. Applying the inverse of the given field transformation to a boosted uniform line charge,Here $\gamma =(1-u^2/c^2{)}^{-1/2}$ and all longitudinal field components vanish. Substituting back verifies ${\mathbf{B}}^{\prime }=\gamma (\mathbf{B}-\mathbf{u}\times \mathbf{E}/c^2)=0$ and ${\mathbf{E}}^{\prime }=\gamma (1-u^2/c^2)\mathbf{E}=\mathbf{E}/\gamma$. The sign of ${\lambda }^{\prime }$ determines the radial and circulation directions.

A material segment of the line with rest length $d{x}^{\prime }$ has laboratory length $dx=d{x}^{\prime }/\gamma$, while its charge $dQ$ is unchanged. ThusEquivalently the line-integrated four-current transforms from $(c{\lambda }^{\prime },0)$ to $(\gamma c{\lambda }^{\prime },\gamma u{\lambda }^{\prime })$. Gauss's and Ampère's laws in $S$ now give $E_r=\lambda /(2\pi {ϵ}_{0}r)$ and $B_{\phi }={\mu }_{0}I/(2\pi r)$, exactly the fields found in part (a). Charge density is frame-dependent even though the charge of the same set of particles is invariant.

In $S$, adding the stationary opposite line cancels charge density and electric field but contributes no current. HenceThe neutral current filament in a boosted frame is not neutral. Transform its line-integrated four-current:The same result follows by adding the original resting line $({\lambda }^{\prime },0)$ to the new negative line, whose primed density and current are $(-\gamma \lambda ,+\gamma \lambda u)$. In particular ${\lambda }^{\prime }-\gamma \lambda =-\gamma \lambda u^2/c^2$ because $\lambda =\gamma {\lambda }^{\prime }$.

Transforming the total fields givesFor positive $\lambda$, $\mathbf{u}\times \mathbf{B}$ points inward, explaining the electric-field sign. The primed Gauss and Ampère fields calculated from ${\lambda }_{\mathrm{tot}}^{\prime }$ and $I_{\mathrm{tot}}^{\prime }$ reproduce these expressions. Both a moving charge distribution and a moving current must be transformed; boosting just one constituent would give an inconsistent total.

An ideal current-carrying wire consists of opposite charge populations, one associated with the stationary lattice and one with drifting carriers. They may cancel in the lattice frame while carrying a net current, as in part (c), up to exchanging the charge signs and current direction. In a carrier-rest description their densities no longer cancel because the two populations undergo different length contractions. The wire then has both electric fields and magnetic fields.

A neutral steady current in one frame generally becomes a charged steady current in another. This illustrates how electric fields and magnetic fields are parts of one relativistic field rather than separately invariant forces. Real resistive wires also have a driving longitudinal electric field and geometry-dependent surface charges; the infinite uniform filament model isolates the transverse fields of a neutral current and does not describe every field of a complete circuit.