Past exam of the optics and electrodynamics course of the University of Cambridge / 2013 / Paper 1 / 1

The wavelength is $\lambda =c/f\simeq 0.50\mathrm{m}$, and the aperture diameter is $D=1.0\mathrm{m}$. A uniformly illuminated circular aperture has normalized power pattern $[2J_1(x)/x{]}^2$, where $x=\pi D\sin \theta /\lambda$. Its first null is at $\sin \theta \simeq 1.22\lambda /D$, while its half-power point has $\sin \theta \simeq 0.514\lambda /D$. ThusThe first-null small-angle estimate is $0.61\mathrm{rad}$, or roughly $35^{\circ }$; using the sine gives about $38^{\circ }$. This is a broad beam because the diameter is only about two wavelengths. Real illumination and aperture efficiency alter the exact beam shape; a required allowable gain loss would specify the pointing tolerance more precisely.

For a matched, correctly polarized half-wave dipole, $A_{e,d}=G_d{\lambda }^2/(4\pi )$ with $G_d\simeq 1.64$. For an efficient dish, $A_{e,s}\simeq \eta \pi (D/2{)}^2$. Received available powers are proportional to these areas. With both voltages referred to the same matched receiver resistance,For ideal aperture efficiency this is about one fifth, or five times greater voltage from the dish. Terminal voltages without a common impedance or matching convention are not fixed by aperture alone: $V_{\mathrm{rms}}^2=P_{\mathrm{available}}{R}_L$ for the respective matched loads. A short-dipole rather than half-wave convention changes the order-one gain factor, not the main estimate.

Spatial coherence compares the electric fields at different positions at a common time; temporal coherence compares the field with itself at a delay. Experimentally, they control whether separated apertures or unequal optical paths can form stable interference fringes. They are correlation properties, not a requirement that every emitted wave have a deterministic phase.

The source diameter is $\alpha =40/206265=1.94\times 10^{-4}\mathrm{rad}$. At a representative visible wavelength $\lambda =550\mathrm{nm}$,For a uniformly bright circular disc, the first visibility zero is at $1.22\lambda /\alpha =3.5\mathrm{mm}$; the coefficient depends on the chosen coherence criterion and brightness distribution. Atmospheric phase fluctuations can further spoil measured coherence.

For an explicitly assumed broad visible band from $400$ to $700\mathrm{nm}$, the frequency span is $\Delta \nu =c(1/{\lambda }_{\min }-1/{\lambda }_{\max })=3.21\times 10^{14}\mathrm{Hz}$. ConsequentlyThus broadband planetary light has millimetre-scale transverse and micrometre-scale longitudinal coherence. The receiver band is not supplied: a narrow filter would increase the temporal coherence length, so this is a stated broadband estimate, not a unique source-only constant.

Use ideal large plates and neglect edge effects. A Lorentz boost normal to the plates does not contract either transverse plate dimension. Charge is invariant, so charge per plate area remains ${\sigma }^{\prime }=\sigma$. Gauss's law then givesThe plate separation contracts, but the infinite-plate field is independent of separation. It is therefore incorrect to infer a stronger field merely from the shorter gap. The ideal field is parallel to the boost, so there is no induced transverse magnetic field from this boost.

For a long solenoid at rest, $B={\mu }_{0}nI$, where n is turns per unit axial length and I is the current around a turn. Under a boost along its axis, longitudinal contraction gives $n^{\prime }=\gamma n$. The circumference of a turn is transverse to the boost and does not contract; time dilation reduces its charge-circulation rate, measured relative to the moving material of that turn, to $I^{\prime }=I/\gamma$. HenceEquivalently, the transverse volume-current density of an ideal continuous current sheet is unchanged by an axial boost. The increase in turn density must not be included while leaving the per-turn current unchanged. End effects and the microscopic winding geometry are outside this ideal-solenoid argument.