Past exam of the physics B course of the University of Cambridge / 2014 / Paper 2 / Section B

The Biot-Savart law gives the magnetic field of a steady thin-wire electric current in vacuum:The cross product fixes the circulation direction by the right-hand rule; fields from different electric current elements superpose. Integration gives $B={\mu }_{0}I/(2\pi r)$ outside a straight infinite wire and $B={\mu }_{0}I/(2a)$ at the centre of a circular loop of radius $a$. A distributed electric current replaces $Id{\mathbf{\ell }}^{\prime }$ by $\mathbf{J}d{V}^{\prime }$. This is a magnetostatic law: arbitrary time-dependent electric currents require retarded electromagnetic fields, not instantaneous substitution.

In magnetostatics, Ampère's circuital law in vacuum isIt is especially useful when symmetry makes the tangential field constant on an Amperian loop. It gives $B={\mu }_{0}I/(2\pi r)$ for an infinite straight wire, $B\simeq {\mu }_{0}nI$ inside a long solenoid, and $B\simeq \mu NI/(2\pi r)$ in an ideal toroid. The long-solenoid and toroidal expressions neglect fringing.

For time-dependent fields, displacement current is essential:In matter, the corresponding macroscopic form is $\oint \mathbf{H}\cdot d\mathbf{\ell }=I_{\mathrm{free}}+d\int \mathbf{D}\cdot d\mathbf{A}/dt$. The enclosed electric current alone is sufficient only in the magnetostatic limit.

Because $\nabla \cdot \mathbf{B}=0$, a magnetic vector potential can be introduced with $\mathbf{B}=\nabla \times \mathbf{A}$. It is not unique: $\mathbf{A}+\nabla \chi$ gives the same field. In magnetostatics, the Coulomb gauge $\nabla \cdot \mathbf{A}=0$ gives ${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$ in vacuum. For localized electric currents,A magnetic scalar potential is more restricted. In a simply connected current-free region of constant permeability, write $\mathbf{B}=-\nabla \psi$; then ${\nabla }^2\psi =0$. Alternatively define the scalar potential for $\mathbf{H}$, differing by a constant permeability factor. A single-valued scalar potential generally cannot cover a domain encircling a net electric current, even where the local curl vanishes. The vector potential does not require the field itself to be curl-free.

Guided electromagnetic waves propagate along a structure whose boundaries select transverse field patterns. At a perfect conductor the tangential electric field vanishes. In a hollow rectangular guide, TE modes have $E_z=0$ and TM modes have $H_z=0$. Their longitudinal wavenumber obeysBelow the waveguide cutoff frequency ${\omega }_c=v{k}_{\perp }$, that mode is evanescent rather than freely propagating. TM modes require both indices positive; TE modes may have one zero but not both. For $a>b$, the lowest cutoff is the TE${}_{10}$ mode, with $f_c=v/(2a)$. A hollow single-conductor guide has no TEM mode, whereas a two-conductor transmission line can support one without this geometrical cutoff.^[ref]

For a propagating TE or TM mode, $v_p=v/\sqrt{1-{\omega }_c^2/{\omega }^2}$ and $v_g=v\sqrt{1-{\omega }_c^2/{\omega }^2}$. The phase velocity can exceed $v$ without superluminal information transfer. Frequency-dependent group delay can broaden pulses; excitation of several modes adds intermodal dispersion.

Electromagnetic induction produces an electromotive force when magnetic flux linkage changes. For an oriented moving loop,For a stationary loop this reduces to Faraday's law of induction, $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$. Motion in a static field can instead supply the motional term. Lenz's law gives the minus sign: the induced electric current opposes the flux change, with the required electrical energy supplied by mechanical work or the field source.

For linear fixed-geometry circuits, flux linkages satisfy ${\Lambda }_1=L_1{I}_1+M{I}_2$ and ${\Lambda }_2=M{I}_1+L_2{I}_2$. Self-inductance gives ${\mathcal{E}}_1=-L_1{\dot{I}}_1$; mutual inductance gives the additional term $-M{\dot{I}}_2$. Reciprocity gives the same mutual coefficient for both circuits, with its sign determined by winding orientation. The stored field energy isThe bound follows from nonnegative energy for arbitrary electric currents. In a linear medium the equivalent field expression is $W_B=\frac{1}{2}\int \mathbf{B}\cdot \mathbf{H}dV$.

An ideal electrical transformer links both windings with one changing core flux, giving $V_1/V_2=N_1/N_2$ for corresponding voltage magnitudes. A loaded secondary drives an opposing magnetomotive force and increases primary electric current; ideally $I_1/I_2=N_2/N_1$ and input and output powers agree. A electrical transformer exchanges voltage for electric current, not for extra power. Real devices have winding resistance, leakage flux, eddy-current and hysteresis losses, and finite magnetizing electric current. A steady DC input gives no continuing secondary voltage after the switching transient and may cause excessive primary electric current or core saturation.

An isotropic linear elastic solid has two independent elastic constants. For infinitesimal strain,The shear modulus $G$ governs shape change and the bulk modulus $K$ governs volume change. Young's modulus $E$ and Poisson's ratio $\nu$ are equivalent parameters:Positive strain energy requires $G,K>0$, hence $-1<\nu <1/2$; $\nu \to 1/2$ is the incompressible limit. Small-amplitude bulk waves have speeds $c_P=\sqrt{(K+4G/3)/\rho }$ and $c_S=\sqrt{G/\rho }$. Isotropy removes directional dependence, not the distinction between compression and shear. These laws apply within the elastic small-strain regime, not after yielding or fracture.

For a rigid body about a chosen origin,The inertia tensor is real and symmetric, so orthogonal principal axes diagonalize it. Angular momentum is parallel to angular velocity for rotation about a principal axis, but need not be parallel in a general direction. Its eigenvalues are the principal moments of inertia.

If the origin is translated from the centre of mass by $\mathbf{a}$, the parallel-axis theorem becomes $I_{ij}=I_{ij}^{\mathrm{CM}}+M(a^2{\delta }_{ij}-a_i{a}_j)$. In body axes, Euler's equation is $\mathbf{\tau }=(d\mathbf{L}/dt{)}_{\mathrm{body}}+\mathbf{\omega }\times \mathbf{L}$. Thus the full tensor is needed whenever the rotation is not confined to one fixed principal axis.

For two bodies interacting through $U({\mathbf{r}}_1-{\mathbf{r}}_2)$, defineTheir kinetic energy separates exactly:Without external forces, the centre of mass moves uniformly while $\mu \ddot{\mathbf{r}}=-\nabla U$. The relative orbit is a one-body problem with reduced mass $\mu$, not generally either individual mass. For $m_2\gg m_1$, $\mu \simeq m_1$; for equal masses $m$, $\mu =m/2$. If the interaction is central, relative angular momentum is conserved and the relative motion is planar.

Dynamic viscosity measures resistance to a velocity gradient. In Newtonian simple shear, ${\tau }_{xy}=\eta \partial u_x/\partial y$; positive viscosity dissipates mechanical energy into heat. Kinematic viscosity is $\nu =\eta /\rho$. For an incompressible fluid of constant viscosity,Scaling velocity by $U$ and length by $L$ gives the Reynolds number $\mathrm{Re}=\rho UL/\eta =UL/\nu$, the ratio of inertial to viscous terms. At small Reynolds number, creeping-flow approximations neglect inertia; at large Reynolds number, inertia dominates much of the bulk but viscosity can remain crucial in boundary layers. A large Reynolds number permits instability and turbulence; it does not by itself specify a universal transition threshold. Geometry and disturbances also matter.

A central potential conserves angular momentum and confines a nonradial orbit to a plane. With reduced mass $\mu$, the radial motion obeysTurning points satisfy $E=U_{\mathrm{eff}}$. A radially bounded allowed interval gives bound motion, whereas an interval reaching infinity permits escape. The energy sign alone classifies these only with suitable assumptions about $U(\infty )$ and barriers. Circular orbits satisfy $U_{\mathrm{eff}}^{\prime }=0$ and are radially stable when $U_{\mathrm{eff}}^{\prime \prime }>0$. Bound orbits in a general central potential need not close: successive periapses can precess.

For attractive inverse-square gravity, $U=-\kappa /r$ andEllipses have $E<0$, $e<1$, and the attracting centre at a focus; parabolic and hyperbolic escape have $e=1$ and $e>1$. Conservation of angular momentum gives equal swept areas in equal times. The inverse-square law additionally yields $T^2=4{\pi }^2{a}^3/[G(m_1+m_2)]$. These are Kepler's laws; a harmonic central potential is another example with closed bound orbits, while generic central laws produce precession.

In a gravitational assist, a spacecraft's asymptotic speed relative to a planet is approximately unchanged but its direction turns. Transforming to the frame in which the planet has velocity $\mathbf{V}$ givesThe spacecraft can gain or lose orbital energy; the planet supplies the compensating energy and momentum, with a very small velocity change because of its large mass.

Repulsive Coulomb scattering instead has $U={\kappa }_C/r$. For incident relative kinetic energy $E$ and impact parameter $b$, the hyperbolic orbit givesThis Rutherford scattering formula follows by equating the incident annulus $2\pi bdb$ to the scattered solid-angle ring. It assumes classical elastic scattering by a point Coulomb centre; screening, finite nuclear size or quantum effects can change the applicable regime.