Past exam of the physics B course of the University of Cambridge / 2015 / Paper 2 / Section B

At an interface with unit normal $\hat{\mathbf{n}}$ pointing from medium 1 to medium 2, the magnetic boundary conditions areThe first follows from a pillbox applied to $\nabla \cdot \mathbf{B}=0$. The tangential magnetic field jump from a free surface current follows from a shrinking rectangular Ampère loop; ${\mathbf{K}}_f$ is the free surface current. With no free sheet current, normal $B$ and tangential $H$ are continuous. In linear isotropic media $\mathbf{B}=\mu \mathbf{H}$, so neither all components of $B$ nor all components of $H$ generally remain continuous.

For magnetic field-line refraction at a permeability interface, with angles measured from the normal, these conditions give $\tan {\theta }_2/\tan {\theta }_1={\mu }_2/{\mu }_1$. They determine the bending of magnetic flux at a material interface. In a slender core electromagnet, flux continuity and Ampère's law lead to the magnetic-circuit estimateAn air gap can dominate the reluctance even when the iron core has high magnetic permeability. Boundary conditions also determine demagnetizing fields; fringing, saturation and hysteresis limit the simple linear circuit model. A high-magnetic permeability material redirects flux, whereas a perfect-conductor boundary is a different physical assumption.^[ref]

An inductor stores magnetic energy and opposes changes of current through electromagnetic induction. For a fixed linear device, the flux linkage is $\Psi =N\Phi =LI$, the self-induced electromotive force is $-LdI/dt$, and the passive terminal voltage is $V=LdI/dt$. HenceThe voltage leads a sinusoidal current by a quarter cycle. A long solenoid has $B\simeq \mu NI/l$ and $L\simeq \mu N^{2}A/l$; its energy agrees with the volume integral of $B^2/(2\mu )$.

For two coupled coils, ${\Psi }_1=L_1{I}_1+\mathcal{M}{I}_2$, ${\Psi }_2=\mathcal{M}{I}_1+L_2{I}_2$, with signs set by coil orientations. Reciprocity gives the same mutual inductance in both equations. For the energy stored in a nonlinear inductor, $U=\int Id\Psi$ replaces the simple quadratic formula, and hysteresis can dissipate energy. Real coils also have resistance, core loss and parasitic capacitance, so an ideal inductance model has a finite useful frequency range.^[ref]

In a linear, nondispersive, lossless medium, the field energy density and energy flux areThe Poynting theorem follows by dotting the two curl Maxwell equations with the other field and subtracting:The right-hand side is power transferred from the field to free charges. In vacuum the energy density is ${ϵ}_0{E}^2/2+B^2/(2{\mu }_0)$.

Static electric or magnetic fields store energy even though their time derivatives vanish. Static Poynting flow occurs because static crossed fields can also have a nonzero Poynting flux: for example, a steady current-carrying coaxial line has radial electric and azimuthal magnetic fields, carrying energy along the line toward the load. Time independence therefore does not imply zero energy flow.

For phasors whose real parts represent the physical fields,For a travelling plane wave in vacuum the electric and magnetic contributions are equal and $⟨S⟩=c⟨u_{\mathrm{em}}⟩$. In dispersive matter the simple macroscopic energy formula needs modification to include the material response; it is not a universal formula for plasma energy.^[ref]

Reflected and transmitted waves arise because the fields must satisfy the electromagnetic boundary conditions. With no free surface charge or current, tangential $E,H$ and normal $D,B$ are continuous. At normal incidence between lossless media, define the real wave impedance $Z_j=\sqrt{{\mu }_j/{ϵ}_j}$. Writing $r=E_r/E_i$ and $t=E_t/E_i$ in a common transverse electric-field direction givesFor the power normalization of Fresnel amplitudes, power fractions are $R=|r{|}^2$ and $T=(Z_1/Z_2)|t{|}^2$, so $R+T=1$. An electric-field amplitude coefficient alone is not a transmitted power fraction.

At oblique incidence, phase matching along the interface gives Snell's law, $n_1\sin {\theta }_i=n_2\sin {\theta }_t$. Applying the same boundary conditions separately to the two linear polarizations gives the Fresnel equations. For nonmagnetic dielectrics the parallel-polarized reflection vanishes at $\tan {\theta }_B=n_2/n_1$. Above the critical angle into a lower-index lossless medium, the transmitted field is evanescent and reflection is total. A perfect conductor instead has zero tangential electric field and unit reflected power; finite conductivity introduces penetration and absorption.^[ref]

Displacement current makes Ampère's law compatible with charge conservation. The macroscopic Maxwell equations areTaking the divergence of the last equation and using Gauss's law gives $\nabla \cdot {\mathbf{J}}_f+{\partial }_t{\rho }_f=0$. Omitting ${\partial }_t\mathbf{D}$ would wrongly require the conduction current to be divergence free during charging. The displacement-current density ${\partial }_t\mathbf{D}$ has current-density units, but need not represent free charges crossing a region.

The surface independence of Ampère circulation during capacitor charging illustrates the need directly. A surface spanning the circuit can cut the wire or pass between the plates. For ideal parallel plates, $Q={ϵ}_{0}AE$ andThe corrected circulation of $\mathbf{H}$ is independent of that surface choice. Free conduction current stops at a plate; displacement current supplies the gap contribution.

In vacuum the two time-dependent curl equations support waves. Taking a curl of Faraday's law in a source-free region yieldsThe corresponding magnetic equation is identical. For a plane wave, $\mathbf{k}\cdot \mathbf{E}=\mathbf{k}\cdot \mathbf{B}=0$ and $\mathbf{B}=\mathbf{k}\times \mathbf{E}/\omega$. The fields carry energy through the Poynting flux. Dropping displacement current everywhere would remove these freely propagating solutions.^[ref]

In a metal, conduction and displacement compete. For a uniform ohmic material with ${\mathbf{J}}_f=\sigma \mathbf{E}$, constant $ϵ,\mu$, and transverse waves,Use $e^{i\mathbf{k}\cdot \mathbf{r}-i\omega t}$. The dispersion relation and conduction-to-displacement current ratio areWhen $\sigma \gg \omega ϵ$, $k\simeq (1+i)/\delta$ with $\delta =\sqrt{2/(\mu \sigma \omega )}$: fields are screened within a skin depth and energy is dissipated by Joule heating. This quasistatic limit neglects displacement current locally compared with conduction; it does not invalidate the full Maxwell law. At sufficiently high frequency, conductivity and permittivity are frequency dependent, so a constant-$\sigma$ approximation must be replaced by a material model.

In an unmagnetized cold plasma, electron inertia gives a cutoff. With immobile ions and negligible collisions, $m_e\dot{\mathbf{v}}=-e\mathbf{E}$ and $\mathbf{J}=-n_{e}e\mathbf{v}$ give $\sigma (\omega )=i{n}_e{e}^2/(m_e\omega )$. For the transverse wave cutoff in a cold unmagnetized plasma, combining this carrier current with vacuum displacement current givesTransverse waves propagate above ${\omega }_p$ and are evanescent below it. The group velocity is $c\sqrt{1-{\omega }_p^2/{\omega }^2}<c$ even though the phase velocity exceeds $c$. The cold model also has longitudinal charge-density oscillations at ${\omega }_p$, distinct from transverse light waves. Collisions add damping, and pressure or magnetic fields modify these simple dispersions. To avoid carrier-current double counting in effective permittivity, once the carrier response is included in ${ϵ}_{\mathrm{eff}}$ it must not be counted a second time as a separate conductivity. Displacement current therefore connects circuit continuity, vacuum radiation, conductor screening and plasma propagation within one consistent field theory.^[ref]

A torque-free rigid body has constant angular momentum $\mathbf{L}$ in inertial space. Its angular velocity need not be parallel to $\mathbf{L}$, because $\mathbf{L}=\mathbf{I}\mathbf{\omega }$. In body principal axes, Euler equations for a torque-free rigid body areFor a symmetric top $I_1=I_2=I_{\perp }$, $I_3=I_{\parallel }$, the component ${\omega }_3$ is constant andThis is the signed rotation frequency of the transverse angular-velocity components in the body frame. In space the symmetry axis instead obeysIts angle to $\mathbf{L}$ is constant, with $\cos \theta =I_{\parallel }{\omega }_3/L$. The body-frame and space-frame frequencies of free symmetric-top precession describe different frames. A spherical body has $\mathbf{\omega }\parallel \mathbf{L}$ and no corresponding wobble. For a triaxial body, rotations about the largest and smallest principal-inertia axes are stable, whereas rotation about the intermediate axis is unstable.^[ref]

For generalized coordinates $q_i$ with Lagrangian $L(q,\dot{q},t)$, the conjugate momentum isA coordinate absent from $L$ is cyclic, and its conjugate momentum is conserved. More generally, Noether's theorem connects continuous symmetries of the action to conserved quantities: spatial translations give linear momentum and rotations give angular momentum. If $L$ has no explicit time dependence, the conserved energy function isThe momentum conjugate to an angle is an angular momentum, not a linear velocity. Canonical momentum can also differ from mechanical momentum: for $L=\frac{1}{2}m{v}^2+q\mathbf{A}\cdot \mathbf{v}-q\varphi$, $\mathbf{p}=m\mathbf{v}+q\mathbf{A}$. Thus a conserved conjugate momentum must be identified from the actual Lagrangian and its symmetry, rather than assumed to equal $m\mathbf{v}$. Gauge-dependent canonical momenta require the corresponding gauge choice or gauge-covariant symmetry interpretation.^[ref]

For positions measured from the chosen origin, the inertia tensor isIt relates angular momentum and rotational kinetic energy to angular velocity:The real symmetric tensor has orthogonal principal axes, in which $\mathbf{I}=\mathrm{diag}(I_1,I_2,I_3)$. Its eigenvalues are nonnegative moments of inertia. Rotation about a principal axis has $\mathbf{L}\parallel \mathbf{\omega }$; a general rotation does not. The moment about a unit axis $\mathbf{n}$ is $I_{\mathbf{n}}=\mathbf{n}\cdot \mathbf{In}$.

Changing from the centre of mass to a parallel origin displaced by $\mathbf{a}$ gives the tensor parallel-axis theorem,For example, a uniform solid sphere has $\mathbf{I}=(2M{R}^2/5)1$, while a thin rod along $x$ has principal moments $0,M{l}^2/12,M{l}^2/12$ in the ideal zero-radius limit. The tensor is needed to describe off-axis rotation and torque-free precession; a single scalar inertia only suffices after an axis is specified.^[ref]

For steady, inviscid, incompressible flow, Euler's equation integrated along a streamline givesThis is Bernoulli's equation. The constant is common to different streamlines if the flow is also irrotational; incompressibility alone does not imply that. Viscous loss or work by a pump requires additional terms.

A Pitot-static tube compares stagnation pressure $p_0$ at a forward-facing opening with nearby static pressure $p$. For equal height, the stagnation point has $v=0$, soThis measures flow speed when compressibility corrections are negligible.

In a flowmeter using the Venturi effect, continuity gives $A_1{v}_1=A_2{v}_2$. For a horizontal constriction with $A_2<A_1$, Bernoulli gives $p_1-p_2=\rho (v_2^2-v_1^2)/2$, henceThe fluid accelerates and static pressure falls in the throat. This is a second practical application of the conversion between pressure and kinetic energy; real instruments need calibration for flow losses.

A normal mode is a motion in which all coordinates have a fixed relative pattern and oscillate at one frequency. Near a stable equilibrium, let the quadratic kinetic and potential energies be $T={\dot{\mathbf{q}}}^{\mathsf{T}}\mathbf{M}\dot{\mathbf{q}}/2$ and $V={\mathbf{q}}^{\mathsf{T}}\mathbf{Kq}/2$. With positive-definite mass matrix $\mathbf{M}$, the equations and harmonic ansatz giveThe stiffness and mass matrices are symmetric. For distinct frequencies, their symmetry gives $({\omega }_s^2-{\omega }_r^2){\mathbf{a}}_r^{\mathsf{T}}\mathbf{M}{\mathbf{a}}_s=0$, so the mode vectors are orthogonal in the mass inner product. A basis can also be chosen in a degenerate eigenspace. Expanding $\mathbf{q}={\sum }_s{Q}_s{\mathbf{a}}_s$ decouples the equations into ${\ddot{Q}}_s+{\omega }_s^2{Q}_s=0$; initial displacements and velocities determine the coefficients. Positive stiffness gives real frequencies, a zero eigenvalue can describe a free symmetry motion, and a negative one signals instability rather than an oscillatory mode.^[ref]

For two equal masses between fixed walls with three equal springs, the stiffness matrix is $k\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$ and the mass matrix is $m1$. The patterns $(1,1)$ and $(1,-1)$ have frequencies $\sqrt{k/m}$ and $\sqrt{3k/m}$. An initial displacement of one mass excites both modes and can produce beats.

For a uniform taut string fixed at both ends, the wave equation ${\partial }_t^{2}y=c_s^2{\partial }_x^{2}y$, with $c_s=\sqrt{\mathcal{T}/{\mu }_{\ell }}$, has patterns and angular frequenciesThe boundary conditions select the spectrum. Fourier expansion of arbitrary admissible initial data provides the mode amplitudes, illustrating the extension from a finite set of coordinates to a field.

For the normal modes of inductively coupled LC circuits, take two identical circuits with self-inductance $L$, capacitance $C$ and mutual inductance $\mathcal{M}$, where $|\mathcal{M}|<L$. The capacitor charges obey $L{\ddot{q}}_1+\mathcal{M}{\ddot{q}}_2+q_1/C=0$ and the exchanged equation. In-phase and opposite-phase charge patterns haveThe positive inductance matrix for two identical coils is the physical counterpart of positive mass. Here inductance plays the role of the mass matrix. Mechanical, wave and electrical examples therefore share the same eigenvalue structure. Weak forcing excites modes according to its spatial overlap with each pattern; damping broadens the resonances, while strong nonlinearity can invalidate a fixed linear-mode decomposition.