Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section A

An infinite sheet of surface charge density $\sigma$ gives an electric field of magnitude $E_s=\sigma /(2{\epsilon }_0)$ on either side, normal to the sheet and directed away from positive charge. Use the superposition principle for the two sheets. Choose normals $\hat{\mathbf{y}}$ for $y=0$ and $\mathbf{n}=(-\hat{\mathbf{x}}+\hat{\mathbf{y}})/\sqrt{2}$ for $y=x$. Away from the charged planes,The minus sign on the second contribution makes its field point toward the negatively charged sheet. Each sector has a uniform electric field; its direction changes on crossing a sheet.

For example, above both sheets the field points right and slightly upward, whereas in the sector $y>0$ but $y<x$ it points left and steeply upward. The other two sectors follow by inversion through the origin. The normal field jumps by the appropriate surface charge divided by ${\epsilon }_0$; a single field value on an ideal zero-thickness charged sheet is not required.

Let the signed axial velocity on the axis be $w(z)$. Smooth axisymmetric flow has $u_z(r,z)=w(z)+O(r^2)$ and regular radial velocity at $r=0$. The incompressible flow condition in cylindrical coordinates isTo leading order near the axis, integrate ${\partial }_r(r{u}_r)=-r{w}^{\prime }(z)$ from zero to $r$. The integration constant vanishes by regularity, giving the near-axis radial velocity in incompressible axisymmetric flow,With the usual signed interpretation $w(z)=A/z^3$,For $A>0$ this is outward: the outward axial flow slows as its distance from the origin grows, and incompressibility requires radial spreading. Here $A$ has units ${\mathrm{m}}^4{\mathrm{s}}^{-1}$, so the answer is a speed.

The source specifies a magnitude rather than an unambiguous signed component. On $z>0$, reversing the axial flow while keeping its magnitude would reverse the radial answer. The general relation $u_r=-r{w}^{\prime }/2$ makes that dependence explicit; the boxed expression adopts $u_z=A/z^3$, outward on both halves of the axis.

A large gas volume can reduce the effective mass density of the water surrounding the hull. In the simple quasihydrostatic model for the buoyancy of an aerated liquid, gas void fraction $\alpha$ givesBy Archimedes' principle, a boat displacing volume $V_{\mathrm{disp}}$ then has buoyant force ${\rho }_{\mathrm{mix}}g{V}_{\mathrm{disp}}$. To support the same boat mass $m$, it needs a larger displaced volume $m/{\rho }_{\mathrm{mix}}$, so its draft increases. If that exceeds the available watertight displacement, it can lose flotation or become swamped:Uneven aeration and strong plume motion can also disturb stability. This explains a possible hazard, not a claim that any bubble plume must sink a boat: upward water flow, turbulence and the distribution of bubbles can exert substantial additional forces.

The conventional electron plasma frequency isUse $n_e=10^{25}{\mathrm{m}}^{-3}$, $e=1.602\times 10^{-19}\mathrm{C}$, $m_e=9.109\times 10^{-31}\mathrm{kg}$ and ${\epsilon }_0=8.854\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$. ThenThe equal proton density ensures charge neutrality; it does not double the electron density in this formula. Protons have their own plasma frequency,If both cold species are allowed to move in the relative charge-separation oscillation, its squared frequency is ${\omega }_{pe}^2+{\omega }_{pi}^2$. This gives $1.7843\times 10^{14}\mathrm{rad}{\mathrm{s}}^{-1}$, only about $0.027\%$ above the electron-only answer, because $m_p\gg m_e$.

Apply equal compressive pressure $p$ in all directions; here $\sigma$ denotes Poisson's ratio. The three principal normal stresses are $-p$. For an isotropic linearly elastic material, Hooke's law givesand similarly for $y,z$. At small strain, the fractional volume change is the sum of these normal strains:The bulk modulus is $B=-p/(\Delta V/V)$, giving the isotropic elastic modulus relation,The result assumes linear elasticity and fixed isotropic moduli. In the incompressible limit $\sigma \to 1/2$ with finite Young's modulus, the bulk modulus tends to infinity.