Find the differential equation which must be satisfied by magnetic potential in a magnetic material the intensity and direction of the magnetisation of which is known. A steel sphere is magnetised so that the direction and intensity of magnetisation is constant. It is surrounded by a close fitting shell of paramagnetic material of permeability \(\mu\). The outer radius of the shell is \(n\) times the radius of the steel sphere. Show that the magnetic potential at external points is \[ \frac{9M\mu n^3}{\{2(\mu-1)^2-n^3(2\mu+1)(\mu+2)\}} \left(\frac{\cos\theta}{r^2}\right) \] where \(M\) is the total magnetic moment of the steel sphere.
A circular wire of radius \(a\) and carrying a current \(i\) is placed so that its centre is at a distance \(c\) from an infinite vertical wire carrying a current \(i'\). The plane of the circular wire is vertical and perpendicular to the vertical plane which contains the straight wire and the centre of the circular wire. Show that a couple of magnitude \[ 4\pi ii'c\left(1-\frac{c}{\sqrt{a^2+c^2}}\right) \] acts on the circular wire tending to turn it about its vertical diameter.
A telegraph cable has resistance \(r\) per unit length and electrostatic capacity \(c\) per unit length. The conductivity between the core and the surrounding sea water through the insulating covering is \(\kappa\) per unit length. Neglecting the effects of self induction find the differential equation which the current must satisfy when a variable E.M.F. is applied at one end. A periodic electromotive force of period \(\frac{2\pi}{p}\) is applied at one end, show that if the wire is infinitely long the amplitude of the electric oscillations will be reduced in the ratio \(1:e\) at a distance \(\frac{1}{2}\sqrt{\kappa r}\left[1+\left(1+\frac{c^2p^2}{\kappa^2}\right)^{\frac{1}{2}}\right]^{-\frac{1}{2}}\) from the end.
Discuss from a thermodynamical point of view the connection between the osmotic pressure of a salt solution and the lowering of the vapour pressure of the solvent due to the presence of the salt.
Investigate the two dimensional motion of an incompressible fluid defined by the stream function \(\psi=A(x^2-y^2)\), and show that it can be produced from rest in the space between any two coaxial elliptic cylinders by rotating them with appropriate angular velocities.
Prove that when a gas flows in steady motion under the action of a pressure gradient only the velocity at a point where the pressure is \(p\) is \[ \left(2\int_p^{p_0} \frac{dp}{\rho}\right)^{\frac{1}{2}}, \] \(p_0\) being the pressure at points where the velocity is zero. Hence show that the maximum possible velocity which the gas can acquire is \[ C_m = \left(\frac{2\gamma p_0}{(\gamma-1)\rho_0}\right)^{\frac{1}{2}}, \] where \(\rho_0\) is the density corresponding with pressure \(p_0\). If the gas passes through a long straight tube in which the velocity is constant and equal to \(q\), show that the pressure at the nose of a projectile moving along the tube with velocity \(U\) in the opposite direction to the stream of gas is \[ p_0\left(1+\frac{2qU+U^2}{C_m^2}\right)^{\frac{\gamma}{\gamma-1}}. \]
Explain carefully how the azimuth of the sun at any given time at a known point on the earth's surface can be calculated from the data given in the Nautical Almanac. The altitude \(\alpha\) of the sun is observed at a place in latitude \(\phi\). If the sun's longitude is \(L\) and the obliquity of the ecliptic is \(\omega\), show that the azimuth \(A\) reckoned from the North is \[ \cos^{-1}\left(\frac{\sin L\sin\omega-\sin\alpha\sin\phi}{\cos\alpha\cos\phi}\right). \]
A cylindrical tin of negligible mass and made of very thin material contains some air and is held down in water by a vertical string attached to the middle point of its base, which is perforated. Show that it cannot be in unstable equilibrium with its axis vertical unless \(h\sqrt{2}
Rationalise the equation \(\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}=0\), and express the result in factors each of which is linear in the square roots of \(x,y,z\) and \(t\). If \(a+b+c+d=0\) and \(x+y+z+t=0\), prove that the results of rationalising the equations \begin{align*} \sqrt{ax}+\sqrt{by}+\sqrt{cz}+\sqrt{dt}&=0, \\ \sqrt{bx}+\sqrt{ay}+\sqrt{dz}+\sqrt{ct}&=0, \end{align*} are equivalent. State the two other equations which lead to equivalent results.
If \(p_r/q_r\) is the \(r\)th convergent of the continued fraction \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_n}, \] prove that \(p_n/p_{n-1}\) is equal to \[ a_n + \frac{1}{a_{n-1} +} \frac{1}{a_{n-2} +} \dots + \frac{1}{a_1}, \] and express \(q_n/q_{n-1}\) as a continued fraction. Prove that \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_{n-1} +} \frac{1}{2a_n +} \frac{1}{a_{n-1} +} \dots + \frac{1}{a_1} \] is equal to the harmonic mean of the convergents \(p_{n-1}/q_{n-1}\) and \(p_n/q_n\), and express the arithmetic mean of the convergents as a continued fraction.