Part IB, 2004, Paper 3

# Part IB, 2004, Paper 3

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3.I.4F

commentLet $X$ and $X^{\prime}$ be metric spaces with metrics $d$ and $d^{\prime}$. If $u=\left(x, x^{\prime}\right)$ and $v=\left(y, y^{\prime}\right)$ are any two points of $X \times X^{\prime}$, prove that the formula

$D(u, v)=\max \left\{d(x, y), d^{\prime}\left(x^{\prime}, y^{\prime}\right)\right\}$

defines a metric on $X \times X^{\prime}$. If $X=X^{\prime}$, prove that the diagonal $\Delta$ of $X \times X$ is closed in $X \times X$.

3.I.7B

commentA wire is bent into the shape of three sides of a rectangle and is held fixed in the $z=0$ plane, with base $x=0$ and $-\ell<y<\ell$, and with arms $y=\pm \ell$ and $0<x<\ell$. A second wire moves smoothly along the arms: $x=X(t)$ and $-\ell<y<\ell$ with $0<X<\ell$. The two wires have resistance $R$ per unit length and mass $M$ per unit length. There is a time-varying magnetic field $B(t)$ in the $z$-direction.

Using the law of induction, find the electromotive force around the circuit made by the two wires.

Using the Lorentz force, derive the equation

$M \ddot{X}=-\frac{B}{R(X+2 \ell)} \frac{d}{d t}(X \ell B)$

3.II.19B

commentStarting from Maxwell's equations, derive the law of energy conservation in the form

$\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S}+\mathbf{J} \cdot \mathbf{E}=0$

where $W=\frac{\epsilon_{0}}{2} E^{2}+\frac{1}{2 \mu_{0}} B^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.

Evaluate $W$ and $\mathbf{S}$ for the plane electromagnetic wave in vacuum

$\mathbf{E}=\left(E_{0} \cos (k z-\omega t), 0,0\right) \quad \mathbf{B}=\left(0, B_{0} \cos (k z-\omega t), 0\right),$

where the relationships between $E_{0}, B_{0}, \omega$ and $k$ should be determined. Show that the electromagnetic energy propagates at speed $c^{2}=1 /\left(\epsilon_{0} \mu_{0}\right)$, i.e. show that $S=W c$.

3.I.10C

commentState Bernoulli's equation for unsteady motion of an irrotational, incompressible, inviscid fluid subject to a conservative body force $-\nabla \chi$.

A long vertical U-tube of uniform cross section contains an inviscid, incompressible fluid whose surface, in equilibrium, is at height $h$ above the base. Derive the equation

$h \frac{d^{2} \zeta}{d t^{2}}+g \zeta=0$

governing the displacement $\zeta$ of the surface on one side of the U-tube, where $t$ is time and $g$ is the acceleration due to gravity.

$\begin{aligned} & \frac{\partial \eta}{\partial t}+U \frac{\partial \eta}{\partial x}=\frac{\partial \phi}{\partial y}, \quad \frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}+g \eta=0 \quad \text { on } \quad y=0 \\ & \frac{\partial \phi}{\partial y}=0 \quad \text { on } \quad y=-h, \end{aligned}$

3.II.21C

commentUse separation of variables to determine the irrotational, incompressible flow

$\mathbf{u}=U \frac{a^{3}}{r^{3}}\left(\cos \theta \mathbf{e}_{r}+\frac{1}{2} \sin \theta \mathbf{e}_{\theta}\right)$

around a solid sphere of radius $a$ translating at velocity $U$ along the direction $\theta=0$ in spherical polar coordinates $r$ and $\theta$.

Show that the total kinetic energy of the fluid is

$K=\frac{1}{4} M_{f} U^{2},$

where $M_{f}$ is the mass of fluid displaced by the sphere.

A heavy sphere of mass $M$ is released from rest in an inviscid fluid. Determine its speed after it has fallen through a distance $h$ in terms of $M, M_{f}, g$ and $h$.

3.I.5E

commentLet $C$ be the contour that goes once round the boundary of the square

$\{z:-1 \leqslant \operatorname{Re} z \leqslant 1,-1 \leqslant \operatorname{Im} z \leqslant 1\}$

in an anticlockwise direction. What is $\int_{C} \frac{d z}{z}$ ? Briefly justify your answer.

Explain why the integrals along each of the four edges of the square are equal.

Deduce that $\int_{-1}^{1} \frac{d t}{1+t^{2}}=\frac{\pi}{2}$.

3.I.3G

commentState Euler's formula for a convex polyhedron with $F$ faces, $E$ edges, and $V$ vertices.

Show that any regular polyhedron whose faces are pentagons has the same number of vertices, edges and faces as the dodecahedron.

3.II.15G

commentLet $a, b, c$ be the lengths of a right-angled triangle in spherical geometry, where $c$ is the hypotenuse. Prove the Pythagorean theorem for spherical geometry in the form

$\cos c=\cos a \cos b$

Now consider such a spherical triangle with the sides $a, b$ replaced by $\lambda a, \lambda b$ for a positive number $\lambda$. Show that the above formula approaches the usual Pythagorean theorem as $\lambda$ approaches zero.

3.I.1E

commentLet $V$ be a finite-dimensional vector space over $\mathbb{R}$. What is the dual space of $V$ ? Prove that the dimension of the dual space is the same as that of $V$.

3.II.13E

comment(i) Let $V$ be an $n$-dimensional vector space over $\mathbb{C}$ and let $\alpha: V \rightarrow V$ be an endomorphism. Suppose that the characteristic polynomial of $\alpha$ is $\Pi_{i=1}^{k}\left(x-\lambda_{i}\right)^{n_{i}}$, where the $\lambda_{i}$ are distinct and $n_{i}>0$ for every $i$.

Describe all possibilities for the minimal polynomial and prove that there are no further ones.

(ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are $(x-1)^{3}(x-3)$.

(iii) Give an example of two matrices $A, B$ with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix $P$ with $P A P^{-1}=B$.

3.I.6D

commentLet

$S[x]=\int_{0}^{T} \frac{1}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right) \mathrm{d} t, \quad x(0)=a, \quad x(T)=b .$

For any variation $\delta x(t)$ with $\delta x(0)=\delta x(T)=0$, show that $\delta S=0$ when $x=x_{c}$ with

$x_{c}(t)=\frac{1}{\sin \omega T}[a \sin \omega(T-t)+b \sin \omega t] .$

By using integration by parts, show that

$S\left[x_{c}\right]=\left[\frac{1}{2} x_{c} \dot{x}_{c}\right]_{0}^{T}=\frac{\omega}{2 \sin \omega T}\left[\left(a^{2}+b^{2}\right) \cos \omega T-2 a b\right]$

3.II.18D

commentStarting from the Euler-Lagrange equations, show that the condition for the variation of the integral $\int I\left(y, y^{\prime}\right) \mathrm{d} x$ to be stationary is

$I-y^{\prime} \frac{\partial I}{\partial y^{\prime}}=\text { constant }$

In a medium with speed of light $c(y)$ the ray path taken by a light signal between two points satisfies the condition that the time taken is stationary. Consider the region $0<y<\infty$ and suppose $c(y)=e^{\lambda y}$. Derive the equation for the light ray path $y(x)$. Obtain the solution of this equation and show that the light ray between $(-a, 0)$ and $(a, 0)$ is given by

$e^{\lambda y}=\frac{\cos \lambda x}{\cos \lambda a},$

if $\lambda a<\frac{\pi}{2}$.

Sketch the path for $\lambda a$ close to $\frac{\pi}{2}$ and evaluate the time taken for a light signal between these points.

[The substitution $u=k e^{\lambda y}$, for some constant $k$, should prove useful in solving the differential equation.]

3.I.11A

commentThe linear system

$\left[\begin{array}{lll} \alpha & 2 & 1 \\ 1 & \alpha & 2 \\ 2 & 1 & \alpha \end{array}\right] \mathbf{x}=\mathbf{b}$

where real $\alpha \neq 0$ and $\mathbf{b} \in \mathbb{R}^{3}$ are given, is solved by the iterative procedure

$\mathbf{x}^{(k+1)}=-\frac{1}{\alpha}\left[\begin{array}{lll} 0 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 0 \end{array}\right] \mathbf{x}^{(k)}+\frac{1}{\alpha} \mathbf{b}, \quad k \geqslant 0$

Determine the conditions on $\alpha$ that guarantee convergence.

3.II.22A

commentGiven $f \in C^{3}[0,1]$, we approximate $f^{\prime}\left(\frac{1}{3}\right)$ by the linear combination

$\mathcal{T}[f]=-\frac{5}{3} f(0)+\frac{4}{3} f\left(\frac{1}{2}\right)+\frac{1}{3} f(1)$

By finding the Peano kernel, determine the least constant $c$ such that

$\left|\mathcal{T}[f]-f^{\prime}\left(\frac{1}{3}\right)\right| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty} .$

3.I.12G

commentConsider the two-person zero-sum game Rock, Scissors, Paper. That is, a player gets 1 point by playing Rock when the other player chooses Scissors, or by playing Scissors against Paper, or Paper against Rock; the losing player gets $-1$ point. Zero points are received if both players make the same move.

Suppose player one chooses Rock and Scissors (but never Paper) with probabilities $p$ and $1-p, 0 \leqslant p \leqslant 1$. Write down the maximization problem for player two's optimal strategy. Determine the optimal strategy for each value of $p$.

3.II.23G

commentConsider the following linear programming problem:

$\begin{aligned} \text { maximize }-x_{1}+3 x_{2} \\ \text { subject to } \quad x_{1}+x_{2} & \geqslant 3 \\ -x_{1}+2 x_{2} & \geqslant 6 \\ -x_{1}+x_{2} & \leqslant 2, \\ x_{2} & \leqslant 5, \\ x_{i} & \geqslant 0, \quad i=1,2 . \end{aligned}$

Write down the Phase One problem in this case, and solve it.

By using the solution of the Phase One problem as an initial basic feasible solution for the Phase Two simplex algorithm, solve the above maximization problem. That is, find the optimal tableau and read the optimal solution $\left(x_{1}, x_{2}\right)$ and optimal value from it.

3.I $9 \mathrm{D} \quad$

commentWrite down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum $L$ is quantised so that

$L=n \hbar,$

for integer $n$. Assuming circular orbits, show that the radius of the $n$ 'th orbit is

$r_{n}=n^{2} a$

and determine $a$. Show that the corresponding energy is then

$E_{n}=-\frac{e^{2}}{8 \pi \epsilon_{0} r_{n}}$

3.II.20D

commentA one-dimensional system has the potential

$V(x)= \begin{cases}0 & x<0 \\ \frac{\hbar^{2} U}{2 m} & 0<x<L \\ 0 & x>L\end{cases}$

For energy $E=\hbar^{2} \epsilon /(2 m), \epsilon<U$, the wave function has the form

$\psi(x)= \begin{cases}a e^{i k x}+c e^{-i k x} & x<0 \\ e \cosh K x+f \sinh K x & 0<x<L \\ d e^{i k(x-L)}+b e^{-i k(x-L)} & x>L\end{cases}$

By considering the relation between incoming and outgoing waves explain why we should expect

$|c|^{2}+|d|^{2}=|a|^{2}+|b|^{2}$

Find four linear relations between $a, b, c, d, e, f$. Eliminate $d, e, f$ and show that

$c=\frac{1}{D}\left[b+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L a\right]$

where $D=\cosh K L-\frac{1}{2}\left(\lambda+\frac{1}{\lambda}\right) \sinh K L$ and $\lambda=K /(i k)$. By using the result for $c$, or otherwise, explain why the solution for $d$ is

$d=\frac{1}{D}\left[a+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L b\right]$

For $b=0$ define the transmission coefficient $T$ and show that, for large $L$,

$T \approx 16 \frac{\epsilon(U-\epsilon)}{U^{2}} e^{-2 \sqrt{U-\epsilon} L}$

3.I $. 8 \mathrm{~B} \quad$

commentWrite down the Lorentz transformation with one space dimension between two inertial frames $S$ and $S^{\prime}$ moving relatively to one another at speed $V$.

A particle moves at velocity $u$ in frame $S$. Find its velocity $u^{\prime}$ in frame $S^{\prime}$ and show that $u^{\prime}$ is always less than $c$.