MATH 282 Field Theory, Partial Differential Equations and Methods of Solution 2015
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MATH 282
JANUARY 2015 EXAMINATIONS
Field Theory, Partial Differential Equations and Methods of Solution
1. (a) A surface is defined by the equation
(x2 + y2 + z2)2 − 4xyz = 25
Calculate the equation of the tangent plane to this surface at the point (0,2,1). [8 marks]
(b) Prove the identity
∇ · (fv) = f∇ · v + v · ∇f
where v is a vector field, and f a scalar field.
[8 marks]
(c) Check that the vector field
u = (cosxsin y − yz, sinxcos y − xz, −xy + 4z) is irrotational. Construct a scalar field φ such that
u = ∇φ .
[9 marks]
2. The three-dimensional heat equation
= κ∇2φ
has a solution of the form
φ0(x,y,z,t) = tp exp −α tq (x2 + y2 + z2) .
(i) Find the values of p,q and α which make this suggested solution satisfy the heat equation.
[10 marks] (ii) Show that if φ(x,y,z,t) is a solution of the heat equation, then so are
φ1(x,y,z,t) =
∂φ
∂t
Use this to find two new solutions of the heat equation.
[7 marks]
(iii) Use your results from parts (i) and (ii) to solve the heat equation with the initial condition
φ(x,y,x,0) = (1 + x)exp[ − (x2 + y2 + z2)] .
Hint: Solutions of the heat equation can be shifted in space or time.
[8 marks]
3. The displacement of a guitar string, y(x,t), obeys the partial differential equation
= V2
∂t2 ∂x2
where V is a constant. The two ends of the string are fixed, giving the boundary conditions
y(0,t) = 0, y(L,t) = 0 for all t .
(i) Use separation of variables to find the general solution of this PDE. [10 marks]
(ii) At t = 0 the guitar string is given an initial velocity proportional to x,
so that
y(x,0) = 0, = cx for 0 < x < L .
Find an expression for y(x,t) at later times.
[15 marks]
4. (i) Consider the two vector fields
(a) u = ,
(b) v = ,
Cx |
x2 + y2 + a2 , Cx |
x2 + y2 + a2 , |
where a and C are constants. Only one of these could be a magnetic field, the other could not. Say which of these fields can not be a magnetic field, justifying your answer.
[9 marks]
(ii) For that field in part (i) which could be a magnetic field, calculate the electrical current density, assuming there are no time-varying electric fields in this problem.
[10 marks]
(iii) From your answer to (ii), calculate the total electric current flowing through the disc
z = 0, x2 +y2 < 1 .
Check your answer by calculating the line integral of the magnetic field around the boundary of the circle.
[6 marks]
5. The temperature T(θ,t) in a metal ring obeys the diffusion equation
∂T ∂2T
∂t ∂θ2
where K is a positive constant. The angular coordinate θ will be chosen to run from −π to π. The temperature obeys the periodic boundary condition
T( −π,t) = T(π,t) .
The initial temperature around the ring is
0
π4 < θ ≤ π
(i) Use separation of variables to find all solutions of the partial differential equation with the form T(θ,t) = P(θ)Q(t)
[7 marks]
(ii) Use your solutions from part (i) to write down T(θ,t), the temperature of the ring for t > 0. What is the limiting temperature as t → ∞?
[18 marks]
6. An array of equally spaced parallel wires are placed in the z = 0 plane and given a charge. The wires run in the y-direction, The resulting electrical potential is
φ(x,y,z) = C ln[cosh(2πβz) − cos(2πβx)]
where C and β are constants.
(i) Calculate the electric field E, and verify that it satisfies ∇ · E = 0, except at the singularities of φ .
[10 marks]
(ii) Find the positions of the charged wires, and the spacing between them. Hint: When is φ singular?
[5 marks] (iii) Calculate the limit of the electric field as z → ∞ and as z → −∞ .
[5 marks]
(iv) Using Gauss’s Law, or otherwise, calculate the charge per unit length on the wires.
[5 marks]
2022-01-26