MA3431
Sample Exam
Sample Exam Solutions
Problem 1
(i) A general expression for the scalar potential with azimuthal symmetry is
A r + B r
(r, , ) =
1
(1)
P (cos ) .
=0
(ii) As there are no charges at the origin we have B = 0 for all and hence
(a, , ) =
(2
Orthogonal functions
Given a real variable over the interval (a, b) and a set of real or complex functions Un (),
n = 1, 2, . . . , which are square integrable and orthonormal
b
Un ()Um ()d = n,m
(1)
a
if the set of of functions is complete an arbitrary s
MA3431
Solution #3
Solution 3
Problem 1
Taking the second derivative of G(x; x ) we nd that
cn n2 2 sin(nx) sin(nx )
G (x; x ) =
(1)
n=1
and combining with the sine-series representation of the delta-function
(x x ) = 2
sin(nx) sin(nx )
(2)
n=1
8
we nd
MA3431
Solution #4
Solution 4
Problem 1
(3pts) The scalar potential at spatial innity is that due to the external electric eld.
Let us choose our coordinate so that the external eld lies along the z-axis then using
E = we have
|r = Eext z = Eext r cos .
(
MA3431
Solutions #2
Solutions 2
Problem 1
Consider Greens theorem with (y) = G(x, y) and (y) = G(x , y) then
y
= 4 (3) (x y) ,
y
= 4 (3) (x y) ,
(1)
where the subscript denotes that y is the variable w.r.t. which we are taking the derivative.
Hence
G(x ,
MA3431
Soln #5
Soln 5
Problem 1
Consider the vector potential
A(x) =
0
g
4
dz z (x z z )
.
3
|x z z |
(1)
This is the Dirac expression for the vector potential of a magnetic monopole located at the
origin (and its associated Dirac string along the negativ
MA3431
Soln #7
Solution 7
Problem 1
(i) For the relativistic particle action:
S = mc
g
d
dx dx
d d
(1)
with the proper time, show that using the Euler-Lagrange equations that one gets
d2 x
=0.
(2)
d 2
Solution: As mentioned in class we apply the variatio
MA3431
Soln #6
Solution 6
Problem 1
Consider the Lorentz transformation:
x0 = (x0 x1 ) , x1 = (x1 x0 ) , x2 = x2 , x3 = x3
(1)
Find the Lorentz transformations for the A00 , A01 , and A03 components of a contravariant
antisymmetric rank two tensor,
A = A
MA3431
Sample Exam
Sample Exam
Policy
Credit will be given for the best three out of four. All problems have equal weight.
Problem 1 (SI)
(i) Write down the general expression for a scalar potential, (r, , ), with azimuthal symmetry as a series of Legendr
MA3431
Soln #9
Solution 9
Problem 1
Starting with the Proca Lagrangian discussed in class,
LP =
1
2
F F +
A A
16
8
(1)
(i) Calculate the canonical stress-energy tensor, T .
Soln: The canonical stress energy tensor is
T
LP
g LP
( A )
1
1 2
1
g F F
g A
MA3431
Solution #8
Solution 8
Problem 1
(i) The transformed Lagrangian is
L =
=
1
2
1
2
n
n
(Mij j ) (Mik k ) V (
i,j,k=1
n
(Mij j )(Mik k )
n
n
j k V (
Mij Mik
i=1
j,k=1
(1)
i,j,k=1
n
(
Mij Mik )j k )
(2)
j,k=1 i=1
As the transformation is orthogonal w
MA3431
Solution #1
Solution 1
Problem 1
We provide a proof at the level of rigour of, say "Mathematical Methods for Physics and Engineering", Riley, Hobson and Bence. See your favourite multi-variable calculus text (e.g. Apostol
Calculus Vol. 2) for a mor