MA3432
Soln #8
Solution 8
Problem 1
(i) We want to consider a particle moving in circular motion in a uniform magnetic eld
B = B z . To nd the radius and frequency we can simply use the formulae from class or
Jackson, however let us briey recall some sali
MA3432
Soln #7
Solution 7
Problem 1
(i) We start from the expression derived in class for the power per unit solid angle emitted
by a particle moving in one dimension:
dP (t )
|v|2 sin2
q2
,
=
3 (1 cos )5
d
4c
(1)
which for the deccelerating particle is
MA3432
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) Show that Maxwells equations in vacuum imply that the E and B elds satisfy the wave
equation.
Soln: The Maxwell e
MA3432
Soln #6
Solution 6
Problem 1 (G)
Jackson 14.4
Solution: For the non-relativistic limit the power radiated per unit solid angle is given by the
formula,
dP
e2
=
|n (n )|2 .
d
4c
(1)
a) Here we are interested in a particle moving with
r(t) = a cos 0
MA3432
Soln #2
Solution 2
Problem 1
Consider the diffusion equation
2
=
t
(1)
where is a scalar function of x and t and is a constant.
A solution to the initial value problem (IVP) in unbounded space-time is of the form
d3 x G(x x , t)(x , 0)
(x, t) =
(2)
MA3432
Soln #3
Solution 3
Problem 1
Consider an arbitrary linear superposition of plan electromagnetic waves in vacuum,
E(x, t) =
1
2
d3 k
E (k)eikxi(k)t + c.c
(2)3
(1)
where the sum over is over two orthogonal polarisations.
Show the total, time average
MA3432
Soln #5
Soln 5
Problem 1 (SI)
Recall from last weeks homework that one can analyse a rotating charge distribution by calculating the time-dependent multipole moments and then use the formula for the sinusoidally
varying charge distributions for eac
MA3432
Soln #4
Soln 4
Problem 1 (SI)
Jackson 9.1 Consider a set of charges with charge density
(x) = (r, , 0 t) ,
(1)
i.e. rotating around the z-axis with angular frequency 0 . The time-dependent multipole moments are given by
q
m (t)
d3 x r Y (, )(r, , 0
MA3432
Soln #1
Solution 1
Problem 1
1. To prove conservation of the magnetic current we take the derivative of the corresponding equation of motion,
4
F =
J
c
(1)
and so using the antisymmetry of F we have that
J = J = 0
(2)
as required.
2. Now we sta