Physics 508
Handout 8
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 8
Prof. M. Stone
2117 ESB
University of Illinois
1) Pantograph Drag: A high-speed train picks up its electrical power via a pantograph
from an overhead line. The lo

Physics 508
Handout 6
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 6
Prof. M. Stone
2117 ESB
University of Illinois
1) Flexible rod again: A flexible rod is supported near its ends by means of knife edges
that constrain its positio

Physics 508
Handout 0
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 0
Prof. M. Stone
2117 ESB
University of Illinois
Revision and Warm-up exercises
These problems are designed to reactivate your mathematical skills after a relaxing

Physics 508
Handout 1
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 1
Prof. M. Stone
2117 ESB
University of Illinois
1) Fermats principle: According to Fermats principle, the path taken by a ray of light
between any two points makes

Solutions to Homework Set 2
1) Bent bars: First some elementary calculus: the curvature , and the radius R of the
osculating circle, at a point (x, y) on a curve y(x) are given by
=
1
d
1
y
d tan1 y
=
=p
=
.
R
ds
dx
(1 + y 2 )3/2
1 + y 2
Here the arc le

Physics 508
Handout 10
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 10
Prof. M. Stone
2117 ESB
University of Illinois
1) Dielectric Sphere: Consider a solid dielectric sphere of radius a and permittivity . The
sphere is placed in a

Solutions to Homework Set 7
1) Critical mass: We expand
n(x, t) =
X
am (t) sin
mx
L
m=1
and also
mx
X 4
=
sin
,
m
L
m,odd
,
0 < x < L.
Substituting in the given equation, and using the linear independence of the sine functions,
then gives
Dm2 2
4
a m

Physics 508
Handout 3
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 3
Prof. M. Stone
2117 ESB
University of Illinois
1) Test functions and distributions:
a) Let f (x) be a smooth function.
i) Show that f (x)(x) = f (0)(x). Deduce th

Solutions to Homework Set 3
Test functions and distributions: For part a) we take any test function (x) and look
at
Z
(, f + f )
(x) cfw_f (x) (x) + f (x)(x) dx
Z
=
cfw_[ (x)f (x) (x)f (x)](x) + (x)f (x)(x) dx
= (0)f (0),
and compare it with
(, f (0) )

Solutions to Homework Set 6
1) Flexible rod again: This is a somewhat tedious, but ultimately rewarding exercise.
a) Look back at your solution for homework set 4, where you showed that
Z 1
Z 1
(4)
(3)
(3) 1
u v dx = [u v (u ) v + (u ) v (u ) v]0 +

Physics 508
Handout 9
Oct 10th 2014
Mathematical Methods in Physics I
Course Material
Homework 9
Prof. M. Stone
2117 ESB
University of Illinois
1) Conducting Strip: A thin insulated conducting strip of width 2a extends infinitely far
in the z direction. I

Solutions to Homework Set 0
Differential calculus: The point of the exercise was to make sure that you know how to
differentiate integrals with respect to their limits:
Z b
Z
d
d b
f (x) dx = f (a),
f (x) dx = f (b),
da a
db a
and in general
d
dt
Z
b(t)
a

Solutions to Homework Set 4
1) Linear Differential operators:
a) Integrating by parts gives us
hu|Lviw
b
d
=
wu i v dx
dx
a
Z b
i d
b
wu v dx
= [iwu v]a +
w
w dx
a
[Q]ba
+
hL u|viw .
Z
Therefore the formal adjoint is
L =
d
i d
w i + i(ln w) ,
w dx
dx
an

Solutions to Homework Set 5
1) Missing state:
The continuum eigenfunctions are are
k = (ik + tanh x)eikx ,
so
k (x) =
(k i)eikx /i
x 0,
(k + i)eikx /i x 0.
It now helps to draw a phasor diagram
Im
i
k
Re
from which we see that (k) = tan1 (1/k) and A = i

Physics 508
Handout 7
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 7
Prof. M. Stone
2117 ESB
University of Illinois
1) Critical Mass: An infinite slab of fissile material has thickness L. The neutron density
n(r) in the material ob

Physics 508
Handout 4
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 4
Prof. M. Stone
2117 ESB
University of Illinois
1) Linear differential operators:
= id/dx. Find the formal adjoint
a) Let w(x) > 0. Consider the differential oper

Physics 508
Handout 2
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 2
Prof. M. Stone
2117 ESB
University of Illinois
1) Elastic Rods. The elastic energy per unit length of a bent steel rod is given by 21 Y I/R2 .
Here R is the radiu

Solutions to Homework Set 1
1) Snellius law: Parts a) and b) are trivial, so I will not write out the solutions. Part c)
has two sub-parts:
i) Setting the variation of F1 [y] to zero gives
!
d
y
n(x) p
= 0.
dx
1 + y 2
Now elementary calculus tells us that

Physics 508
Handout 10
Oct 10th 2014
Mathematical Methods in Physics I
Course Material
Homework 11
Prof. M. Stone
2117 ESB
University of Illinois
Here are some optional problems on integral equations. They are taken verbatim from
Paul Goldbarts homework s

Physics 508
Handout 5
Fall 2014
Mathematical Methods in Physics I
Course Material
Homework 5
Prof. M. Stone
2117 ESB
University of Illinois
1) Missing State: In Homework Set 4 you found that the Schrodinger equation
d2
2
2 2 sech x = E
dx
has eigensolut