Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers
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126
THE
DIRAC
6-FUNCTION
(a)
Ax
co)
Figure
5.24
Discrete
Sum
Representation
of
an
Integral
area of nb
6
-
function
=
f(nAx)Ax
-
Ax
Figure
5.25
The Construction
of
f(y)
from
an
Infinite
Sum
of
&Functions
Equation 5.102 makes a very interesting statement:
Any
continuous function can be
viewed as the sum of an infinite number of 6-functions. Both
f(y)
and the
RHS
of
Equation 5.102 are functions of
y,
and
are
plotted in Figure 5.25 for finite Ax. In
this
figure, the 6-functions are located at
y
=
nAx and have an area f(nAx)Ax.
The function
f(y)
is generated by
this
sum of 6-functions
as
Ax
+
0,
i.e., as the
spacing between the 6-functions and their areas go to zero. Thus an infinite number
of infinitesimal area 6-functions, spaced arbitrarily close together, combine to form
the continuous function
f(y)!
EXERCISES FOR CHAPTER
5
1.
Simplify the integral
where
a
and
b
are real, positive constants.
EXERCISES
127
2.
3.
4.
Perform these integrations which involve the Dirac &function:
i.
J_9dx
6(x
-
1).
ii.
1:
dx (x2
+
3)6(x
-
5).
iii.
1:
dx x 6(x2
-
5).
iv.
/:5
dx x
6(x2
+
5).
r2v
v.
dx
~(COSX).
vi.
c’4
dx
x2
~(COSX).
d8(x
-
5)
10
viii.
/,,d.
(x2
+
3)
[
dx
]
Determine the integral properties of
the
“triplet” d26(t)/dt2
by
evaluating the
integrals
The function h(x)
is
generated from the function
g(x)
by
the integral
If
h(x)
is the triangular pulse shown below, find and plot
g(x).
-1
1
128
5.
Given that
THE
DIRAC
6-FUNCTION
find and plot the first and second derivatives of
f(x).
6.
An ideal impulse moving in space and time can be described by the function,
f(x,t)
=
106(x
-
vat),
where
v,
is a constant. Make a three-dimensional plot for
f(x,
t)
vs.
x
and
t
to
show how this impulse propagates. What are the dimensions of
vo?
How does
the plot change if
where
a,
is a constant? What are the dimensions of
a,?
7.
Consider the sinc function sequence:
&(t)
=
sin (nn)/(rx).
(a)
On the same graph, plot three of these functions with
n
=
1,
n
=
10,
and
(b)
Prove that the limit
of
the sinc sequence functions acts like a 6-function by
n
=
100.
deriving the relation:
As a first step,
try
making the substitution
y
=
nt.
You will need to use the
identity:
8.
The function G(cosx) can be written as a sum of Dirac 6-functions
n
Find the range for
n
and the values for the
a,
and the
xn.
so
that its charge density can be expressed as
9.
A
single point charge
qo
is located at
(1,
1,O)
in a Cartesian coordinate system,
pc(x,
y,
z)
=
qo
6(x
-
1)
wy
-
1)
&z).
EXERCISES
129
(a)
What is
pc(p,
8,
z),
its charge density in cylindrical coordinates?
(b)
What is
pJr,
8,
+),
its charge density in spherical coordinates?
10.
An infinitely long, one-dimensional wire of mass per unit length
A,
is
bent to
follow the curve
y
=
A
cosh
(Bx),
as shown below.
11
Find
an
expression for the mass
per
unit volume
p(x,
y,
z).
Express your answer
two ways:
(a)
As
the product of two &functions.
(b)
As
a
sum of two terms, each the product
of
two &functions.
Be
sure
to
check the dimensions of your answers.
An infinitely long, one-dimensional wire of mass per unit length
A,
is bent to
follow the line formed
by
the intersection of the surface
x
=
y
with the surface
y
=
z2,
as shown in
the
figure below. Find an expression for
pm(x,
y,
z),
the mass
per unit volume of the wire.
Z
Y
130
THE DIRAC &FUNCTION
12.
An
infinitely long, one-dimensional wire with
a
constant mass per unit length
A,
is bent to follow the curve y
=
sinx
in
the
z
=
0
plane.
Determine the mass density
p,(x,
y,
z)
that describes
this
mass distribution.
13.
A wire of mass per unit length
A,
is bent to follow the shape of a closed ellipse
that lies in the xy-plane and is given by the expression
x2
+
2y2
=
4.
Express
p,(x,y,z),
the mass per unit volume of
this
object, using Dirac
6-
functions.
Show
that your expression has the proper dimensions. There
is
more
than one way to express the answer to
this
problem. Identify the most compact
form.
14.
An infinite, one-dimensional bar of mass per
unit
length
A,
lies along the line
y
=
m,x
in the
z
=
0
plane.
Y
(a)
Determine the mass per unit volume
p(x,
y,
z)
of
this
bar.
(b)
Now consider the situation where
the
bar is rotating about the z-axis at
a
constant angular velocity
o,
so
that the angle the bar makes with respect to
the x-axis is given by
8
=
mot,
as shown below. Find
an
expression for the
time-dependent mass density
p(x,
y,
z,
t).
EXERCISES
131
15.
A charge
Q,
is evenly distributed along the x-axis fromx
=
-L,/2
to
x
=
L,/2,
as shown below.
(a)
Using the Heaviside step function, what is the charge density
p&,
y,
z),
(b)
What is this charge density in cylindrical coordinates?
expressed using Cartesian coordinates?
Z
16.
Using &functions and the Heaviside step function, express the charge density
p,(f)
of a uniformly charged cylindrical shell of radius
r,
and length
Lo.
The
total charge on the surface of
the
shell is
Q,.
17.
An infinite, two-dimensional sheet with mass per unit area
cr,
is bent to follow
the surface
y
=
x3.
(a)
Make
a
plot
of
the curve made by the intersection of this sheet and the plane
given
by
z
=
0.
(b)
Determine the mass per unit volume
pm(x,
y,
2).
18.
Express the mass density
pm(p,
8,
z)
of a conical surface that is formed by cutting
a
pie-shaped piece from
an
infinite, uniform two-dimensional sheet of mass per
unit area
cr,
and joining the cut edges. The conical surface that results lies
on
the
surface
p
=
a,
z
where
(p,
8,
z)
are the standard cylindrical coordinates.
Z
132
THE
DIRAC
&FUNCTION
19.
An infinite, two-dimensional sheet with mass per unit area
a,
is bent to follow
the surface
xy
=
1
in a Cartesian coordinate system.
(a)
Using the hyperbolic coordinates developed in Exercise
13
of Chapter
3,
express the mass density
p,(u,
u,
z)
for
this
sheet.
(b)
Using the equations relating the coordinates, convert your answer to part (a)
above to Cartesian coordinates.
(c)
Now, working from scratch in a Cartesian system, obtain
p,(x,y,z)
by
requiring that
this
density function take the volume integral over all space to
a surface integral over the hyperbolic surface.
20.
Express the mass density
p,@)
for a spherical sheet of radius
r,,
with constant
mass per unit area
a,.
21.
A
dipole electric field
is
generated outside the surface of a sphere, if the charge
per unit
area
on the surface of that sphere is distributed proportionally to cos(
0).
If the sphere has a radius
r,
and there is a total charge
of
+Qo
on the upper
hemisphere and
-Q,
on the lower hemisphere, what is the expression for the
charge density
pc(r,
0,4)
in spherical coordinates?
+
+++
;.-.:
/
+
/
/
+
+
22.
In a two-dimensional Cartesian coordinate system, the mass density
pm(Q
of a
pair of point masses is given by
EXERCISES
133
iii.
S_:
1”
cixldx2
[F
*
FI
pm(x1,
x2).
iv.
1:
1:
dxld~2
IF
I;]
pm(x1,xd.
23.
Prove that the monopole and quadrapole moments
of
any &pole (“physical” or
24.
A
quadrapole charge distribution consists
of
four
point charges in the xlx2-plane
“ideal”) are zero.
as shown below.
(a)
Using
Dirac &functions express the charge density,
pc(xl,
x2,
x3),
of this
(b)
The quadrapole moment
of
this
charge distribution is a second rank tensor
distribution
.
given by
- -
Q
=
Qtj
el@,.
The elements
of
the quadrapole tensor are given by the general expression
Q,
=
Im
dxl
/:
dx2
1;
dx3
p&l,
x2,
x3)
[~xJ,
-
(-Q%)~J]
--m
where
a,,
is
the Kronecker delta. In particular,
Q22
=
/=
dxl
Irn
dx2
Sy
dx3
pc(xl,
x2,
x3)
[kxz
-
(x?
+
~2’
+
x?)]
.
Evaluate all the elements
of
the
quadrapole tensor for the charge distribution
shown in the figure above.
-x
-‘x
--m
(c)
Does this charge distribution have
a
dipole moment?
(d)
Find the coordinate system in which
this
quadrapole tensor is diagonal.
25.
An
ideal quadrapole has a charge density
p,(x,
y, y)
that
is
zero everywhere
except at the origin. It has zero total charge, zero dipole moment, and
a
nonzero
quadrapole moment.
Express the elements
of
Q
in this system.
THE
DIRAC
6-FUNCTION
134
(a)
Show that
if
pc(x,
y,
t)
has the
form
it
satisfies the above requirements
for
an
ideal quadrapole. Evaluate
[?I
SO
that the elements
of
the
quadrapole moment tensor
for
this distribution are
the same as the quadrapole elements in Exercise
24.
(b)
Determine
the
electric field produced
by
this ideal quadrapole.
INTRODUCTION
TO
COMPLEX VARIABLES
One of the most useful mathematical skills to develop for solving physics and en-
gineering problems is the ability to move with ease in the complex plane.
For
that
reason, two chapters of this book are devoted to the subject of complex variable
theory. This first chapter is primarily restricted to the fundamentals of complex vari-
ables and single-valued complex functions. Some knowledge of complex numbers is
assumed.
6.1
A
COMPLEX NUMBER REFRESHER
Complex numbers are based on the imaginary number
“i”,
which is defined as the
positive square
root
of
-
1
:
In general, a complex number
z
has a real and
an
imaginary part,
z
=
x
+
iy,
(6.2)
where
x
and
y
are both real numbers. If
y
is zero
for
a particular complex number,
that number is called pure real. If
x
is
zero, the number
is
pure imaginary.
6.1.1
The Complex Plane
A
complex number can be plotted as a point in the
complex
plane.
This
plane is
simply a two-dimensional orthogonal coordinate system, where the real
part
of the
135