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Drill Problems
PHY2206 (Electromagnetic Fields)
Drill Problems
1
© Copyright CDH Williams
University of Exeter 1996-8
CW980505/1
Drill Problems
Introduction
Drill problems are short exercises to help you to practise the techniques discussed in the lectures. To
derive real benefit from these questions you should try to do them as soon as possible after the lecture
that covers the material. If you have difficulty doing these problems it might indicate that you are not
understanding what is going on in the lectures; first read the textbook and check the hints on the
separate handout and if you are still stuck seek help from your study group or tutor without delay.
Problems marked with an asterisk * are more interesting, but much harder than the usual standard;
there is no need to worry if you cant do them!
Notation
Vectors are conventionally indicated by boldface type in printed material, e.g. B. In hand-written
materials under- or over-lining is used, e.g. B, B, or r<i>B. The unit vector in the direction B is printed B
and hand-written
B or
B. Some authors use i, j, k for the unit vectors x, y, z in the Cartesian
coordinate system; avoid this notation as j is more often used for the current density field. PHY2206 (Electromagnetic Fields)
Drill Problems
2
DP1
In rectangular Cartesian coordinates (x,y,z) a scalar field has the form =
xyz
2 x
2
yz .
Calculate
at the point P having coordinates (1,2,3).
DP2
The vector r is defined in spherical polar coordinates by r
=
rr and in Cartesian
coordinates by r
=
xx
+
yy
+
zz .
(a)
Express the function =
A r , where A is constant, in terms of (a) r and (b) x,y,z.
(b)
Use the definitions of given on the Vector Analysis Formulae handout to find
in both coordinate systems. Are the expressions equal, and if so, why?
DP3
A frame of reference S is rotated by 45° clockwise about the z</b>-axis of another frame S.
This means that the frames are related by the transformations
x
=
1
2 x
+
y
(
)
;
y
=
1
2 x
+
y
(
)
;
z
=
z.
Find the 3
×
3 matrix that transforms the components in S of a vector C into its components
in S . If the components of C in S are (0.7,2,1.5) what are its components in S?
DP4
What is the divergence of the vector fields when a and b are constants:
(a)
G
=
ax
2
x
+
byy
+
z in Cartesian coordinates?
(b)
H
=
ar
2
r
+
br + in spherical polar coordinates?
DP5
Charge is moving in a system causing a current density j
=
ax
2
x
+
byy. What is happening
to the charge density at the point (2 x
+
3y) m if the parameters have the values:
a
=
12 Am 4
;
b
=
4 Am 3
.
DP6
What is the net electric flux through the surface of an empty 1
i
m cube, centred on the
origin, in an electric field (90 x
+
75z) Vm
-1
? PHY2206 (Electromagnetic Fields)
Drill Problems
3
DP7
A cube of side length L, centred on the origin of the coordinates, is bounded by the closed
surface S. Find the total flux of the vector field F
=
xx
+
yy
+
zz out of the surface S by:
(a)
Evaluating the flux through each face of the cube separately.
(b)
Using the divergence theorem.
DP8
A circle of radius R in the z
i
=
i
0 plane is defined by x
2
+
y
2
=
R
2
.
(a)
Obtain an expression for the vector line element d<b>l of the circle in terms of d<i>x the
associated differential change in x.
(b)
Evaluate the closed line integral F d<b>l around the circle, for the case
F
=
yx xy
x
2
+
y
2
(
)
.
DP9
The velocity field of a fluid in cylindrical polar coordinates is v r, ,
z
(
)
= r
where is a
constant. What are the units of and how could such a flow be established in water?
Calculate (a) the curl of the velocity field, and (b) the circulation around a circular path of
radius r around the z</i>-axis. How are the two quantities related?
DP10
Use the Cartesian form of to find expressions for Curl F
= ×
F
for the fields
(a) F
=
yx xy
x
2
+
y
2
(
)
3 2
and
(b) F
=
yx xy
x
2
+
y
2
.
DP11
Use the Cartesian form of to show that, when A is a vector field and f is a scalar field:
(a)
× ×
A
(
)
=
A
(
)

2
A
, (b)
×
A
(
)
=
0, (c)
×
f
( )
=
0.
DP12
The potential r
( )
at a position r due to a point charge q at position r
is r
( )
=
q
4 0
1
r

r .
Calculate
and hence find the electric field E at any position r.
DP13
Find an approximation for V r
( )
that is valid when d
<<
r and
V r
( )
=
1
r d . DP15 (a) Find the monopole and dipole moments of this arrangement
of charges.
(b) Does it have a non-zero quadrupole moment?
(c) At what distance has the maximum dipole contribution to its
E
field dropped to 10% of the monopole field?
DP16
What are the
SI
units of D, E, P and 0
?
DP17
If x
( )
is the Dirac delta function evaluate
(a)
sin x
( ) +
x 0.34
(
)
d<i>x
(b)
sin x
( )
d
d
x x 0.34
(
)






d<i>x + .
DP18
Two charges, +q and q are positioned at +d/2 and d/2 respectively. Calculate the change
in the potential energy of this system when a uniform field E is applied and hence confirm
the expression stated in the lectures for the potential energy of a point dipole p in a uniform
electrostatic field.
DP19
Two pieces of dielectric material are bonded together with the interface lying in the plane
x
i
=
i
0. Find the equivalent bound surface charge density at the interface if the polarisations
in the materials are:
P x
<
0
(
)
=
10 x Cm 2
and
P x
>
0
(
)
=
3x Cm 2
.
DP20
A sphere of dielectric material is uniformly polarised in the z</i>-direction, i.e. P
=
z</b><i>P . Find
the equivalent bound surface charge density ( , ) at the point on its surface specified by
spherical polar coordinates ( , ) . PHY2206 (Electromagnetic Fields)
Drill Problems
5
DP21
Find the field at the centre of the sphere described in the previous problem by using a
surface integral to calculate the field due to the charge distribution ( , ).
DP22
If liquid helium has a density i
=
i
145
i
kgm
3
and relative permittivity r
i
=
i
1.0556 what is the
the atomic polarisability of helium?
d/2
d/2
d
+ DP23
A slab of dielectric, relative permittivity r
, is placed between the plates of a
capacitor as shown. A potential
difference V is maintained between the
plates. Ignore edge effects and calculate
the values of D, E and P (a) inside the
dielectric (b) in the gaps between the
dielectric and the plates.
DP24
Calculate the charge density on the plates of the capacitor described in the previous
problem. Multiply this by the area of each plate A to find the total charge and hence deduce
the capacitance of the structure.
DP25
Deduce Poissons equation for electrostatics from the differential equations relating the
electric-field intensity E to the charge density and the potential .
DP26
A long coaxial cable consists of an inner wire, radius r, inside a metal tube of internal
radius R, the space in between being completely filled with a dielectric of relative
permittivity r
. Use Gausss law to find the electric displacement flux D and hence show
that the cable capacitance is 2 r 0
(
)
ln R / r
(
)
i
.
DP27
In free space D
= 0
E
so the expression for the energy associated with a field E in a
region V simplifies to
U
=
1
2 0
E
2
d
3
r
V .
Use this expression to find the electrostatic energy stored in terms of the plate area A,
separation d and the internal electric field strength E in a parallel-plate air capacitor. Hence
find the energy stored in a capacitor in terms of the potential difference and its capacitance. PHY2206 (Electromagnetic Fields)
Drill Problems
6
DP28
Six identical spherical drops of mercury are each charged to 10
i
V above earth potential and
then made to coalesce into a single spherical drop. If the drops were initially widely
separated, what is the potential of the new drop? Has the electrostatic energy changed?
DP29
Use Laplaces equation in spherical polar coordinates to find the potential as a function of
radius in the space between two concentric conducting sphere of radius a and b and at
potentials V
1
and V
2
respectively.
DP30*
Find an expression for the potential at a perpendicular distance a from the centre of a
straight line of total charge Q spread along a length l. Hence show that the computer
program that calculated the values in table 3-3 of Reitz Milford and Christy must have had
a bug in it.
DP31*
Cork is a dielectric with relative permittivity 3.6 and density of 2.5
i
×
i
10
2
i
kgm
3
. A small
sphere of it is suspended a distance d vertically below a point electric charge of 10
7
i
C.
Estimate the value of d at which the cork sphere will be picked up by the point charge.
DP32
A 15
i
pF parallel-plate air capacitor is charged by connecting it to a 75
i
V
PSU
. How much
work must be done to double the separation of the plates of the capacitor (a) with the
PSU
disconnected and the capacitor fully charged (so the charge stays constant), and (b) with
the
PSU
connected (so the potential difference stays constant)?
DP33*
A parallel plate capacitor has rectangular plates of length a and width b spaced d apart
connected to a battery of
EMF
V. A slab of dielectric of relative permittivity r
that would
just fill the space between the capacitor plates is slid part way between them. Show that the
force pulling it into the plates is 0 r 1
(
)
V
2
b 2<i>d . Why is the answer independent of a?
DP34
A conducting sphere of radius R is placed in a uniform electric field E
0
. Check that the
potential at a point (r, ) outside the sphere is given by
= E
0
r cos 1 R / r
(
)
3
{
}
as follows: (a) Does the potential satisfy Laplaces equation? (b) Is the potential the same at
all points on the surfac