Download Homework Section 1

Homework Section 1
1) Analytically prove the following. (Do this for arbitrary dimension)
a) The commutative law: A + B = B + A
b) The Associative Law: A + (B + C) = (A+B) + C
c) A + B = C if and only if B = C - A
d) A + 0 = A and A - A = 0
e) Scalar product is commutative [A•B=B•A] and
f) Scalar product is distributive [A•(B+C)=A•B+A•C].
2) Prove that the area of a parallelogram with sides A and B is |A x B|. Note that
the surface area has a direction associated with it.
3) Prove that the volume of a parallelepiped with side A, B and C is A•(B x C).
4) Find the magnitude of the following vectors.
a) (1, 4)
b) (4, 3, 0)
c) (0, -1, 1)
d) (6, 1, 0, -1, 2)
5) Which of the following vectors are unit vectors?
a) (1, 0)
b) (1, 1/2)
c) (1, -1)
d) (1/√2, -1/√2)
e) (1/2, 0, √3/2)
f) (1, 0, 0, 0)
6) Express the vector A = (2, 7) as a linear combination of the vectors:
a) B1 = (2, 4), B2 = (-1, 3).
b) C1 = (4, 4), C2 = (5, 5).
7) Express the vector A = (2, -1, 3) as a linear combination of the vectors:
a) B1 = (2, 4, 1), B2 = (3, 7, 1).
b) C1 = (1, 0, 0), C2 = (0, 1, 0), C3 = (0, 0, 1).
c) D1 = (2, 4, 1), D2 = (3, 7, 1), D3 = (-1, 2, 2).
8) Show that in a 3-dimensional space, a set of three vectors A, B and C are
linearly independent if and only if
a1 a2 a3
b1 b2 b3  0 ;
c1 c2 c3
(Linear independence requires that: a A + b B + c C = 0 if and only if a, b, c
= 0).
9) Determine if the vector R1 = (2, 4, 1), R2 = (0, 3, -1) and R3 = (2, 1, 1) are
linearly independent.
10) Consider a system of n electric charges, e1 through en. Let ri be the position
vector of charge ei. The dipole moment of the system of charges is defines as
n
p   ei ri
i 1
and the center of the charge of the system is
n
R
n
e
i
i 1
e r
i i
p

i 1
n
e
i
i 1
where
n
e
i 1
i
0
The system is called neutral is
n
e
i 1
i
0
a) Show that the dipole moment of a neutral system is independent of the
origin.
b) Express this moment in terms of the centers of the systems of negative and
positive charges making up the original system.
11) Find the scalar product of the following vectors.
a) (2, 3)•(1, -1)
b) (4, 1)•(6, -5)
c) (1, 2, -3)•(-1, 1, 2)
d) (2, 4)•(1, 5, 3)
e) (0, sin, 1, 3)•(2, 4, -2, 1)
f) (sin(t), cos(t))•( sin(t), cos(t))
12) Find the angles that the vector (2, 4, -5) makes with the coordinate axes.
13) Find the projection of the vector (2, 5, 1) on the vector (1, 1, 3).
14)
a) Using the dot product, prove the law of cosines.
b) Let U1 and U2 be two vectors in the x-y plane with angles  and 
between U1 and x and U2 and x. Using the dot product show that cos(-)
= cos()cos()+sin()sin()
15) Determine the value of  such that A and B are perpendicular and C and D
are perpendicular.
a) A = (2, 3, 1), B = (4, 2, 4)
b) C = (2, 4, 3, 1), D = (, 2, -1, 2)
16) Let A = (-1, 2, 4), B = (3, 2, 7). Find the unit vector perpendicular to the
plane determined by A and B.
17) The force F = (2, 3, 1) is applied to an object which move along a vector r =
(1, 4, 1). What is the work done?
18) Determine the magnitude, phase angle, real and imaginary parts of the
following
a) 3+i
b) 3
c) 3ei2/3
d) 2(cos (/6) +i sin (/6))
e) 345°
19) A Force F = 2i - 3j + k acts at the point (1, 5, 2). Find the torque due to F
about
a) The origin
b) The y axis
c) The line x/2=y/1=z/(-2)
Del operator questions
20) Find the gradient of w=x2y3z at (1, 2, -1)
21) Find the gradient of
Compute the divergence and the curl of each of the following Vector Fields.
ˆ
22) r  z ˆi  y ˆj  x k
23) r  x2 ˆi  y2 ˆj  z2 kˆ
2
Calculate the Laplacian   of each of the following
2
2
24) x  y
25) x 2  y 2  z 2 
1/ 2

26) For r  x 2  y2  z 2
1
r̂
  2
r
r

1/2
, Prove
ˆ , evaluate    r     r̂ .
27) For r  x ˆi  y ˆj  z k
 
r
Divergence, Stokes and Green’s Theorem Problems
2
2
28) Evaluate the integral  x  y dx  2xydy along each of the following


paths from (0,0) to (1,2)
a) y = 2x2
b) x = t2, y = 2t
c) y = 0, for x = 0 -> 1 and then x = 1 for y = 0 -> 2.
29) Evaluate the integral  xydx  x dy  along each of the following paths –
each path.
a) a) (0,0) to (1,2)
b) b) (0, 0) to (3, 0) to (1,2).
30) Determine if the following force fields are conservative. Then determine a
scalar potential for each field.
a) F  ˆi   zˆj   ykˆ
2
b) F  z sinh yˆj  2z cosh y kˆ
31) Which, if either, of the following force fields is conservative? Calculate the
work done moving a particle around a circle of x = cos t, y = sin t in the x-y
plane.
a) F  y ˆi  x ˆj  z kˆ
b) F  yˆi  x ˆj  zkˆ
Explain why you have gotten these answers.
32) In spherical coordinates, show that the electric field E of a point charge is
conservative. Determine and write the electric potential  in rectangular
(cartesian) and cylindrical coordinates. Find E   using both cartesian
and cylindrical coordinates and show that the results are the same as in
spherical coordinates.
P(x, y)
33) Derive  P(x,y) dx  
dxdy using methods similar to that used in
y
c
class.
34) Evaluate
A
 x dy  xydx, around the curve (1, 0) to (4, 0) to (4, .5) along
2
c
y  1 x to (1, 1).
35) For a simple closed curve C in a plane show by Green’s theorem that the area
enclosed is A  12  xdy  ydx .
c
36) Find the area inside the curve x  a cos, y  bsin , 0    2 .
Evaluate the following three problems using either a surface or a volume integral,
whichever is easier.
37)    Vd ; V  x 2  y 2  x ˆi  yˆj , over the volume bounded by x2 + y2 ≤


V
4, 0 ≤ z ≤ 5. (Remember the top and bottom!)
38)  V  nd ; V  x ˆi  yˆj  zkˆ , over the surface of the cone with base
A
2
x  y  16 and vertex at (0,0,3).
39)    Fd ; F  x 3  y2 y ˆi  y 3  2y 2  yx ˆj  z 2 1kˆ , over the unit
2


V
cube in the first octant.
Using either Stoke’s Theorem or the Divergence Theorem evaluate each of the
following.
40)  V  nd ; V  2xy ˆi  y 2 ˆj  z  xy kˆ , where  is a tin can defined by
A
x2 + y2 ≤ 9, 0 ≤ z ≤ 5. (Remember the top and bottom!).
41)  (  V)  nd ; V  x  x 2z ˆi  yz 3  y 2  ˆj  x 2y  xz kˆ , where  is
A
any surface with a bounding curve entirely in the x-y plane.
42)  (  V)  nd ; V  x 2 y ˆi  xz kˆ , where  is the closed surface of the
A
ellipsoid
1 = x2/4 + y2/9 + z2/16.
Electro and Magneto static point source
43) Determine the electric potential of a point charge from the electric field
q r̂
E
4 r 2
44) Determine the magnetic potential of a current element.
 dI  r̂
B
4 r 2
45) Need to add A cross B = magnitude times sin of angle
Homework Section 2
Static electric fields using Coulomb’s Law – notice that symmetry is
lacking in most of these problems.
Line charges
(These might approximate what you would find on a line if it were exposed to an
external charge. Also note that these represent – as best I can tell – all of the
possible problems of this form that one can solve analytically.)
1) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
l  0
2) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
z a
 0

l   1
z a
 0 z

3) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
z a
 0

l  
1
z a
 0 z 2

4) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
z a
 0

l  
1
z a
 0 z 3

5) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
z a
 0

l  
1
z a
 0 z m

6) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
z a
 z
l   0
z a
 0
7) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
 0 z 2 z  a
l  
z a

 0
Surface charges
(These might approximate what you would find on a surface if it were exposed to an
external charge. Again note that these represent – as best I can tell – all of the
possible problems of this form that one can solve analytically.)
8) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
 s  0
9) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
 0 r  a

s   1
 0 r r  a
10) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
ra
 0

s   1
 0 r 2 r  a
11) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
ra
 0

s   1
 0 r 3 r  a
12) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
ra
 0

s   1
 0 r m r  a
13) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
 r r  a
s   0
 0 ra
14) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
 0 r 2 r  a
s  
ra
 0
Static electric fields using Gauss’ Law – notice the symmetry in these
problems.
15) Use cylindrical coordinates to calculate the electric field in the x-y plane for a
line charge where the charge density is
l  0
16) Use cylindrical coordinates to calculate the electric field on the z-axis for a
surface of charges on the x-y plane where the charge density is
 s  0
Cylindrical Volume charges
(These might approximate what you would find in a volume of a material. Under
some conditions it might be an insulator with charges distributed around the
volume, in others it might be a wire (or two) with charge carriers near surfaces.)
17) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r  a
v   0
- in cylindrical coordinates
0 ra
18) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 1
ra

v   0 r
- in cylindrical coordinates
 0
ra
19) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 0
ra

1

v   0 2 a  r  b - in cylindrical coordinates
r

 0
rb
20) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 0
ra

1

v   0 m a  r  b - in cylindrical coordinates
r

 0
rb
21) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r r  a
v   0
- in cylindrical coordinates
 0 ra
22) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r 2 r  a
v   0
- in cylindrical coordinates
ra
 0
23) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
0 r m r  a
v  
- in cylindrical coordinates
ra
 0
24) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
0

v   0 a  r  b - in cylindrical coordinates
0
br

25) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0

v   0 r a  r  b - in cylindrical coordinates
 0
br

26) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 m
v   0 r a  r  b - in cylindrical coordinates
 0
br

27) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 1
v   0
a  r  b - in cylindrical coordinates
 r
br
 0
28) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 1
v   0 m a  r  b - in cylindrical coordinates
 r
br
 0
Spherical Volume charges
(These might approximate what you would find in a volume of a material. Under
some conditions it might be an insulator with charges distributed around the
volume.)
29) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r  a
v   0
- in spherical coordinates
0 ra
30) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 1
ra

v   0 r
- in spherical coordinates

ra
 0
31) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 1
ra

v   0 r 2
- in spherical coordinates
 0
ra
32) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 0
ra

1

v   0 2 a  r  b - in spherical coordinates
r

 0
rb
33) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r r  a
v   0
- in spherical coordinates
 0 ra
34) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
 r 2 r  a
v   0
- in spherical coordinates
0
r

a

35) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
0 r m r  a
v  
- in spherical coordinates
ra
 0
36) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
0

v   0 a  r  b - in spherical coordinates
0
br

37) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0

v   0 r a  r  b - in spherical coordinates
 0
br

38) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 m
v   0 r a  r  b - in spherical coordinates
 0
br

39) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 1
v   0
a  r  b - in spherical coordinates
 r
br
 0
40) Calculate the electric field EVERYWHERE for a volume of charges where
the charge density is
ra
 0
 1
v   0 m a  r  b - in spherical coordinates
 r
br
 0
Physically realistic systems
Need to add problems
Capacitors etc
Static Magnetic fields using Biot-Savart Law – notice that symmetry is
lacking in most of these problems.
Line currents
(These might approximate what you would find on a few loops of wire. There
perhaps a few more problems that can be solve analytically, but not many. )
41) Use Cartisean coordinates to calculate the magnetic field on the z-axis for a
line current where the charge density is
x  a; b  y  b
 I 0 ŷ
 I x̂ a  x  a; y  b
 0
Il  
I 0 ŷ x  a; b  y  b
 I 0 x̂ a  x  a; y  b
42) Use cylindrical coordinates to calculate the magnetic field on the x-y plane for
a line current where the charge density is
 I 0 ẑ r  0
Il  
 0 r0
43) Use cylindrical coordinates to calculate the magnetic field on the z-axis for a
line current where the charge density is
 I 0̂ r  a
Il  
I 0̂ r  b
44) Use cylindrical coordinates to calculate the magnetic field on the z-axis for a
line current where the charge density is
 I 0̂
ra
Il  
2I 0̂ r  2a
45) The magnetic field around a strip cable (2 parallel wires) can be examined if
one considers where the current is flowing. Assume that for the specific strip
cable that the wires are filamentary strands of metal (typically Cu) and that the
net flow on one wire is balanced by an opposite net flow on the other wire.
First show that the surface current density on the wires should be given by:
xa
 I 0 x  a ẑ

I  I 0 x  a ẑ
x  a - in cartisean coordinates

0
elsewhere

Then calculate the magnetic field EVERYWHERE.
Static magnetic fields using Ampere’s Law – notice the symmetry in
these problems.
Cylindrical current densities
(These might approximate what you would find in a volume of a material carrying a
current.)
46) Calculate the magnetic field EVERYWHERE where the current density is
 J 0 ẑ r  a
J
- in cylindrical coordinates
 0 ra
47) Calculate the magnetic field EVERYWHERE where the current density is
 J 0 rẑ r  a
J
- in cylindrical coordinates
ra
 0
48) Calculate the magnetic field EVERYWHERE where the current density is
 J 0 r 2 ẑ r  a
J
- in cylindrical coordinates
ra
 0
49) Calculate the magnetic field EVERYWHERE where the current density is
 J r m ẑ r  a
J 0
- in cylindrical coordinates
0
r

a

50) Calculate the magnetic field EVERYWHERE where the current density is
ra
 0
 1
J   J 0 ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
51) Calculate the magnetic field EVERYWHERE where the current density is
ra
 0
 1
J   J 0 2 ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
52) Calculate the magnetic field EVERYWHERE where the current density is
ra
 0
 1
J   J 0 m ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
53) Calculate the magnetic field EVERYWHERE where the current density is
ra
 J 0 rẑ
 0
arb

J
- in cylindrical coordinates
1
J
ẑ
b

r

c
 0r

rc
 0
54) Calculate the magnetic field EVERYWHERE where the current density is
 J 0 r 2 ẑ
ra

arb
 0
J
- in cylindrical coordinates
1
J 0 r 2 ẑ b  r  c

rc
 0
Physically realistic systems
These problems are intended to approximate physically real situations. Use
Ampere’s Law to solve for the magnetic fields everywhere.
Need to add problems
Parallel wires, coaxial wires, electromagnets, speaker coils, (Long coils vs. short
coils) etc
55) The magnetic field around a strip cable (2 parallel wires) can be examined if
one considers where the current is flowing. Assume that for the specific strip
cable that the wires are filamentary strands of metal (typically Cu) and that the
net flow on one wire is balanced by an opposite net flow on the other wire.
First show that the surface current density on the wires should be given by:
xa
 I 0 x  a ẑ

I  I 0 x  a ẑ
x  a - in cartisean coordinates

0
elsewhere

Then calculate the magnetic field EVERYWHERE.
56) The magnetic field around a coaxial cable can be examined if one considers
where the current is flowing. Assume that for the specific coaxial cable that
the wires are made from single strands of metal (typically Cu) and that the net
flow on one wire is balanced by an opposite net flow on the other wire. First
show that the surface current density on the wires should be given by:
0
ra

 J  r  a ẑ
ra
 s
0
arb

J
- in cylindrical coordinates
J a  r  b ẑ
rb
 sb

0
br

Then calculate the magnetic field EVERYWHERE.
57) Speakers/microphone coils, transformers and inductors rely on the magnetic
field produced by or induced from coils. The simplest geometry is that of an
infinitely long coil. (In fact this is one of only a few geometries that can be
solved analytically.) Assume the coil has a radius of ‘a’ and N loops per
length ‘l’. Calculate the magnetic field EVERYWHERE.
Material issues
For more information look at:
http://www.matweb.com
http://www.nist.gov/srd/materials.htm
http://search.globalspec.com/Search/MaterialsSearch
http://home.san.rr.com/bushnell/Material%20Properties.htm
http://www.memsnet.org/material
http://www.kayelaby.npl.co.uk/general_physics/2_6/2_6_3.html
Material
Al
r ()
µr ()
 (S
1.000021
3.5E7
m
Al2O3
4.5-8.4
0.999963
1e-16
Au (gold)
0.99996
4.1E7
Ag (silver)
0.9999976
6.1E7
Ag2O
0.999866
Cu
0.99999
CuO
1.000250
W
1.00008
WO2
1.000057
5.7E7
1.8E7
Teflon
2.1
~1
1E-20
Wax (paraffin)
2.0-2.5
0.99999942
1E-17
Sea water
70
0.9999901
4
Distilled H2O
81
0.999987
1E-4
Mica
5.4
Air
1.006
1E-15
~1
Mineral Oil
<1E-10
Si (pure)
0.9999961
3.9E-4
Quartz (SiO2)
3.8
0.9999704
1e-16
Glass
4-10
~”
1E-12
Pyrex
4-6
~”
5E-12
Dry Earth
7
Wet Earth
30
1e-3
Nickel
600
1.6E8
Cobalt
250
1.6E8
Mild Steel
2000
6.3E7
Iron
5000
1.1E7
58) Calculate the E and D fields for a parallel plate capacitor assuming
a) Air filled gap
b) Quartz filled gap
c) Mica filled gap
d) Mineral oil filled gap
e) Wax filled gap
Assume that the capacitor has a gap of ‘d’ and is made of two infinitely large
planes (Thus, you are ignoring fringing fields.)
59) Calculate the E and D fields and surface charge densities for a parallel plate
capacitor assuming that the capacitor has a structure of
where a1 and a2 are areas and d is the gap. (Ignore fringing fields.)
60) Calculate the E and D fields and surface charge densities for a parallel plate
capacitor assuming that the capacitor has a structure of
where a is the area and d1 and d2 are the dielectric thicknesses. (Ignore
fringing fields.) HINT REMEMBER TO GET ALL OF THE SURFACE
CHARGES.
61) Calculate the E and D fields resulting from a surface charge at the interface
between two semi-infinite slabs of dielectric materials. Thus a structure of
62) Calculate the B and H fields resulting from a surface current at the interface
between two semi-infinite slabs of magnetic materials. Thus a structure of
63) Calculate the B and H fields resulting from currents coaxial wire with a series
of different magnetic materials. Thus a structure of
Assume that the current on the inner and outer wires are balanced and that the
radii of the wires are a (inner) and b (outer).
a) Teflon
b) Quartz
c) Mineral Oil
d) Mica
64) Calculate the B and H fields resulting from currents coaxial wire with a series
of different magnetic materials. Thus a structure of
Assume that the current on the inner and outer wires are balanced and that the
radii of the wires are a (inner) and b (outer) and the inner magnetic material,
µ1, has a radius of c.
65) Calculate the conductivity for the following materials
a) Au (There is no AuOx)
b) Ag and Ag2O
c) Cu and CuO
d) Al and Al2O3
e) Explain why cooper is used in household wires.
66) Assuming an electric field of 1 V/m, calculate the terminal velocity of
electrons in the following materials
a) Au
b) Ag
c) Cu
d) Al
e) Explain why Au is used for ‘expensive’ connectors in electronics
applications.
67) Assuming an electric field of 1 V/m, calculate the current density for the
following materials
a) Au (There is no AuOx)
b) Ag and Ag2O
c) Cu and CuO
d) Al and Al2O3
e) Explain why cooper is used in household wires.
68) Boundary problems
69) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
E1  2 x̂  3ŷ  5 ẑ
 1  2 0
 2  4  0

x0
x0
 s  0 0
70) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
D1  2 0 x̂  3 0 ŷ  5 0 ẑ
 1  4  0
 2  2  0

x0
x0
s  3 0
71) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
E1  4 x̂  8 ŷ  10 ẑ
 1  1 0
 2  3 0

y0
y0
 s  2 0
72) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
D1  5 0 x̂  3 0 ŷ  2 0 ẑ
 1  2  0
 2  3 0

y0
y0
 s  5 0
73) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
E1  2 x̂  5 ŷ  6 ẑ
  2 0
 1
  2  1 0
s  3 0
z0
z0
74) Calculate the electric fields (E and D) everywhere assuming that you have two
semi-infinite dielectric materials such that
D1  7 0 x̂  2 0 ŷ  3 0 ẑ
 1  4  0
  2  1 0

z0
z0
 s  2 0
75) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
B1  7 0 x̂  2 0 ŷ  30 ẑ
 1  4 0
 2  10

z0
z0
J s  2 ŷ
76) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
H1  7 x̂  2 ŷ  3ẑ
 1  4 0
 2  10

z0
z0
J s  2 ŷ
77) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
B1  7 0 x̂  2 0 ŷ  30 ẑ
 1  4 0
 2  10

x0
x0
J s  2 ŷ
78) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
H1  2 x̂  6 ŷ  3ẑ
 1  3 0
 2  5 0

J s  5 ẑ
79)
x0
x0
80) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
B1  2 0 x̂  5 0 ŷ  6 0 ẑ
 1  2 0
 2  30

y0
y0
J s  1ẑ
81) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
B1  30 x̂  5 0 ŷ  6 0 ẑ
 1  5 0
 2  2 0

z0
z0
J s  2 x̂  3ŷ
82) Calculate the magnetic fields (B and H) everywhere assuming that you have
two semi-infinite magnetic materials such that
H1  4 x̂  3ŷ  2 ẑ
 1  30
 2  2 0

z0
z0
J s  1x̂  2 ŷ
Energy held in Static electric fields using Gauss’ Law – notice the
symmetry in these problems.
83) Use cylindrical coordinates to calculate the energy held by a line of charges
where the charge density is
l  0
Is this system physically possible?
84) Use cylindrical coordinates to calculate the energy held by a surface of
charges where the charge density is
 s  0
Is this system physically possible?
Cylindrical Volume charges
(These might approximate what you would find in a volume of a material. Under
some conditions it might be an insulator with charges distributed around the
volume, in others it might be a wire (or two) with charge carriers near surfaces.)
85) Calculate the energy DENSITY for a volume of charges where the charge
density is
 0 r  a
- in cylindrical coordinates
0 ra
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
v  
86) Calculate the energy DENSITY for a volume of charges where the charge
density is
 1
ra

v   0 r
- in cylindrical coordinates
 0
ra
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
87) Calculate the energy DENSITY for a volume of charges where the charge
density is
 1
ra

v   0 r 2
- in cylindrical coordinates

ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
88) Calculate the energy DENSITY for a volume of charges where the charge
density is
 1
ra

v   0 r m
- in cylindrical coordinates

ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
89) Calculate the energy DENSITY for a volume of charges where the charge
density is
 r r  a
v   0
- in cylindrical coordinates
 0 ra
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
90) Calculate the energy DENSITY for a volume of charges where the charge
density is
0 r 2
v  
ra
- in cylindrical coordinates
ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
91) Calculate the energy DENSITY for a volume of charges where the charge
density is
 r m r  a
v   0
- in cylindrical coordinates
ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
92) Calculate the energy DENSITY for a volume of charges where the charge
density is
ra
0

v   0 a  r  b - in cylindrical coordinates
0
br

Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
93) Calculate the energy DENSITY for a volume of charges where the charge
density is
ra
 0

v   0 r a  r  b - in cylindrical coordinates
 0
br

Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
94) Calculate the energy DENSITY for a volume of charges where the charge
density is
ra
 0
 m
v   0 r a  r  b - in cylindrical coordinates
 0
br

Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
95) Calculate the energy DENSITY for a volume of charges where the charge
density is
ra
 0
 1
v   0
a  r  b - in cylindrical coordinates
 r
br
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
96) Calculate the energy DENSITY for a volume of charges where the charge
density is
ra
 0
 1
v   0 m a  r  b - in cylindrical coordinates
 r
br
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
Spherical Volume charges
(These might approximate what you would find in a volume of a material. Under
some conditions it might be an insulator with charges distributed around the
volume.)
97) Calculate the TOTAL energy for a volume of charges where the charge
density is
 r  a
v   0
- in spherical coordinates
0 ra
98) Calculate the TOTAL energy for a volume of charges where the charge
density is
 1
ra

v   0 r
- in spherical coordinates

ra
 0
99) Calculate the TOTAL energy for a volume of charges where the charge
density is
 1
ra

v   0 r 2
- in spherical coordinates
 0
ra
100) Calculate the TOTAL energy for a volume of charges where the charge
density is
 1
ra

v   0 r m
- in spherical coordinates

ra
 0
101) Calculate the TOTAL energy for a volume of charges where the charge
density is
 r r  a
v   0
- in spherical coordinates
 0 ra
102) Calculate the TOTAL energy for a volume of charges where the charge
density is
0 r 2 r  a
v  
- in spherical coordinates
ra
 0
103) Calculate the TOTAL energy for a volume of charges where the charge
density is
0 r m r  a
v  
- in spherical coordinates
ra
 0
104) Calculate the TOTAL energy for a volume of charges where the charge
density is
ra
0

v   0 a  r  b - in spherical coordinates
0
br

105) Calculate the TOTAL energy for a volume of charges where the charge
density is
ra
 0

v   0 r a  r  b - in spherical coordinates
 0
br

106) Calculate the TOTAL energy for a volume of charges where the charge
density is
ra
 0
 m
v   0 r a  r  b - in spherical coordinates
 0
br

107) Calculate the TOTAL energy for a volume of charges where the charge
density is
ra
 0
 1
v   0
a  r  b - in spherical coordinates
 r
br
 0
108) Calculate the TOTAL energy for a volume of charges where the charge
density is
ra
 0
 1
v   0 m a  r  b - in spherical coordinates
 r
br
 0
Energy held in Static magnetic fields using Ampere’s Law – notice the
symmetry in these problems.
109)
Cylindrical current densities
(These might approximate what you would find in a volume of a material carrying a
current.)
110) Calculate the energy DENSITY where the current density is
 J 0 ẑ r  a
J
- in cylindrical coordinates
 0 ra
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
111) Calculate the energy DENSITY where the current density is
 J 0 rẑ r  a
J
- in cylindrical coordinates
ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
112) Calculate the energy DENSITY where the current density is
 J r 2 ẑ r  a
J 0
- in cylindrical coordinates
ra
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
113) Calculate the energy DENSITY where the current density is
 J r m ẑ r  a
J 0
- in cylindrical coordinates
0
r

a

Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
114) Calculate the energy DENSITY where the current density is
ra
 0
 1
J   J 0 ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
115) Calculate the energy DENSITY where the current density is
ra
 0
 1
J   J 0 2 ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
116) Calculate the energy DENSITY where the current density is
ra
 0
 1
J   J 0 m ẑ a  r  b - in cylindrical coordinates
 r
rb
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
117) Calculate the energy DENSITY where the current density is
ra
 J 0 rẑ
 0
arb

J
- in cylindrical coordinates
1
J 0 r ẑ b  r  c

rc
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
118)
Calculate the energy DENSITY where the current density is
 J 0 r 2 ẑ
ra

arb
 0
J
- in cylindrical coordinates
1
J 0 r 2 ẑ b  r  c

rc
 0
Why have you not been asked to calculate the TOTAL energy?
Could you build a real system that ‘looks’ like this?
Describe a situation in which this might be a ‘good’ approximation.
Homework Section 3
Motors and rails
1) Assume that you have a single turn square coil that rotates around the z-axis at
60 Hz. The side of the coil is 10 cm. The coil passes through a magnetic
field of
B  100x̂ G
How many loops would you need to have in order for you to use this as an AC
generator (120 V RMS)?
2) Assume that you have a single turn square coil that rotates around the z-axis at
60 Hz. The side of the coil is 100 cm. The coil passes through a magnetic
field of
B  100x̂ G
How many loops would you need to have in order for you to use this as an AC
generator (120 V RMS)?
3) Assume that you have a single turn square coil that rotates around the z-axis at
60 Hz. The side of the coil is 100 cm. The coil passes through a magnetic
field of
B  10x̂ G
How many loops would you need to have in order for you to use this as an AC
generator (120 V RMS) ?
4) Assume that you have a single turn square coil that rotates around the x-axis at
60 Hz. The side of the coil is 50 cm. The coil passes through a magnetic
field of
B  50 ẑ G
How many loops would you need to have in order for you to use this as an AC
generator (120 V RMS) ?
5) Assume that you have a single turn square coil that rotates around the y-axis at
60 Hz. The coil passes through a magnetic field of
B  50 x̂ G
How big (area) would the coil have to be in order for you to use this as an AC
generator (120 V RMS) ?
6) The permanent magnet in a stereo speaker has a strength of
B  10ẑ G
XXXX ?
Figure of Problems 7-XXX
7) A bar conductor completes a Rail gun as in the above figure. Let
B  10 G ẑ
d  1.0m x̂
v  10m / s ŷ
What is the voltage?
8) Rail gun
B  10 G x̂
d  1.0m x̂
v  10m / s ŷ
What is the voltage?
9) Rail gun
B  7 G ẑ
d  1.5m x̂
v  30m / s ŷ
What is the voltage?
10) Rail gun
B  1 G x̂
d  0.5m x̂
v  1m / s ŷ
What is the voltage?
11) What are the wavelength, frequency and speed of light for wave in free space
the following
H  H 0 cos 0.2cm1 x  3.14 kHz t ẑ


12) What are the wavelength, frequency and speed of light for wave in free space
the following
H  H 0 cos 0.1 cm1 x  0.9 GHz t ẑ


13) What are the wavelength, frequency and speed of light for wave in free space
the following
H  H 0 cos 1.5 mm 1 x  3 MHz t ẑ


14) Assuming  r  1 , what is the dielectric constant of the material through the
following wave in free space is traveling
H  H 0 cos 0.2 m 1 x  3.14 MHz t ẑ


15) Can the following
H  H 0 cos 0.5 km 1 x  5 GHz t ẑ
describe an electromagnetic wave in free space? Why or why not?


16) Can the following
H  H 0 cos 1 cm1 x  1 GHz t x̂
describe an electromagnetic wave in free space? Why or why not?


17) Calculate the related field (E or H) for
H  5 cos 0.4 cm 1 x  6 GHz t ẑ


18) Calculate the related field (E or H) for
E  5 cos 2 cm 1 x  1 GHz t ŷ


19) Calculate the related field (E or H) for
H  5 cos 150 cm 1 r  0.5 MHz t ̂


20) Determine the skin depth of 1 MHz radiation in sea water. What implications
does this have for communications with submarines?
21) Determine the skin depth of 633 nm radiation in sea water. What implications
does this have for being able to see in the sea?
22) Determine the skin depth of 1500 nm radiation in glass and in quartz. What
implications does this have for communications in fiber optics?
23) Determine the skin depth of 1 MHz radiation in Teflon. As Teflon is
sometimes used in coax cables, what implications does this have for
communications along coax?
24) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
H  H 0 cos 0.1 cm1 x  0.9 GHz t ẑ
  3.4E5(S/m)
r  1


25) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
H  H 0 cos 1.5 mm 1 x  3 MHz t ẑ
  7E3(S/m)
r  1


26) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
H  H 0 cos 0.2 m 1 x  3.14 MHz t ẑ
  3E2(S/m)
r  1


27) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
H  H 0 cos 1 cm1 x  1 GHz t x̂
  3E8(S/m)
r  1


28) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
H  5 cos 0.4 cm 1 x  6 GHz t ẑ
  3E3(S/m)
r  1


29) Is this wave propagating in a perfect dielectric, good dielectric, good
conductor or perfect conductor
E  5 cos 2 cm 1 x  1 GHz t ŷ
  8E6(S/m)
r  1


30) Determine the Poynting vector for
H  H 0 cos 0.2 m 1 x  3.14 MHz t ẑ


31) Determine the Poynting vector for
H  H 0 cos 0.1 cm1 x  0.9 GHz t ẑ


32) Determine the Poynting vector for
H  H 0 cos 1.5 mm 1 x  3 MHz t ẑ


33) Determine the Poynting vector for
H  H 0 cos 0.2 m 1 x  3.14 MHz t ẑ


34) Determine the Poynting vector for
H  5 cos 0.4 cm 1 x  6 GHz t ẑ


35) Determine the Poynting vector for
E  5 cos 2 cm 1 x  1 GHz t ŷ


36) Determine the Poynting vector for
H  5 cos 150 cm 1 r  0.5 MHz t ̂


37) Determine the Poynting vector for
Homework Section 4
Transmission lines
L
Z0
S1
S2
ZL
V0
Figure for problems 1 to 11
1) At t= 0 close switch S1. (S2 is open.) Plot the voltage at the load and the
source as a function of time for t  0 . (Batteries have perfect match.)
2) At t= 0 close switch S1. S2 is closed and ZL = Z0. Plot the voltage at the load
and the source as a function of time for t  0 . (Batteries have perfect match.)
3) At t= 0 close switch S1. S2 is closed and ZL > Z0. Plot the voltage at the load
and the source as a function of time for t  0 . (Batteries have perfect match.)
4) At t= 0 close switch S1. S2 is closed and ZL < Z0. Plot the voltage at the load
and the source as a function of time for t  0 . (Batteries have perfect match.)
5) At t= 0 close switch S1. S2 is closed and ZL = 0. Plot the voltage at the load
and the source as a function of time for t  0 . (Batteries have perfect match.)
6) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL = Z0. (Batteries
have perfect match.)
7) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL > Z0. (Batteries
have perfect match.)
8) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL < Z0. (Batteries
have perfect match.)
9) At t= -  switches S1 and S2 are closed. At t = 0 S2 is opened. Plot the voltage
at the load and the source as a function of time for t  0 . Assume that ZL =
Z0. (Batteries have perfect match.)
10) At t= -  switches S1 and S2 are closed. At t = 0 S2 is opened. Plot the voltage
at the load and the source as a function of time for t  0 . Assume that ZL >
Z0. (Batteries have perfect match.)
11) At t= -  switches S1 and S2 are closed. At t = 0 S2 is opened. Plot the voltage
at the load and the source as a function of time for t  0 . Assume that ZL <
Z0. (Batteries have perfect match.)
L1
L2
Z1
Z2
S1
S2
Zd
ZL
V0
Figure for problems 12 to 22
12) At t= 0 close switch S1. (S2 is open.) Plot the voltage at the load and the
source as a function of time for t  0 . Zd = Z1 (Batteries have perfect match.)
13) At t= 0 close switch S1. S2 is closed and ZL = Z2, Zd = 2Z1, Z2 = Z1. Plot the
voltage at the load, damage point and the source as a function of time for
t  0 . (Batteries have perfect match.)
14) At t= 0 close switch S1. . S2 is closed and ZL = Z2, Zd = ½ Z1, Z2 = Z1. Plot the
voltage at the load, damage point and the source as a function of time for
t  0 . (Batteries have perfect match.)
15) At t= 0 close switch S1. S2 is closed and ZL < Z2, Zd = 10Z1, Z2 = Z1. Plot the
voltage at the load, damage point and the source as a function of time for
t  0 . (Batteries have perfect match.)
16) At t= 0 close switch S1. S2 is closed and ZL < Z2, Zd = ½ Z1, Z2 = Z1. Plot the
voltage at the load, damage point and the source as a function of time for
t  0 . (Batteries have perfect match.)
17) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL = Z2, Zd = 2Z1,
Z2 = Z1. (Batteries have perfect match.)
18) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL = Z2, Zd = ½
Z1, Z2 = Z1. (Batteries have perfect match.)
19) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL < Z2, Zd =
10Z1, Z2 = Z1. (Batteries have perfect match.)
20) At t= -  switch S1 is closed. At t = 0 S2 is closed. Plot the voltage at the load
and the source as a function of time for t  0 . Assume that ZL > Z2, Zd =
10Z1, Z2 = Z1. (Batteries have perfect match.)
21) At t= -  switches S1 and S2 are closed. At t = 0 S2 is opened. Plot the voltage
at the load and the source as a function of time for t  0 . Assume that ZL <
Z2, Zd = ½ Z1, Z2 = Z1. (Batteries have perfect match.)
22) At t= -  switches S1 and S2 are closed. At t = 0 S2 is opened. Plot the voltage
at the load and the source as a function of time for t  0 . Assume that ZL >
Z0, Zd = ½ Z1, Z2 = Z1 . (Batteries have perfect match.)
Homework Section 5
Antennas
1) Calculate and plot the radiation pattern for a ¼ wave antenna.
2) Calculate and plot the radiation pattern for a ½ wave antenna.
3) Calculate and plot the radiation pattern for a full wave antenna.
4) Calculate and plot the radiation pattern for a 1/3 wave antenna.
5) Calculate and plot the radiation pattern for a 2/3 wave antenna.
6) Calculate and plot the radiation pattern for a 3/2 wave antenna.
7) Derive the array factor for an array of n antennae along the axis of the
antenna. (e.g. both the antenna and the array are along the z axis.) Assume ½
wave antenna and a constant phase (  ) and distance (d) separation between
adjoining antenna. Determine the array factor angle (  ) in terms of the
spherical coordinate angles,  and  .
8) Derive the array factor for an array of n antennae along an axis that is at right
angles to the axis of the antenna. (e.g. the antenna is along z and the array is
along the x axis.) Assume½ wave antenna and a constant phase (  ) and
distance (d) separation between adjoining antenna. Determine the array factor
angle (  ) in terms of the spherical coordinate angles,  and  .
9) Identical to 8 except assume the array is along y.
10) Calculate and plot the radiation pattern for an array of n antennae along the
axis of the antenna. (e.g. both the antenna and the array are along the z axis.)
Assume ½ wave antenna and a constant phase (  ) and distance (d) separation
between adjoining antenna.
11) Calculate and plot the radiation pattern for an array of n antennae along an
axis that is at right angles to the axis of the antenna. (e.g. the antenna is along
z and the array is along the x axis.) Assume ½ wave antenna and a constant
phase (  ) and distance (d) separation between adjoining antenna.
12) Identical to 11 except assume the array is along y.
13) Cell phone companies typically use tri-sided antenna arrays to cover a ‘cell
site’. Each side of the antenna typically has three antenna in that part of the
array. (The sides operate more-or-less independently.) Use a honeycomb to
explain why they do this.
14) Along open highways cell phone companies typically use bi-sided antenna
arrays to cover a ‘cell site’. Each side of the antenna typically has three
antenna in that part of the array. (The sides operate more-or-less
independently.) Explain why they do this.
15) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 2 and d   / 2
16) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 2 and d   / 4
17) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 4 and d   / 2
18) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 4 and d   / 4
19) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for   3 / 4 and d   / 2
20) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for   3 / 4 and d   / 4
21) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 4 and d  
22) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    / 2 and d  
23) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for    and d  
24) For an array of two antenna along the x-axis – with the antenna axis along the
z-axis, calculate and plot the radiation pattern for   0 and d  