Homework Section 1
1)
2)
3)
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].
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.
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.
g) (1, 4)
h) (4, 3, 0)
i) (0, -1, 1)
j) (6, 1, 0, -1, 2)
5)
Which of the following vectors are unit vectors?
k) (1, 0)
l) (1, 1/2)
m) (1, -1)
n) (1/√2, -1/√2)
o) (1/2, 0, √3/2)
p) (1, 0, 0, 0)
6)
Express the vector A = (2, 7) as a linear combination of the vectors:
q) B1 = (2, 4), B2 = (-1, 3).
r) C1 = (4, 4), C2 = (5, 5).
Express the vector A = (2, -1, 3) as a linear combination of the vectors:
s) B1 = (2, 4, 1), B2 = (3, 7, 1).
t) C1 = (1, 0, 0), C2 = (0, 1, 0), C3 = (0, 0, 1).
u) D1 = (2, 4, 1), D2 = (3, 7, 1), D3 = (-1, 2, 2).
7)
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).
8)
Determine if the vector R1 = (2, 4, 1), R2 = (0, 3, -1) and R3 = (2, 1, 1) are linearly
independent.
9)
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
v) Show that the dipole moment of a neutral system is independent of the origin.
w) Express this moment in terms of the centers of the systems of negative and
positive charges making up the original system.
10)
Find the scalar product of the following vectors.
x) (2, 3)•(1, -1)
y) (4, 1)•(6, -5)
z) (1, 2, -3)•(-1, 1, 2)
aa) (2, 4)•(1, 5, 3)
bb) (0, sin, 1, 3)•(2, 4, -2, 1)
cc) (sin(t), cos(t))•( sin(t), cos(t))
11)
Find the angles that the vector (2, 4, -5) makes with the coordinate axes.
12)
Find the projection of the vector (2, 5, 1) on the vector (1, 1, 3).
13)
dd) Using the dot product, prove the law of cosines.
ee) 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()
14)
Determine the value of  such that A and B are perpendicular and C and D are
perpendicular.
ff) A = (2, 3, 1), B = (4, 2, 4)
gg) C = (2, 4, 3, 1), D = (, 2, -1, 2)
15)
Let A = (-1, 2, 4), B = (3, 2, 7). Find the unit vector perpendicular to the plane
determined by A and B.
16)
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?
17)
Determine the magnitude, phase angle, real and imaginary parts of the following
1) 3+i
2) 3
3) 3ei2/3
4) 2(cos (/6) +i sin (/6))
5) 345°
18)
A Force F = 2i - 3j + k acts at the point (1, 5, 2). Find the torque due to F about
6) The origin
7) The y axis
8) The line x/2=y/1=z/(-2)
Del operator questions
19) Find the gradient of w=x2y3z at (1, 2, -1)
20)
Find the gradient of
Compute the divergence and the curl of each of the following Vector Fields.
ˆ
21) r  z ˆi  y ˆj  x k
22) r  x2 ˆi  y2 ˆj  z2 kˆ
2
Calculate the Laplacian   of each of the following
23)
24)
25)
26)
x2  y2
x
2
 y2  z 2 
1/ 2

For r  x 2  y2  z 2
1
r̂
  2
r
r

1/2
, Prove
ˆ , evaluate    r     r̂ .
For r  x ˆi  y ˆj  z k
 
r
Divergence, Stokes and Green’s Theorem Problems
27)
Evaluate the integral
 x
2

 y dx  2xydy along each of the following paths
2
from (0,0) to (1,2)
hh) y = 2x2
ii) x = t2, y = 2t
jj) y = 0, for x = 0 -> 1 and then x = 1 for y = 0 -> 2.
28)
Evaluate the integral
 xydx  x dy  along each of the following paths – each
path.
kk) a) (0,0) to (1,2)
ll) b) (0, 0) to (3, 0) to (1,2).
29)
Determine if the following force fields are conservative. Then determine a scalar
potential for each field.
mm)
F  ˆi   zˆj   ykˆ
2
nn) F  z sinh yˆj  2z cosh y kˆ
30)
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.
oo) F  y ˆi  x ˆj  z kˆ
pp) F  yˆi  x ˆj  zkˆ
Explain why you have gotten these answers.
31)
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.
32)
Derive
 P(x,y)
c
dx  
A
P(x, y)
dxdy using methods similar to that used in
y
class.
33)
Evaluate
 x dy  xydx, around the curve (1, 0) to (4, 0) to (4, .5) along y 
2
c
to (1, 1).
34)
For a simple closed curve C in a plane show by Green’s theorem that the area
enclosed is A  12  xdy  ydx .
c
35)
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.
1x
36)
   Vd ;
x
V
2


 y 2  x ˆi  yˆj , over the volume bounded by x2 + y2 ≤ 4, 0 ≤
V
z ≤ 5. (Remember the top and bottom!)
37)
 V  nd
2
2
; V  x ˆi  yˆj  zkˆ , over the surface of the cone with base x  y  16
A
and vertex at (0,0,3).
38)    Fd ; F  x 3  y2 y ˆi  y 3  2y 2  yx ˆj  z 2 1kˆ , over the unit cube


V
in the first octant.
Using either Stoke’s Theorem or the Divergence Theorem evaluate each of the
following.
39)  V  nd ; V  2xy ˆi  y 2 ˆj  z  xy kˆ , where  is a tin can defined by x2 +
A
y2 ≤ 9, 0 ≤ z ≤ 5. (Remember the top and bottom!).
40)
 (  V)  nd
; V  x  x 2z ˆi  yz 3  y 2  ˆj  x 2y  xz kˆ , where  is any
A
surface with a bounding curve entirely in the x-y plane.
41)
 (  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
42) Determine the electric potential of a point charge from the electric field
q r̂
E
4 r 2
43)
Determine the magnetic potential of a current element.
 dI  r̂
B
4 r 2
44)
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
1
l   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
1
l  0 2
z
4) Use cylindrical coordinates to calculate the electric field in the x-y plane for a line
charge where the charge density is
1
l   0 3
z
5) Use cylindrical coordinates to calculate the electric field in the x-y plane for a line
charge where the charge density is
1
l  0 m
z
6) Use cylindrical coordinates to calculate the electric field in the x-y plane for a line
charge where the charge density is
l  0 z
7) Use cylindrical coordinates to calculate the electric field in the x-y plane for a line
charge where the charge density is
l  0 z 2
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
1
 s  0
r
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
1
 s  0 2
r
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
1
 s  0 3
r
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
1
 s  0 m
r
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
 s  0 r
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
s  0 r 2
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
 1
ra

v   0 r 2
- in cylindrical coordinates
 0
ra
20) Calculate the electric field EVERYWHERE for a volume of charges where the charge
density is
 1
ra

v   0 r m
- in cylindrical coordinates
 0
ra
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
0 r 2 r  a
v  
- 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
 0
ra
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
 1
ra

v   0 r m
- in spherical coordinates
 0
ra
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
 r m r  a
v   0
- 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