Homework
1.
TOTAL 120 pts
1. A sphere of radius a is located at the back corner of a cube (length of each side L) in
air as shown in the figure. The center of the sphere coincides with one corner of the
cube at (0,0,0). Assume a total charge of q is uniformly distributed on the sphere,
calculate the total flux of through the shaded side. (15 pts)
2. The diagram below shows some of the equipotential lines in a plane perpendicular to
two parallel charged metal cylinders. The potential of each line is labeled.
(a) If the left cylinder is positively charged, determine the sign of charge of the right
cylinder. (2 pts)
(b) Sketch the electric field lines produced by the charged cylinders, you have to also
include the equipotential lines in your plot. (2 pts)
(c) Determine the potential difference VA-VB, between points A and B (3 pts)
(d) Calculate the work done by the electric field if a charge of 1 coulomb is moved
along a path from point A to point C then to point B and finally to point D. (3 pts)
3. Given a uniform sphere of charge of radius b and volume density in air;
(a) Find the electric field intensity in the regions R
and R
. (10 pts)
E
E
(b) Find the electric potential V in the regions R
and R
. (16 pts)
4. Two spheres, each of radius b and ca
ying uniform charge densities + and -,
espectivley, are placed so that they partially overlap as shown in the figure below.
Assume the vector from the positive center to negative center to be . Derive the
electric field in the overlap region. (15 pts)
5. Four charges are located at the corners of a square, of length 2L on each side, and
symmetrically located relative to the origin as shown,
(a) Derive an expression for the potential V(x) at a position x along the x-axis. (6 pts)
(b) If a test charge q is to be placed at a position x on the x-axis, what is the force
exerted on this test charge? (6 pts)
6. For two conducting cylinders placed parallel to each other as shown, assuming the
adii are a and b, with spacing between the two conductors S (S
a and S
).
Given uniform line charge density on both conductors to be l , find the capacitance per
unit length of the structure. (10 pts)
x
Q
Q
-Q
-Q
2L
2L
x = 0
x
Q
Q
-Q
-Q
2L
2L
x = 0
x
a
S
x
+ -
P
x
a
S
x
+ -
P
d
7. A capacitor is formed from a segment of two coaxial cylinders as shown in the figure
elow, neglect the fringing field at the edge of the plates,
(a) Using Cylindrical Coordinates, at z = 0 plane, draw the electric field lines and
equipotential lines within the dielectric region (use a
ows to indicate the prope
directions whenever necessary). (4 pts)
(b) Express the potential Vo in terms of the electric field inside the region. (6 pts)
(c) Applying boundary condition at plate, express the surface charge density
s of the co
esponding surface. (6 pts)
(d) Integrate the surface charge density you obtained in (c) to find the total charge Q
induced on the plate. (8 pts)
(e) Calculate the capacitance of the structure. (8 pts)
(Cartesian)
(Cylindrical)
(Spherical)
sin
x y z
z
R
V V V
V a a a
x y z
V V V
a a a
R r z
V V V
a a a
R R R
2
2
(Cartesian)
1
( ) (Cylindrical)
1 1 1
( ) ( sin ) (Spherical)
sin sin
yx z
z
R
AA A
A
x y z
A A
A
r r z
A
R A A
R R R R
2
XXXXXXXXXXCartesian)
1
(Cylindrical)
sin
1
(Spherical)
sin
sin
y yx xz z
x y z
z
z
R
R
A AA AA A
A a a a
y z z x x y
a a r a
r z
A A A
a a R a R
R R
A RA R A