HW-5
1. For a point charge q at a distance a from a spherical conductor of radius R (R<a), we
obtained in the lecture the result of the image charge inside the conductor. Starting from that
result, calculate the following quantities.
(a) The surface charg
HW-5
1. For a point charge q at a distance a from a spherical conductor of radius R (R<a), we
obtained in the lecture the result of the image charge inside the conductor. Starting from that
result, calculate the following quantities.
(a) The surface charg
HW-1
1. For the two vectors A x 2 y z and B 2 x 3 y z , find (a) A B and A B , (b)
component of B along A , (c) angle between A and B , (d) A B , (e) A B A B .
2. A , B , and C are three vectors pointing from the origin of a coordinate
HW-1 solution
1. For the two vectors A x 2 y z and B 2 x 3 y z , find (a) A B and A B , (b)
component of B along A , (c) angle between A and B , (d) A B , (e) A B A B .
Solution:
(a) A B ( x 2 y z ) (2 x 3 y z ) 3x y 2 z
| A B
HW-2
1. (a) Let F1 x 2 z and F2 xx yy zz . Calculate the divergence of and curl of F1 and
F2 Which one can be written as the gradient of a scalar? Find a scalar that does the
job. Which one can be written as a curl of a vector? Find a suitable vector po
HW-2
1. (a) Let F1 x 2 z and F2 xx yy zz . Calculate the divergence of and curl of F1 and
F2 Which one can be written as the gradient of a scalar? Find a scalar that does the
job. Which one can be written as a curl of a vector? Find a suitable vector po
HW-3
1. Find the electric field a distance z above the center of circular disk of radius R which
carries a uniform surface charge density . What does your formula give in the limit
R and R z .
2. Suppose the electric field in some region is found to be E
HW-3
1. Find the electric field a distance z above the center of circular disk of radius R which
carries a uniform surface charge density . What does your formula give in the limit
R and R z .
Solution:
Obviously the electric field is in the z direction s
Homework #4
1. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric
metal shell (inner radius a, outer radius b). The shell carries no net charge.
(a) Find the surface charge density at R, at a, and at b.
(b) Find the potenti
Homework #4
1. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric
metal shell (inner radius a, outer radius b). The shell carries no net charge.
(a) Find the surface charge density at R, at a, and at b.
(b) Find the potenti
HW-6
1. A rectangular pipe, running parallel to the z axis (from -∞ to +∞), has three ground metal
sides at y=0, y=a, and x=0. The fourth side, at x=b, is maintained at a specific potential
V0(y).
(a) Develop a general formula for the potential within the
HW-6
1. A rectangular pipe, running parallel to the z axis (from - to +), has three ground metal
sides at y=0, y=a, and x=0. The fourth side, at x=b, is maintained at a specific potential
V0(y).
(a) Develop a general formula for the potential within the p
HW-7
P(r ) kr , where k is a constant.
1. A sphere of radius R carries a polarization
(a) Calculate the bound charge b and b.
(b) Find the field inside and outside the sphere.
2. The space between the plate a parallel-plate capacitor is filled with two
HW-7
P(r ) kr , where k is a constant.
1. A sphere of radius R carries a polarization
(a) Calculate the bound charge b and b.
(b) Find the field inside and outside the sphere.
Solution:
(a) At the surface of the sphere ,
b P n kR R kR .
Within the
HW-8
1. Find the magnetic field at point P for each of the following steady current configurations.
2. Find the force on a square loop and a triangle near in infinite straight wire.
J Jx .
3. A thick slab extending from z=-a to z=+a carries a uniform volu
HW-8
1. Find the magnetic field at point P for each of the following steady current configurations.
Solution:
(a) The magnetic field at P are contributed by the current from 4 segments. But the
dl r 0 so they do not contribute to
two segments in the rad
HW-9
1. Starting from the Lorentz force law, show that the torque on any steady current loop in a
m B .
uniform magnetic field is
A ( B C ) ( A C ) B ( A B)C
Hint:
1
( A r )dl r dl A for any constant vector A .
2
M parallel to its axis. Find
2. In
HW-10
1. (a) Two metal objects are embedded in weakly conducting material of conductivity . Show
that the resistance between them is related to the capacitance of the arrangement by
R
0
C
(b) Suppose you connected a battery between the two conductors and
HW-11
1. Consider a capacitor consisting two metal circular disks (radius R) separated by a distance d.
Assuming d<R and ignore the edge effect. The capacitor starts to be charged at t=0 with a
constant current I, and the charges distribute uniformly over
HW-11
1. Consider a capacitor consisting two metal circular disks (radius R) separated by a distance d.
Assuming d<R and ignore the edge effect. The capacitor starts to be charged at t=0 with a
constant current I, and the charges distribute uniformly over
HW-12
1. In the complex notation there is a clever device for finding the time average of a product.
f (r , t ) A cos(k r t a ) and g (r , t ) B cos(k r t b ) .
Suppose
Show that
~
~
~
fg (1/ 2) Re( fg *) , where f and g are the complex of f and g and
HW-12
1. In the complex notation there is a clever device for finding the time average of a product.
f (r , t ) A cos(k r t a ) and g (r , t ) B cos(k r t b ) .
Suppose
Show that
~
~
~
fg (1/ 2) Re( fg *) , where f and g are the complex of f and g and