101 EE Homework 3:
DUE NOVEMBER 3TH Thursday 5PM;
(THERE IS A COLLECTION HW CABINET MARKED EE101 IN ROOM 67-112 ON THE 6TH FLOOR OF ENGR IV.)
1. Charge is distributed with constant surface charge density on a circular disc of radius a
lying in the xy-plane with center at the origin. Show that the potential at a point on the zaxis is given by
=
2 + 2 ||
20
What is the electric field at that point? What does V become as a becomes very large?
What is the smallest value of z for which the potential due to this disc can be calculated
as if it were a point charge without making an error greater than 1 percent?
2. a) An infinitely long cylinder has a circular cross section of radius a. It is filled with charge of
constant volume density ch. Find E for all points both inside and outside the cylinder.
b) An infinitely long hollow cylinder with an inner radius a and an outer radius b is filled with
-r
charge whose volume density in cylindrical coordinates is ch = Ae . Find E everywhere.
3. A cylindrical capacitor consists of an inner conductor of radius a and an outer conductor
whose inner radius is b. The space between the conductors filled is with a dielectric of
permittivity , and the length of the capacitor is L. Assume that the outer conductor of
cylindrical capacitor is grounded and that the inner conductor is maintained at a potential
V 0.
a) Find the electric field intensity, E, at the surface of the inner conductor (E=E(a)=?).
b) With the inner radius, b, of the outer conductor fixed, find a so that E(a) is minimized.
4. An early model of atomic structure of a chemical element was that the atom was a spherical
cloud of uniformly distributed positive charge Ne, where N is the atomic number and e is
the magnitude of electronic charge. Electrons, each carrying a negative charge e, were
considered to be imbedded in the cloud. Assuming the spherical charge cloud to have a
radius R0 and neglecting collision effects, find the force experienced by an imbedded
electron at a distance r from the center.
5. Assume that the z=0 plane separates two lossless dielectric regions with r1 = 2 and r2 =3. If
we know that E1 in region 1 is 2 3 + (5 + ) , what do we also know about E2
and D2 in region 2? Can we determine E2 and D2 at any point in region 2? Explain.