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This is a three hour examination. Answer all ?ve questions. Each question is worth 20 points. This is the ?rst of 2 pages. 1. Point charge in a conducting spherical shell. A point charge q is placed at the origin. It is surrounded by an uncharged spherical conducting shell that is centered on the origin and has inner radius a and outer radius b. Thus, the conducting shell has uniform thickness b a, while all other space is in vacuum. a) Find the electric ?eld in the region r > b outside the shell. b) Find the electrostatic potential (r > b) everywhere outside the shell if the potential vanishes at in?nity, (r ! 1) ! 0. c) What is the electric ?eld within the wall of the shell, a < r < b, and what is the electrostatic potential within this region? d) Find the potential (r < a) inside the shell. 2. Line charge in a dielectric cylinder. A long thin uniformly charged wire has linear charge density . This wire runs down the central axis of a long straight dielectric cylinder of radius a that has dielectric constant ?. Outside the dielectric cylinder is vacuum with permittivity ?0 . (Thus, we have a wire surrounded by a dielectric sheath.) Find the electric ?eld and potential both inside and outside the dielectric in the following steps. a) Find the electric ?eld in the region r < a inside the dielectric cylinder (but outside the thin wire). b) Find the electric ?eld in the region r > a outside the dielectric cylinder. c) Find the electrostatic potential both inside and outside the dielectric, and sketch the form of your result if the charge density is positive. 2 Page 2 of 2 3. Ampere’s law. ~ One of Maxwell’s equations gives the curl of H as ~ ~ @ ~ r?H =J + D . @t (1) a) Consider only the steady currents and ?elds so that the displacement current ~ @D @t vanishes. Use Stokes’s theorem to show that this Maxwell equation (1) then gives Ampere’s law in the form I d~ · H = I , ` ~ (2) H where the line integral d~ · H is around a closed loop, and I is the current that ` ~ passes through the surface enclosed by the loop. b) Use the form (2) of Ampere’s law that you found in (a) to determine the magnetic ~ ?eld H at distance r from the axis of a long straight wire that carries a current I. (Here, r is greater than the radius of the wire, because only want the ?eld in the region outside the wire itself.) c) If the region outside the wire is empty space with permeability µ0 , what is the ~ magnetic ?eld B (magnetic induction) at distance r from the wire? 4. A magnetized sphere. ~ A sphere of radius a is uniformly magnetized with a magnetization M = M z , where ˆ M is a constant. a) Find the (volume) magnetic pole density ?M within the sphere. b) Find the surface magnetic pole density M everywhere on the surface of the sphere. ( M is also called the surface density of magnetic pole strength.) c) Evaluate the total magnetic pole strength of this magnetized sphere. (This is the sum of the appropriate volume and surface integrals of ?M and M .) (You may wish to recall the analogies between ?M and de?ned for electric polarization.) M and the similar quantities 5. Short answers. Answer each part in several complete English sentences. Include a picture if you think it helps. a) What is the purpose of a multipole expansion of the electrostatic potential of a charge distribution? In what situation(s) is it applicable? What do the various terms in the expansion represent? b) What is the method of (electrostatic) images? What kinds of problems is it designed to solve? Brie?y describe a speci?c example of the use of the method of images. c) What is electric polarization? What are its microscopic origins? In linear dielectric materials, polarization is proportional to the electric ?eld. Why does this behavior occur?

This is a three hour examination. Answer all ?ve questions. Each question is worth
20 points. This is the ?rst of 2 pages.
1. Point charge in a conducting spherical shell.
A point charge q is placed at the origin. It is surrounded by an uncharged spherical
conducting shell that is centered on the origin and has inner radius a and outer radius
b. Thus, the conducting shell has uniform thickness b a, while all other space is in
vacuum.
a) Find the electric ?eld in the region r > b outside the shell.
b) Find the electrostatic potential (r > b) everywhere outside the shell if the
potential vanishes at in?nity, (r ! 1) ! 0.
c) What is the electric ?eld within the wall of the shell, a < r < b, and what is the
electrostatic potential within this region?
d) Find the potential (r < a) inside the shell.
2. Line charge in a dielectric cylinder.
A long thin uniformly charged wire has linear charge density . This wire runs down
the central axis of a long straight dielectric cylinder of radius a that has dielectric
constant ?. Outside the dielectric cylinder is vacuum with permittivity ?0 . (Thus, we
have a wire surrounded by a dielectric sheath.) Find the electric ?eld and potential
both inside and outside the dielectric in the following steps.
a) Find the electric ?eld in the region r < a inside the dielectric cylinder (but
outside the thin wire).
b) Find the electric ?eld in the region r > a outside the dielectric cylinder.
c) Find the electrostatic potential both inside and outside the dielectric, and sketch
the form of your result if the charge density is positive.

2

Page 2 of 2
3. Ampere’s law.
~
One of Maxwell’s equations gives the curl of H as
~
~ @ ~
r?H =J + D .
@t

(1)

a) Consider only the steady currents and ?elds so that the displacement current
~
@D
@t vanishes. Use Stokes’s theorem to show that this Maxwell equation (1) then
gives Ampere’s law in the form
I

d~ · H = I ,
` ~

(2)

H
where the line integral d~ · H is around a closed loop, and I is the current that
` ~
passes through the surface enclosed by the loop.
b) Use the form (2) of Ampere’s law that you found in (a) to determine the magnetic
~
?eld H at distance r from the axis of a long straight wire that carries a current
I. (Here, r is greater than the radius of the wire, because only want the ?eld in
the region outside the wire itself.)
c) If the region outside the wire is empty space with permeability µ0 , what is the
~
magnetic ?eld B (magnetic induction) at distance r from the wire?

4. A magnetized sphere.
~
A sphere of radius a is uniformly magnetized with a magnetization M = M z , where
ˆ
M is a constant.
a) Find the (volume) magnetic pole density ?M within the sphere.
b) Find the surface magnetic pole density M everywhere on the surface of the
sphere. ( M is also called the surface density of magnetic pole strength.)
c) Evaluate the total magnetic pole strength of this magnetized sphere. (This is
the sum of the appropriate volume and surface integrals of ?M and M .)
(You may wish to recall the analogies between ?M and
de?ned for electric polarization.)

M

and the similar quantities

5. Short answers.
Answer each part in several complete English sentences. Include a picture if you think
it helps.
a) What is the purpose of a multipole expansion of the electrostatic potential of a
charge distribution? In what situation(s) is it applicable? What do the various
terms in the expansion represent?
b) What is the method of (electrostatic) images? What kinds of problems is it
designed to solve? Brie?y describe a speci?c example of the use of the method
of images.
c) What is electric polarization? What are its microscopic origins? In linear dielectric materials, polarization is proportional to the electric ?eld. Why does this
behavior occur?

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