Download 2012. 10. 23. (화) 7:00 - 9:30 p.m. ** 1

<전기와 자기 중간 고사>
2012. 10. 23. (화) 7:00 - 9:30 p.m.
** 1-4번 문제는 필수, 5, 6번 문제 중 택일. 총 5문제
1. Let us consider a sphere of radius R, which carries a uniform volume charge density ρ.
Answer the following questions. (Total 30 points).
(a) Use Coulomb's law to find the field inside and outside the sphere.
(b) Use Gauss's law to find the field inside and outside the sphere..
(c) Use the electric field obtained in (a) or (b) to find the potential inside and outside
the sphere. (Use infinity as your reference point).
(d) Use the solution of Poisson's equation to find the potential inside and outside the
sphere.
(e) Two spheres, each of radius R and carrying uniform charge densities +ρ and -ρ,
respectively, are placed so that they partially overlap as shown in the figure.
Call the vector from the negative center to the positive center
. Show that the field
in the region of overlap is constant, and find its value.
2.
An uncharged metal sphere of radius R is placed in an otherwise uniform electric field
. (Total 30 points).
(a) Determine the electric potential inside and outside of the sphere.
(b) Find the induced surface charge density.
(c) Determine the induced dipole moment of this sphere from the form of the potential
outside the sphere.
(d) Determine the induced dipole moment of this sphere by direct calculation of the
dipole moment.
(e) Calculate the polarizability of the induced dipole moment.
3. A spherical shell of radius R carries a uniform surface charge
opposite charge -
on the lower half. (Total 30 points).
(a) Find the electric potential inside and outside the sphere.
on the upper half and an
(b) Find the direction and magnitude of the force exerted on the northern hemisphere by
the southern half.
(c) Use the electric pressure to obtain the magnitude of the force between the two
hemispheres (as discussed during class).
4. An electric dipole
is situated a distance z above an infinite grounded conducting plane.
The dipole makes an angle θ with the perpendicular to the plane. (Total 30 points).
(a) Find the electric field due to the grounded conducting plane.
(b) Find the torque
on the dipole
will it come to rest? (Hint: torque on
. If the dipole is free to rotate, in what orientation
by
;
)
(c) What is the work you have to exert to rotate the dipole from
to
.
5. We want to show that the average field inside a sphere of radius R, due to all the charge
within the sphere, is
where
is the total dipole moment in the
following several different ways (Total 30 Points).
(a) Show that the average field due to a single charge q at point
same as the field at
inside the sphere is the
due to a uniformly charged sphere with
, namely
.
(b) The latter, i.e. field at
due to a uniformly charged sphere with
, can be found from
Gauss's law. Express the answer in terms of the dipole moment of q.
(c) Use the superposition principle to generalize to an arbitrary charge distribution.
(d) Show that the average field over the sphere due to all the charges outside is the same as
the field they produce at the center.
6. A
conducting plate has a semispherical boss of radius R with a center on the plate as
shown in the figure. The plate is grounded and a point charge q is brought next to it at a
distance D>R. The charge is on the normal to plate passing through the center of the boss
(Total 30 points).
(a) Determine the image charge needed to replace the plate.
(b) Determine the potential on the side of the charge.
(c) Determine the charge induced on the boss.
(d) Determine the force between the charge and the plate.