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2004 Electricity and Magnetism I
The figure above left shows a hollow, infinite, cylindrical, uncharged conducting shell of inner radius r1,
and outer radius r2 . An
infinite line charge of
linear charge density
+λ is parallel to its axis
but off center. An
enlarged cross section of
the cylindrical shell is
shown above right.
(a) On the cross section
above right,
i.
sketch the electric
field lines, if any, in each
of regions I, II, and III
and
ii. use + and - signs to
indicate any charge
induced on the conductor.
(b) In the spaces below, rank the electric potentials at points a, b, c, d, and e from highest to lowest
(I = highest potential). If two points are at the same potential, give them the same number.
____Va ____Vb _____Vc
_____Vd
_____Ve
(c) The shell is replaced by
another cylindrical shell
that has the same
dimensions but is
nonconducting and carries a
uniform volume charge
density +ρ. The infinite
line charge, still of charge
density +λ is located at the
center of the shell as shown
above. Using Gauss's law,
calculate the magnitude of
the electric field as a
function of the distance r
from the center of the shell
for each of the following
regions. Express your answers in terms of the given quantities and fundamental constants.
i.
r < r1
ii. r1 < r < r2
iii. r>r2
1992- Electricity and Magnetism I
A positive charge distribution exists within a nonconducting spherical region of radius a. The volume
charge density p is not uniform but varies with the distance r from the center of the spherical charge
distribution, according to the relationship ρ = βr for 0 < r < a, where β is a positive constant, and
ρ=0, for r >a.
a. Show that the total charge Q in the spherical region of radius a is βπa 4
b. In terms of β, r, a, and fundamental constants, determine the magnitude of the electric field at a point a
distance r from the center of the spherical charge distribution for each of the following cases. i. r> a
ii. r =a
iii. 0 < r <a
c. In terms of β, a, and fundamental constants, determine the electric potential at a point a
distance r from the center of the spherical charge distribution for each of the following cases
i. r= a
ii. r =0
1993- Electricity and Magnetism I
The solid nonconducting cylinder of radius
R shown above is very long. It contains a
negative
charge
evenly
distributed
throughout the cylinder, with volume
charge density ρ. Point P1 is outside the
cylinder at a distance r, from its center C
and point P2 is inside the cylinder at a
distance r 2 from its center C. Both points are in the same plane, which is
perpendicular to the axis of the cylinder.
a. On the following cross-sectional diagram, draw vectors to indicate the directions of the electric
field at points P1 and P2
b. Using Gauss's law, derive expressions for the magnitude of the electric field E in terms of r, R. p,
and fundamental constants for the following two cases.
i.
r > R (outside the cylinder)
ii.
r < R (inside the cylinder)
Another cylinder of the same dimensions,
but made of conducting material, carries a
total current I parallel to the length of the
cylinder, as shown in the diagram above.
The current density is uniform throughout
the cross-sectional area of the cylinder.
Points P, and P2 are in the same positions
with respect to the cylinder as they were for the nonconducting cylinder.
c. On the following cross-sectional diagram in which the current is out of the
plane of the page (toward the reader), draw vectors to indicate the directions of the
magnetic field at points P 1, and P2.
d. Use Ampere's law to derive an expression for the magnetic field B inside the
cylinder in terms of r, R,I, and fundamental constants
Homework Tuesday 4-24-2012
A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and
angled at 30° , as shown above. The ball rolls along the roof without slipping. The rotational inertia of a
sphere of mass M and radius R about its center of mass is MR2.
(a) On the figure below, draw and label the forces (not components) acting on the ball at their points
of application as it rolls along the roof.
(b)
Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw
anything other than what you have shown in part (a) to assist in your solution, use the space below. Do
NOT add anything to the figure in part (a).
(c)
Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the
roof.
(d)
A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical
distance of 3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box
is 12 kg. Calculate the horizontal speed of the wagon immediately after the ball lands in it.