1988E1. The isolated conducting solid sphere of radius a shown above is charged to a potential V.
a. Determine the charge on the sphere.
Two conducting hemispherical shells of inner radius b are then brought up and, without contacting the solid sphere are connected to form a spherical shell surrounding and concentric with the solid sphere as shown below. The outer shell is then grounded.
b. By means of Gauss's law, determine the electric field in the space between the solid sphere and the shell at a distance r from the center.
c. Determine the potential of the solid sphere (in the center) relative to ground.
d. Determine the capacitance of the system in terms of the given quantities and fundamental constants.
1989E1. A negative charge ‑ Q is uniformly distributed throughout the spherical volume of radius R shown above. A positive point charge + Q is at the center of the sphere. Determine each of the following in terms of the quantities given and fundamental constants.
a. The electric field E outside the sphere at a distance r > R from the center
b. The electric potential V outside the sphere at a distance r > R from the center
c. The electric field inside the sphere at a distance r < R from the center
d. The electric potential inside the sphere at a distance r < R from the center
1991E1. Two equal positive charges Q are fixed on the x‑axis. one at +a and the other at -a, as shown above. Point P is a point on the y‑axis with coordinates (0, b ). Determine each of the following in terms of the given quantities and fundamental constants.
a. The electric field E at the origin O
b. The electric potential V at the origin O.
2002E1. A rod of uniform linear charge density l = +1.5 x 10‑5 C/m is bent into an arc of radius R = 0.10 m. The arc is placed with its center at the origin of the axes shown above.
a. Determine the total charge on the rod.
b. Determine the magnitude and direction of the electric field at the center O of the arc.
c. Determine the electric potential at point O.
1995E1. A very long nonconducting rod of radius a has positive charge distributed throughout its volume. The charge distribution is cylindrically symmetric, and the total charge per unit length of the rod is l .
a. Use Gauss's law to derive an expression for the magnitude of the electric field E outside the rod.
b. The diagrams below represent cross sections of the rod. On these diagrams, sketch the following.
i. Several equipotential lines in the region r > a
ii. Several electric field lines in the region r > a
c. In the diagram above, point C is a distance a from the center of the rod (i.e., on the rod's surface), and point D is a distance 3a from the center on a radius that is 90° from point C. Determine the following.
i. The potential difference Vc ‑ VD between points C and D
ii. The work required by an external agent to move a charge + Q from rest at point D to rest at point C
Inside the rod (r < a), the charge density r is a function of radial distance r from the axis of the rod
and is given by r=ro(r/a)½, where ro is a constant.
d. Determine the magnitude of the electric field E as a function of r for r < a. Express your answer in terms of ro, a, and fundamental constants.
1974 B5
. The diagram above shows some of the equipotentials in a plane perpendicular to two parallel charged metal cylinders. The potential of each line is labeled.
(a) The left cylinder is charged positively. What is the sign of the charge on the other cylinder?
(b) On the diagram above, sketch lines to describe the electric field produced by the charged cylinders.
(c) Determine the potential difference, VA — VB, between points A and B.
(d) How much work is done by the field if a charge of 0.50 coulomb is moved along a path from point A to point E and then to point D?
The lines above represent electric field lines in the directions indicated. Points A, B, and D lie roughly on the same line in space.
a) At which of these three points (A,B,D) is the magnitude of the electric field the greatest? Justify your answer,
b) At which of these three points(A,B,D) is the electric potential the greatest? Justify your answer.
c) An electron is released from rest at point B. Qualitatively describe its motion. Be sure to discuss direction, speed and acceleration. (side topic: If a proton were released at point B instead, how would answer c) change?)
d) Calculate the electron's speed after it has moved through a potential difference of 20 V.
e) Points B and D are separated by a potential difference of 50 V. Estimate the magnitude of the electric field midway between them and state any assumptions that you make.
f) On the diagram, draw an equipotential line that passes through point C and intersects at least five electric field lines.
1986E1. Three point charges produce the electric equipotential lines shown on the diagram above.
a. Draw arrows at points L, N. and U on the diagram to indicate the direction of the electric field at these points.
b. At which of the lettered points is the electric field E greatest in magnitude? Explain your reasoning.
c. Compute an approximate value for the magnitude of the electric field E at point P.
d. Compute an approximate value for the potential difference, VM ‑ VS, between points M and S.
e. Determine the work done by the field if a charge of +5 x 10‑12 coulomb is moved from point M to point R.
f. If the charge of +5 x 10‑12 coulomb were moved from point M first to point S, and then to point R, would the answer to (e) be different, and if so, how?