Physics 208 ST Quiz 2

Lecturer: Dr. Min Xu

Apr 9, 2002
LAST NAME: ___________________________________________________________________________________________________ FIRST NAME: __________________________________________________________________________________________________ SS# (last 4 digits): _____________________________________________________________________________________________

General Instruction

You have 1 hour 30 minutes to answer 4 questions on this exam. Work each problem only on the sheet provided for this problem, using the back if necessary. Please write the name and recitation section on each sheet at the top of the page.

You must indicate the reasoning and show all the steps.

If you are unable to obtain the answer to part of a problem and you need the answer to solve the subsequent parts, use an appropriate symbol for the answer to the previous part and solve in terms of that 1symbol.

1. (25 points) Two point charges Q1 = Q2 = +25nC are located on (0, 3m) and (0,-3m) on the y-axis. A testing charge q = +1nC is put at position (4m, 0) on the x-axis.


(a) (5 points) Draw a free-body diagram for the testing charge.
(b) (5 points) Find the electric force (magnitude and direction) acting on the testing charge.
(c) (5 points) Find the potential energy of the above testing charge.
(d) (5 points) If the testing charge is released from rest at its initial position, it will escape. Why? How about a negative testing charge?
(e) (5 points) Find the escaping speed when this testing charge moves to infinity. Assume the mass of the testing charge is 10-10g.
PIC
Figure 1: Figure for question 1.

2. (25 points) The Gauss’s law. Charge Q is uniformly distributed in a sphere with radius R.


(a) (5 points) Find the net electric flux through the surface of a concentric sphere with radius r when i) r < R and ii) r > R.
(b) (5 points) Find the electric field at a distance r away from the center when i) r < R and ii) r > R.
(c) (5 points) Draw a figure showing the radial component of the electric field as a function of the distance r.
(d) (10 points) Find the electric potential at a distance r away from the center when i) r < R and ii) r > R. Take the potential at infinity as zero. Hint: Using the relation
dV E = - ---. dr
(1)

3. (25 points) A capacitor is a device which is able to store electric energy. In a network of capacitors shown in the figure below, the capacitor C1 = 4mC has an initial voltage 100V . The switch is then closed such that the capacitor C2 = 2mF is charged by C1.


(a) (5 points) Find the charge Q in C1 and the electric energy U stored in this network of capacitors before the switch is closed.
(b) (5 points) After the switch is closed, what is the equivalent capacitance of this network of capacitors? Are capacitors connected in serial or parallel?
(c) (5 points) Find the charge Q1 in C1 and Q2 in C2 after the switch is closed.
(d) (10 points) Find the electric energy U' stored in this network of capacitors after the switch is closed. Explain the energy loss if any.
PIC
Figure 2: Figure for question 3.

4. (25 points) A metal sphere with charge +q and radius ra is placed at the center of a metal spherical shell with charge -q, inner radius rb and outer radius rc. Here ra < rb < rc.


(a) (10 points) Find the potential V (r) for r < ra, ra < r < rb, rb < r < rc and r > rc.
(b) (5 points) Find the capacitance C formed by the sphere and the shell.
(c) (5 points) Find the surfaces charge density at the surface of the sphere, the inner and outer surfaces of the shell.
(d) (5 points) Draw the electric field lines in all regions.
(e) (extra credit, 10 points) If the space between the sphere and the shell is filled by a dielectrics with a dielectric constant K = 3, find the electric field in the regions ra < r < rb and the induced surface charges at r = ra and r = rb.
(f) (extra credits, 5 points) Following (e), redraw the electric field lines in all regions. Label the free surface charges and induced surface charges in your graph.