102 PHYS General Physics II
Midterm Exam II, April 26, 2008 Duration: 100 minutes NAME:.................................................................... Section:......................
Q.1 (35)
Q.2 (35)
Q.3 (35)
Total (105)
Suggestions:
1. 2. 3. 4. Read the questions carefully. State the solutions clearly and with necessary comments (explanations). Write legibly. Check your results in terms of dimensions, units, and special limits of the problem.
Note: Solutions will be available just after the exam at http://www.fen.bilkent.edu.tr/~phys102
Q.1 (35 points) RC Circuit
An EMF source with potential difference and internal resistance r is connected to a resistor R and a capacitor with capacitance C in series connection through a switch S as in the figure. Initially there is no charge in the capacitor and at t=0 switch S is closed. Write your answers in terms of , r, R, C and t.
R
S
C
r
(a) (5pts) Show the direction of the positive current flow and write down the equation for charge on the capacitor for t>0, using Kirchhoff's rules. (Note that your equation should include charge q, only (not current) as a dependent variable. (b) (5pts) Show that q (t ) = C 1 - e-t / , is the solution for the equation you write in part (a) by
directly substituting it into equation provided that is chosen properly. Determine (also specify its SI unit).
(
)
(c) (5pts) What is the initial current at t=0 when switch S is closed? What is the current passing through the circuit at t>0? (d) (5pts) What is the maximum amount of charge that can be stored in the capacitor and when is this amount of charge accumulated in the capacitor? (e) (5pts) Plot current versus time and charge versus time graphs. Show the initial and final current/charge values in the graphs. (f) (5pts) How much energy is stored in the capacitor at some time t1>0? (g) (5pts) How much heat (energy) has been generated in the resistor R from t=0 to t=t1?
Q.2 (35 points) Magnetic Field
Two current loops with equal current I (both oriented the in same direction) and radii R are separated by a long distance L as shown in the figure. L is much larger than R. (a) Find the magnetic moments of both current loops. (b) Find the magnetic field generated by the loop on the left at the center of the other loop on the right (c) Show that, in order to separate these loops infinitely far apart, one has to do a positive work given by
W = 0
2 ( L2 + R 2 )
1 2
3/ 2
.
I R
I R
L
Q.3 (35 points) Electromagnetic Induction
A rectangular loop of wire with mass m, width w, vertical length l, and resistance R falls out of a magnetic field under the influence of gravity. The magnetic field is uniform and out of the paper ) within the area shown (see sketch) and zero outside of that area. At the time shown in ( , where the sketch, the loop is exiting the magnetic field at speed (meaning the loop is moving downward, not upward). Suppose at time t the distance from the top of the loop to the point where the magnetic field goes to zero is z(t) (see sketch).
(a) What is the relationship between V(t) and z(t)? Be careful of your signs here, remember that z(t) is positive and decreasing with time, so dz(t)/dt < 0. (b) If we define the area vector to be out of the page, what is the magnetic flux B through our circuit at time t (in terms of z(t), not V(t)). (c) What is dB / dt ? Is this positive or negative at time t? Be careful here, your answer should include V(t) (not z(t) and remember that V(t) < 0 ). (d) What is the direction (clockwise or counterclockwise) and magnitude of the induced current on the loop of wire? (e) What is the direction (into the page or out of the page) of the self-magnetic field due to the induced current inside the circuit loop? (f) Besides gravity, what other force acts on the loop in the z-direction? Give the magnitude
and direction of this force in terms of the quantities given. (Hint: use dF = Idl B )
(g) What is the magnitude of the terminal velocity?
(Extra blank sheet; please indicate the problem number you continue)