AP Physics C Ch 25, 26 Take Home Due 03/19/2013

1. A spherical capacitor consists of two thin concentric spherical shells of radii R1 and R2.

(a) Show that the capacitance is given by C  4 0 R1R2 R2  R1  . (b) Show that when the radii of the shells are nearly equal, the capacitance is given approximately by the expression 2 for the capacitance if a parallel-place capacitor, C   0 A d , where A is the area of the sphere (4R where

R  R1  R2 and d  R2  R1 .

- 1 - AP Physics C Ch 25, 26 Take Home Due 03/19/2013

2. A spherical capacitor has an inner sphere of radius R1 with a charge of +Q and an outer concentric spherical shell of radius R2 with a charge of –Q. (a) Find the electric field and the energy density at any point in space; that is for r R2. (b) Calculate the energy in the electrostatic field in a spherical shell of radius r, thickness dr, and volume 4r2dr between the conductors. (c) Integrate your expression in part (b) to find the total energy stored in the capacitor, and compare your result with that obtained using U =(1/2)QV. (This is problem #49 on page 780 in the text).

3. Determine the values for all the currents in the circuit (I1, I2, …). You may solve using Kirchhoff’s laws or the method of mesh currents. (For five bonus points solve the problem with both methods – of course in the end your answer should be the same with either method). Preliminary directions have already been assigned to the currents (and mesh currents) – this is not so much to help you as it is to assure that the answers will be easier to check!! Although the resistors are labeled, assume they all have the same value R. The potential source on the left (battery) has a potential of 2V, twice the potential of the source on the right (V). Your answers for each of the currents should be in the form of a number, or numerical fraction, multiplied by the ratio V/R.

4. The RC circuit shown was used as a lab, in which the charging equation for the capacitor was given as

 R1  R2    R   t / R2C    2   R1   Q   VC1 e  ,  R1  R2   

And some of the steps in the derivation of this expression were suggested in the lab instructions. Fill in the details of the derivation.

5. In the circuit at right, I1 = 8 A, and I3 = 1 A. Find the values of V, R, I2, and I4.

V: ______

R: ______

I2: ______

I4: ______

6. Solve for the currents in the circuit at right. Given that point B is grounded (at zero volts electric potential, determine the potential at points A, C, D, E, F & G. . (Hint: It’s not as complex as it looks. You can determine one of the current values and several of the potentials just by looking …).

I1: ______

I2: ______

I3: ______

I4: ______

I5: ______

VA: ______

VB: ______

VC: ______

VD: ______

VE: ______

VF: ______

7. In the figure at right switch S1 (only) is closed. (a) How much time must elapse before the capacitor is charged to 90% of its maximum value? (b) When the capacitor is fully charged, how much energy (in joules) is stored on the capacitor, and how much energy has been dissipated in the resistor? (c) Switch S1 is now opened and S2 is closed. How much time elapses before the capacitor is 90% discharged? (d) How much energy has been dissipated in the resistor when it is 90% discharged?