CAPACITANCE EXERCISE COMBINATION OF CAPACITORS 1. In this exercise you are to find the voltage across each capacitor and the charge on each capacitor in the circuit below. Use V = 12 volts, C1 = 3 μF, C2 = 2 μF, and C3 = 4 μF. 1.1. Before solving the problem, answer the following: (a) Do you expect the voltages across each capacitor to be the same? Are there any capacitors that you know will have the same voltage across them? (b) Do you expect the charges on each capacitor to be the same? Are there any capacitors that you know will have the same charge? 1.2. Draw a simplification of the circuit by reducing the parallel combination of C2 and C3 to a single capacitor Cp. What is the value of Cp? 1.3. Now, are the charges the same on C1 and Cp? What about the voltages? 1.4. Now reduce the series combination C1 and Cp to the equivalent Cs. Find Cs and redraw the simplified circuit. 1.5. Find the charge on Cs using Q = CV. Explain why this is the same as the charge on C1 and Cp. 1.6. Use the relationship V = Q/C to find the voltages across C1 and Cp. Is V1 + Vp = 12 volts? 1.7. Use the relationship Q = CV to find the charges on C2 and C3. Does Q2 + Q3 = Qs? CAPACITOR CONNECTED TO A SPRING 2. Consider an air‐filled parallel‐plate capacitor with one plate (labeled a) connected to a spring having a force constant k, and another plate (b) held fixed as shown in the figure. If the charges placed on plates a and b are +Q
and ‐Q, respectively, how much does the spring expand from the position (x 0) with no charges present? Hint: What is the electric field generated by the charges on plate b? CAPACITOR NETWORK 3. Evaluate the equivalent capacitance of the configuration shown in the figure. All the capacitors are identical, each with a capacitance C. C
C
C
C
C
C
IMPOSSIBLE NETWORK 4. Determine the equivalent capacitance of the combination of capacitors shown in the figure. Hint: You will have to take advantage of the symmetry of the problem! C
2C
3C
C
2C
SPHERICAL CAPACITOR 5. A conducting solid sphere of radius a, carrying a charge +Q is surrounded by a thin conducting spherical shell with radius b (b>a) carrying a charge of –
Q. This is a spherical capacitor. a) What is the direction and magnitude of the electric field in the following regions: 1.) r<a 2.) a<r<b and 3.) r>b a
b
+Q
‐Q
b) What is the potential difference ∆ between the spherical shell and the solid sphere? What is the capacitance of the spherical capacitor?