AP Physics – Capacitance Examples

1. A capacitor of 25 μF is connected to a 1.50 V battery. The battery is then disconnected and in its place is connected a resistor of 33 Ω. (a) Find the initial charge on the capacitor.  (b) What is the initial power output of the capacitor?  (c) What is the current through the resistor when the capacitor has lost half its original charge?  (d) Determine the total energy dissipated by the resistor.

2. A 470 μF capacitor is given a charge of 3.0 mC.  It is then connected to a 220 μF capacitor.  (a) How much charge will be transferred between the capacitors?  (b) What will be the final voltage?  (c) How much heat is generated in the transfer process?

3. Two capacitors C₁ = 330 μF and C₂ = 220 μF are connected in series with a battery with emf 6.00 V and internal resistance 2.3 Ω and a SPST switch, initially open.  (a) Determine the initial current when the switch is closed (assuming both capacitors are initially uncharged).  (b) Find the eventual charge, voltage, and energy for each capacitor.  (c) Repeat for a parallel connection of the capacitors.

4. Capacitors C₁ = 1.5 mF, C₂ = 4.7 mF, and C₃ = 6.8 mF are connected to a 1.50 V battery as shown.  The switch is closed; this charges the capacitors.  Find the eventual charge, voltage, and energy for each capacitor. 

5. Capacitors C₁ = 4.7 mF and C₂ = 1.2 mF are connected to a 9.0 V battery as shown.  Capacitor C₁ is charged to 12 V as shown before the switch is closed.  Find the eventual charge and voltage on each capacitor.

6. A parallel plate capacitor has circular metal plates of diameter 25.0 cm separated by 2.50 mm.  (a) Determine the capacitance.  (b) Find the eventual charge on each plate if it is connected to a 6.00 V battery.  (c) If the battery remains connected determine the work necessary to double the separation of the plates.

7. A capacitor has parallel plates, each with area 30.0 cm², separated by 0.500 cm.  Sandwiched between the plates is a layer of paraffin, which has dielectric constant 2.3 and dielectric strength of 11 MV/m.  (a) Find the capacitance.  (b) What maximum charge could be held?  (b) Find the charge and electric field strength when connected to a 12.0 V battery.  (c) If the battery is then disconnected, find the work necessary to remove the paraffin layer.

8. A capacitor has parallel plates, each with area 0.050 m², separated by 7.00 mm. A metal plate of the same cross section and thickness 3.00 mm is inserted precisely midway between the plates.  (a) Find the original capacitance.  (b) Determine the modified capacitance.  (c) What would be the effect of placing it "off-center"?  (d) Repeat if a pyrex plate (κ = 5.0) of the same dimensions is inserted instead of the metal plate.

9. Derive a formula for the capacitance of a spherical capacitor consisting of two metallic shells separated by air, as shown below.

10. Derive a formula for the capacitance of the capacitor shown below in which there are alternating charged plates, equally spaced.  Let A be the area of one plate and d be the separation.

11. The capacitor in the circuit below is charged and discharged using the SPDT switch.  Find expressions for voltage and current as functions of time.

12. Resistors R₁ = 330 Ω, R₂ = 470 Ω, capacitor C = 2.2 mF, and a 1.50 V battery are connected as shown.  The switch is closed at t = 0 and then opened at t = 100.0 s.  (a) Find the times at which the voltage of the capacitor is 0.75 V.  (b) Make a careful sketch of current vs. time for the battery and for R₂.

13. The capacitor in the circuit below is charged and discharged by closing and opening the switch.  (a) Sketch voltage vs. time graphs for each resistor and the capacitor.  Find the time constant for (b) discharging and (c) charging.