Physics 1102 In-Class Problems: Capacitors

1. Use the fact that the field due to a plane of charge is E = Σ/2ε₀ and the Principle of Conservation of Charge to prove the following.

(a) For very large thin conducting plate with total charge Q and area A, show that the charge and charge densities on the two sides are equal to ½Q.
(b) Consider two very large conducting plates which are initially far apart. One has total charge Q₁ and the other has total charge Q₂. Show that when the plates are brought closer together so that only a small gap separates them, that the charge on each plate redistributes itself so that the sides facing the gap have equal and opposite charges and charge densities while the outer sides must have equal charges and charge densities. Determine the net electric field to the left, between, and to the right of the plates.
2. Two very large thin square conducting plates are very far apart. The area of the plates is A. A total charge of +Q is added to the first plate; −Q to the second. The two plates are now brought closer together so that only a small gap separates them.
   (a) Use the results of question 1 to find the charge on each side of each plate.
   (b) Determine the electric field
       (i)   just to the left of the plates,
       (ii)  between the plates, and
       (iii) just to the right of the plates.
3. Two very large thin square conducting plates are very far apart. The area of the plates is 5.0 m². A total charge of 7.0 μC is added to the first plate; −5.0 μC to the second. The two plates are now brought closer together so that only a small gap separates them.
   (a) Use the results of question 1 to find the charge on each side of each plate.
   (b) Determine the electric field
       (i)   just to the left of the plates,
       (ii)  between the plates, and
       (iii) just to the right of the plates.
4. Two capacitors, one with charge Q_A and the other with charge Q_B, are connected by a single wire as shown in the diagram below. Prove that the charge on the plates does not change when the switch S is closed. Use the results of questions 1 and 2.

   

5. Two capacitors are charged separately to the same potential V. They are then connected in parallel with positive plate to positive plate and negative to negative. Prove (a) that the charge on the each capacitor resides only on the inner faces of the plates and (b) that the charge on the capacitors does not change when the switch S is closed. Use the results of questions 1 and 2.

   

6. A capacitor consists of two parallel flat plates of area A with equal and opposite charge density Σ and separation d. The electric field between the plates is constant at E = Σ/ε₀ and is directed from the positive plate to the negative. Find an expression for the capacitance of the parallel plate capacitor.
7. A capacitor consists of two concentric spherical shells. The inner shell at radius R has positive charge Q.  The outer shell at radius R + d has charge −Q. The distance d is much smaller than R. As a result the electric field is radial and has a magnitude E(r) = Q/4pe₀r². Find an expression for the capacitance.
8. A 4.00 μF capacitor and an 8.00 μF capacitor are separately charged by a 20.0 Volt power supply. The capacitors are then placed in the circuit shown below.
   (a) What is the charge on each plate when both switches are open?
   (b) What will be the charge on each plate when both switches are closed?

   

9. A 4.00 μF capacitor and an 8.00 μF capacitor are separately charged by a 20.0 Volt power supply. The capacitors are then placed in the circuit shown below. What will be the charge on each plate when both switches are closed? (Note that the polarities are switched from the previous question.)
   
10. In the figure below, C₁ = C₅ = 3.00 μF and C₂ = C₃ = C₄ = 2.00 μF. What is the equivalent capacitance of the circuit?

    

11. What is the equivalent capacitance of the circuit shown below?

    

12. A capacitor consists of two parallel flat plates of area A with equal and opposite charge density Σ and separation d. The electric field between the plates is constant at E = Σ/ε₀ and is directed from the positive plate to the negative. A dielectric of thickness ¹/₃d and constant k is inserted next to the positive plate. Find an expression for the capacitance of the parallel plate capacitor.
13. A capacitor consists of two concentric spherical shells. The inner shell at radius R has positive charge Q.  The outer shell at radius R + d has charge −Q. The distance d is much smaller than R.  As a result the electric field is radial and has a magnitude E(r) = Q/4pe₀r². A spherical shell dielectric of radius ½d is inserted next to the negative shell. Find an expression for the capacitance.
14. Two oppositely charged conducting plates, with equal magnitude of charge per unit area, are separated by a dielectric 3.00 mm thick, with a dielectric constant of 4.50. The resultant electric field in the dielectric is 1.60 × 10⁶ V/m. Compute (a) the charge per unit area on the conducting plates, and (b) the charge per unit area on the surfaces of the dielectric.
15. Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electric field is 3.60 × 10⁶ V/m. When the space is filled with a dielectric, the electric field is 1.20 × 10⁵ V/m. (a) What is the charge density on the surface of the dielectric? (b) What is the dielectric constant?
16. A capacitor that has air between its plates is connected across a potential difference of 12 V and stores 48 μC of charge. It is then disconnected from the source while still charged. (a) Find the capacitance of the capacitor. (b) A piece of Teflon (k = 2.1) is inserted between the plates. Find the voltage and charge on the capacitor. (c) Find its new capacitance.
17. Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 1.75 cm² and dielectric thickness of 0.04 mm. Breakdown occurs when the electric field exceeds 60 × 10⁶ V/m.
18. In question 10, a potential of 600 V is applied across points A and B. What is the charge on each capacitor? What is the energy stored in each capacitor?
19. In question 11, a potential of 24 V is applied across points A and B. What is the charge on each capacitor? What is the energy stored in each capacitor?
20. Three 10-μF capacitors are connected in parallel. A dielectric k = 2.0 is inserted into one of the capacitors. The capacitors are then connected to a 4.0 V battery. (a) What is the charge on each capacitor and what is the energy stored by each capacitor? (b) The battery is disconnected. The dielectric is then removed from the capacitor. What is the new charge on each capacitor and what now is the energy stored by each capacitor? (c) What happened to the lost energy?

Questions?mike.coombes@kwantlen.ca