AP Physics C Questions: Parallel Plate Capacitors and Electric Fields

📚 Quick Study Guide

• ⚡ Capacitance (C): A measure of a capacitor's ability to store electric charge. For a parallel plate capacitor: $C = \frac{\epsilon_0 A}{d}$, where $\epsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12} \text{ F/m}$), $A$ is the area of the plates, and $d$ is the separation between the plates.
• 📊 Electric Field (E): The electric field between the plates of a parallel plate capacitor is uniform and is given by $E = \frac{V}{d}$, where $V$ is the potential difference between the plates. Also, $E = \frac{\sigma}{\epsilon_0}$, where $\sigma$ is the surface charge density.
• 🔋 Charge (Q), Voltage (V), and Capacitance (C) Relationship: The charge stored on a capacitor is related to the voltage across it by the equation $Q = CV$.
• 💡 Energy Stored (U): The energy stored in a capacitor is given by $U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{1}{2}\frac{Q^2}{C}$.
• 📐 Dielectrics: Inserting a dielectric material between the plates increases the capacitance by a factor of the dielectric constant $\kappa$: $C' = \kappa C$. The electric field decreases: $E' = \frac{E}{\kappa}$.
• 🧮 Surface Charge Density (σ): $\sigma = \frac{Q}{A}$, where Q is the charge on the plate and A is the area of the plate.

🧪 Practice Quiz

1. A parallel plate capacitor has a capacitance of $C$ with air between the plates. If the separation between the plates is doubled, what is the new capacitance?
   1. $2C$
   2. $C/2$
   3. $4C$
   4. $C/4$
2. The electric field between the plates of a parallel plate capacitor is $E$. If the charge on the plates is doubled, what is the new electric field?
   1. $E/2$
   2. $2E$
   3. $4E$
   4. $E$
3. A parallel plate capacitor is connected to a battery with a constant voltage $V$. If the distance between the plates is decreased, what happens to the charge on the capacitor?
   1. Decreases
   2. Increases
   3. Stays the same
   4. Becomes zero
4. A parallel plate capacitor is fully charged and then disconnected from the battery. A dielectric material with a dielectric constant $\kappa > 1$ is inserted between the plates. What happens to the voltage across the capacitor?
   1. Increases by a factor of $\kappa$
   2. Decreases by a factor of $\kappa$
   3. Stays the same
   4. Becomes zero
5. What is the effect on the capacitance of a parallel plate capacitor if both the area of the plates and the distance between them are doubled?
   1. Increases by a factor of 4
   2. Decreases by a factor of 4
   3. Stays the same
   4. Doubles
6. A parallel plate capacitor has a charge $Q$ and a voltage $V$. If the charge is tripled while keeping the capacitance constant, what is the new voltage?
   1. $V/3$
   2. $3V$
   3. $9V$
   4. $V$
7. Which of the following changes will increase the energy stored in a parallel plate capacitor (assuming it remains connected to a voltage source)?
   1. Decreasing the plate separation
   2. Decreasing the area of the plates
   3. Inserting a material with a dielectric constant of less than 1
   4. Increasing the plate separation