#AP Physics C: E&M - Capacitors Study Guide ⚡️

Hey there, future physics master! Let's dive into capacitors, those amazing little devices that store charge and energy. This guide is designed to be your go-to resource for acing the AP exam, especially when you're doing some last-minute review. Let's get started!

#What is a Capacitor?

A capacitor is a device that stores electrical charge and potential energy. Think of it like a tiny rechargeable battery, but instead of chemical reactions, it uses electric fields to store energy. Capacitors are everywhere, from your phone's flash to complex circuits. We'll focus on the parallel-plate capacitor, which is the most common type you'll encounter.

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Parallel Plate Capacitor

Quick Fact

A parallel plate capacitor consists of two conductive plates separated by a small distance, often with a dielectric material in between.

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Different types of capacitors

#Parallel Plate Capacitor Theory 🎓

Let's imagine charging a capacitor by connecting it to a battery. The battery moves charge, creating an electric field between the plates.

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Capacitor connected to a battery

Key Concept

The potential difference (V) between the plates is proportional to the charge (Q) stored. The more charge, the stronger the electric field and the higher the voltage.

We define capacitance (C) as the constant of proportionality between Q and V:

$C = \frac{Q}{V}$

Quick Fact

The unit for capacitance is the Farad (F), where 1 F = 1 C/V.

#Capacitance and Physical Dimensions

We can also define capacitance based on the physical properties of the capacitor. Recall that the surface charge density is given by $\sigma = \frac{Q}{A}$ and the electric field of a conductive plate is given by $E = \frac{\sigma}{\epsilon_0}$ .

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Key Concept

For a parallel plate capacitor with air (or vacuum) between the plates, the capacitance is given by: $C = \frac{\epsilon_0 A}{d}$ where:

$\epsilon_0$ is the permittivity of free space
A is the area of the plates
d is the distance between the plates

Exam Tip

Remember: Capacitance is directly proportional to the plate area (A) and inversely proportional to the distance between the plates (d). Think: bigger plates = more room for charge, closer plates = stronger attraction.

#Other Capacitor Geometries

You might encounter other capacitor shapes on the AP exam, such as spherical or cylindrical capacitors. The process to find their capacitance is the same:

Find the electric field using Gauss's Law.
Calculate the potential difference using $\Delta V = - \int E \cdot dr$ .
Use $C = \frac{Q}{V}$ to find the capacitance.

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Spherical Capacitor

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Cylindrical Capacitor

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#Energy in a Capacitor 💡

Capacitors store energy in the electric field between their plates. To calculate this energy, we imagine moving a tiny bit of charge (dq) from one plate to the other.

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Key Concept

The energy stored in a capacitor (Uc) can be expressed in three equivalent ways: $U_c = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$

Memory Aid

Remember: $U_c = \frac{1}{2}QV$ (it's like the area of a triangle: $\frac{1}{2} \times base \times height$ ). Then, use $Q=CV$ to derive the other two forms.

Exam Tip

Choose the formula that uses the variables you know. If you have Q and V, use $U_c = \frac{1}{2} QV$ . If you have C and V, use $U_c = \frac{1}{2} CV^2$ . If you have Q and C, use $U_c = \frac{Q^2}{2C}$ .

Check out this Phet Simulation to visualize how changing the physical properties affects voltage, field strength, and stored energy.

#Final Exam Focus

Here's what to focus on for the exam:

Capacitance Calculations: Be comfortable calculating capacitance for parallel-plate, spherical, and cylindrical capacitors.
Energy Storage: Master the energy equations and understand how energy is affected by changes in charge, voltage, and capacitance.
Dielectrics: Understand how dielectrics affect capacitance and energy storage (we will cover this in the next section).
Conceptual Understanding: Know the relationship between charge, voltage, electric field, and capacitance.

Exam Tip

Time Management: Start with what you know. If a question looks tricky, move on and come back to it later.

Common Mistake

Common Pitfall: Forgetting units! Always include units in your calculations and answers. Also, make sure to use consistent units (e.g., meters, not centimeters).

#Practice Questions 👍

Practice Question

#Multiple Choice Questions

A 20 µF parallel-plate capacitor is fully charged to 20 V. The energy stored in the capacitor is most nearly __________.- Answer:
A capacitor with circular parallel plates of radius R that are separated by a distance d has a capacitance of C. What would the capacitance (in terms of C ) be if the plates had radius 2R and were separated by a distance d/2?- Answer:

#Free Response Question

An isolated conducting sphere of radius a = 0.20 m is at a potential of -2,000 V.

(a) Determine the charge Q_0 on the sphere.

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The charge sphere is then concentrically surrounded by two uncharged conducting hemispheres of inner radius b = 0.40 m and outer radius c = 0.50 m, which are joined together as shown above, forming a spherical capacitor. A wire is connected from the outer sphere to ground, and then removed.

(b) Determine the magnitude of the electric field in the following regions as a function of the distance r from the center of the inner sphere.

i. r < a

ii. a < r < b

iii. b < r < c

iv. r > c

(d) Determine the capacitance of the spherical capacitor.