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2.2 Capacitors

4 min readdecember 29, 2022

Peter Apps

Peter Apps

Peter Apps

Peter Apps

What's a Capacitor?

A is a device that can be used to store charge, and therefore, electrical potential energy. They are used in a wide range of electrical devices including the flash on your cell phone camera. There are several different ways to construct a , but we're going to focus on the parallel-plate version.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-M8yUNdiWK0IX.jpg?alt=media&token=cc66f589-5189-4690-9bcd-ba5f6569d5ee

Image from Wikipedia

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-G7NPO3hbKvbI.png?alt=media&token=f07c2b20-f23c-4ecd-95cc-08257815fa66

Image from Wikipedia

The parallel plate is created by taking two conductive plates and separating them by a small distance. A is often added to increase the amount of charge a can store. We'll discuss more about dielectrics in the next section.

Theory Time - Parallel Plate Capacitor

Let's create a simple using two metal plates and connect them to a battery to charge them up. Recall from Unit 1, that the strength of the electric field is proportional to the amount charge.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4nupvF7qsIhj.PNG?alt=media&token=1dbf4fe1-f56e-4964-affd-bac1d9b6542a

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-VCGsLIDHkWPy.jpg?alt=media&token=60e2f47b-ee5e-48e0-b274-492407fcfd01

Image from opentextbc.ca

We also know that the potential difference (V) between the plates is related to the electric field through ∆V = -∫E*dr. Following this thought process, we can see that V is proportional to Q as well. The more charge that gets stored on each plate, the stronger the field, and the higher the voltage between the plates will be. We'll define a new quantity, capacitance (C), as the constant of proportionality between V and Q such that:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jdl7GdpqeFKN.PNG?alt=media&token=e7e9ef23-162a-491a-b9c5-ffa8ad057c92

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

We can also define capacitance in terms of the physical dimensions of the . Recall that σ = Q/A (area charge density for a sheet of charge, and E = σ / e0 for a conductive plate.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%203.04-x9TN0dgmmK9B.png?alt=media&token=49988805-4efd-4e93-9c08-f9cc4bb16c77

From here, we can see that capacitance is directly proportional to the area of the plates (A) and inversely proportional to the distance between them. This should make sense since a larger plate has more room for the charge to occupy and, therefore, moe should be able to fit on it.

Other Types of Capacitors

You might also be asked to derive capacitance equations for other geometrically shaped capacitors on the AP exam (even though the most common are parallel plate and spherical). The basic process is the same for each of these. We'll need to use the appropriate electric field equation (from Unit 1) and use it to find the change in voltage (∆V = ∫E*dr) from which we can find the capacitance equation C = Q/V. You may also need to use ρ, σ, λ for volume, area, and linear charge densities respectively.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-8coUfaVK5oo3.png?alt=media&token=4bfd4b18-fef6-4079-89ab-0e03f0bb43a1

Image from Wikipedia.org

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-v9TT6mhZAzGm.png?alt=media&token=a850d52c-12d1-45dd-aa72-4dc8d5ad709e

Image from toppr.com

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dwnzY4tDCJfo.PNG?alt=media&token=4ee2021e-ce03-4e13-810a-3b1cbb5cb5c8

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-k39g3A0KV6o7.PNG?alt=media&token=452cbab2-58ef-4388-91bf-2451fac48804

Energy in a Capacitor

Because the stores charge, it also stores electric potential energy (Uc). The amount of energy stored can be determined through the following derivation. We're going to imagine moving a tiny bit of charge (dq) from one side of the to the other. The charge will gain energy that's proportional to the potential difference passed through.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rVWjdhiwYJ8a.PNG?alt=media&token=76ed26a2-cebc-4e38-8ba9-bfe7ade2f29b

Take a few minutes and check out this Phet Simulation where you can alter the physical properties of a parallel plate and see the effect on voltage, field strength and energy stored

Practice Questions

  1. A 20 µF parallel-plate is fully charged to 20 V. The energy stored in the is most nearly __________.

    • Answer:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6F8e7uob1GJK.PNG?alt=media&token=1673be69-dec4-4c7a-b2dc-9c3c5b991840

  2. A 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:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vkH0yf9gS065.PNG?alt=media&token=ce734fe0-89b2-4651-95a6-6cbd3d572514

  3. Practice FRQ

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.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%203.10-wxqBN1Q8Hq1n.png?alt=media&token=d17d94f9-ffaa-4fc8-a3e8-e79958542acc

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 . 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

(c) Determine the magnitude of the potential difference between the sphere and the conducting shell.

(d) Determine the capacitance of the .

Answer:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-QKpY2nh9YIft.PNG?alt=media&token=cbbae534-8414-4f5c-8865-fb39bcadad35

Key Terms to Review (7)

Area charge density (σ)

: Area charge density refers to the amount of electric charge per unit area. It is a measure of how much charge is distributed over a given surface.

Capacitor

: A capacitor is an electronic component that stores and releases electrical energy. It consists of two conductive plates separated by an insulating material, known as a dielectric.

Dielectric

: A dielectric is an insulating material that can be placed between the plates of a capacitor to increase its capacitance. It reduces the electric field inside the capacitor, allowing for more charge to be stored.

Electric field (E)

: The electric field is a vector quantity that describes the force experienced by a charged particle at any point in space. It is created by other charged particles and can exert forces on other charges placed within its influence.

Energy stored in a capacitor (Uc)

: The energy stored in a capacitor refers to the amount of electrical energy that can be stored in the electric field between its plates. It is determined by the capacitance and voltage across the capacitor.

Permittivity of free space (ε0)

: The permittivity of free space, denoted as ε0 (pronounced "epsilon naught"), is a fundamental constant in physics that represents the ability of a vacuum to permit the formation of an electric field. It quantifies how easily electric fields can be established in empty space.

Spherical Capacitor

: A spherical capacitor is a device that consists of two concentric conducting spheres, with the inner sphere acting as the positive plate and the outer sphere acting as the negative plate. It stores electric charge and has capacitance.

2.2 Capacitors

4 min readdecember 29, 2022

Peter Apps

Peter Apps

Peter Apps

Peter Apps

What's a Capacitor?

A is a device that can be used to store charge, and therefore, electrical potential energy. They are used in a wide range of electrical devices including the flash on your cell phone camera. There are several different ways to construct a , but we're going to focus on the parallel-plate version.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-M8yUNdiWK0IX.jpg?alt=media&token=cc66f589-5189-4690-9bcd-ba5f6569d5ee

Image from Wikipedia

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-G7NPO3hbKvbI.png?alt=media&token=f07c2b20-f23c-4ecd-95cc-08257815fa66

Image from Wikipedia

The parallel plate is created by taking two conductive plates and separating them by a small distance. A is often added to increase the amount of charge a can store. We'll discuss more about dielectrics in the next section.

Theory Time - Parallel Plate Capacitor

Let's create a simple using two metal plates and connect them to a battery to charge them up. Recall from Unit 1, that the strength of the electric field is proportional to the amount charge.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4nupvF7qsIhj.PNG?alt=media&token=1dbf4fe1-f56e-4964-affd-bac1d9b6542a

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-VCGsLIDHkWPy.jpg?alt=media&token=60e2f47b-ee5e-48e0-b274-492407fcfd01

Image from opentextbc.ca

We also know that the potential difference (V) between the plates is related to the electric field through ∆V = -∫E*dr. Following this thought process, we can see that V is proportional to Q as well. The more charge that gets stored on each plate, the stronger the field, and the higher the voltage between the plates will be. We'll define a new quantity, capacitance (C), as the constant of proportionality between V and Q such that:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jdl7GdpqeFKN.PNG?alt=media&token=e7e9ef23-162a-491a-b9c5-ffa8ad057c92

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

We can also define capacitance in terms of the physical dimensions of the . Recall that σ = Q/A (area charge density for a sheet of charge, and E = σ / e0 for a conductive plate.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%203.04-x9TN0dgmmK9B.png?alt=media&token=49988805-4efd-4e93-9c08-f9cc4bb16c77

From here, we can see that capacitance is directly proportional to the area of the plates (A) and inversely proportional to the distance between them. This should make sense since a larger plate has more room for the charge to occupy and, therefore, moe should be able to fit on it.

Other Types of Capacitors

You might also be asked to derive capacitance equations for other geometrically shaped capacitors on the AP exam (even though the most common are parallel plate and spherical). The basic process is the same for each of these. We'll need to use the appropriate electric field equation (from Unit 1) and use it to find the change in voltage (∆V = ∫E*dr) from which we can find the capacitance equation C = Q/V. You may also need to use ρ, σ, λ for volume, area, and linear charge densities respectively.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-8coUfaVK5oo3.png?alt=media&token=4bfd4b18-fef6-4079-89ab-0e03f0bb43a1

Image from Wikipedia.org

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-v9TT6mhZAzGm.png?alt=media&token=a850d52c-12d1-45dd-aa72-4dc8d5ad709e

Image from toppr.com

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dwnzY4tDCJfo.PNG?alt=media&token=4ee2021e-ce03-4e13-810a-3b1cbb5cb5c8

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-k39g3A0KV6o7.PNG?alt=media&token=452cbab2-58ef-4388-91bf-2451fac48804

Energy in a Capacitor

Because the stores charge, it also stores electric potential energy (Uc). The amount of energy stored can be determined through the following derivation. We're going to imagine moving a tiny bit of charge (dq) from one side of the to the other. The charge will gain energy that's proportional to the potential difference passed through.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rVWjdhiwYJ8a.PNG?alt=media&token=76ed26a2-cebc-4e38-8ba9-bfe7ade2f29b

Take a few minutes and check out this Phet Simulation where you can alter the physical properties of a parallel plate and see the effect on voltage, field strength and energy stored

Practice Questions

  1. A 20 µF parallel-plate is fully charged to 20 V. The energy stored in the is most nearly __________.

    • Answer:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6F8e7uob1GJK.PNG?alt=media&token=1673be69-dec4-4c7a-b2dc-9c3c5b991840

  2. A 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:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vkH0yf9gS065.PNG?alt=media&token=ce734fe0-89b2-4651-95a6-6cbd3d572514

  3. Practice FRQ

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.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%203.10-wxqBN1Q8Hq1n.png?alt=media&token=d17d94f9-ffaa-4fc8-a3e8-e79958542acc

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 . 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

(c) Determine the magnitude of the potential difference between the sphere and the conducting shell.

(d) Determine the capacitance of the .

Answer:

      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-QKpY2nh9YIft.PNG?alt=media&token=cbbae534-8414-4f5c-8865-fb39bcadad35

Key Terms to Review (7)

Area charge density (σ)

: Area charge density refers to the amount of electric charge per unit area. It is a measure of how much charge is distributed over a given surface.

Capacitor

: A capacitor is an electronic component that stores and releases electrical energy. It consists of two conductive plates separated by an insulating material, known as a dielectric.

Dielectric

: A dielectric is an insulating material that can be placed between the plates of a capacitor to increase its capacitance. It reduces the electric field inside the capacitor, allowing for more charge to be stored.

Electric field (E)

: The electric field is a vector quantity that describes the force experienced by a charged particle at any point in space. It is created by other charged particles and can exert forces on other charges placed within its influence.

Energy stored in a capacitor (Uc)

: The energy stored in a capacitor refers to the amount of electrical energy that can be stored in the electric field between its plates. It is determined by the capacitance and voltage across the capacitor.

Permittivity of free space (ε0)

: The permittivity of free space, denoted as ε0 (pronounced "epsilon naught"), is a fundamental constant in physics that represents the ability of a vacuum to permit the formation of an electric field. It quantifies how easily electric fields can be established in empty space.

Spherical Capacitor

: A spherical capacitor is a device that consists of two concentric conducting spheres, with the inner sphere acting as the positive plate and the outer sphere acting as the negative plate. It stores electric charge and has capacitance.


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© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.