Stored Energy and Dielectrics

Original source retained for reference; this draft does not establish AP coverage.

Original author draft. Original wording and recorded answer states are preserved without claiming reviewed scientific correctness or AP alignment.

Physics 2 original textbook · Chapter 3

Reviewed corrections and authored clarifications are applied to this displayed edition. The original manuscript remains preserved.

Archived original reading · contains source errors and unfinished material

Section 3

Stored Energy and Dielectrics

Arguably the most important and practical concept of the capacitor is that the capacitor can be considered to store energy. This arises because when you charge a capacitor you are in essence moving charges onto the plates. When these charges are put onto the plate they begin to create an electric field and each charge after the first then has to work against this field to get onto the plate which then creates an even greater electric field. When the capacitor discharges this work done charging the capacitor is then released as usable energy.

To get an expression for this energy we must consider the capacitance of the capacitor.

Since the capacitance is a property of the geometry of the capacitor and not the charge and voltage this ratio holds for any amount of charge and voltage on the capacitor so that at any point in time

Recall that potential energy and voltage are related by the fact that

If we move charges onto the capacitor we are then creating potential energy. If we solve the capacitance equation for v we then get that

If we were to then add an infinitesimal amount of charge dq to the capacitor plate we would be increasing the potential energy by an infinitesimal amount du

Finally when we completely charge the capacitor we then creating a potential energy

Example 3.3.1

A parallel plate capacitor has a capacitance of 2200 μf and maximum voltage of 25 V. What is the maximum amount of energy the capacitor may hold?

In this problem we are given the capacitance and maximum voltage of a capacitor and asked for the maximum amount of energy the capacitor can store. Unfortunately we do not yet have a formula which relates the appropriate quantities.

Let’s recall what we do known very quickly. For one we know how that capacitance relates to the charge on a capacitor and the voltage across the plate. This relation is the definition of capacitance.

We also know how the capacitance and charge relates to the energy stored.

Notice that Q appears in both equations and is the only thing (besides the energy) that we don’t have. We can easily cancel is out by substituting for Q in the energy equation. Solving the capacitance equation for Q and substituting that in for Q in the energy equation gives us the following.

Now all that’s left to do is make the appropriate numerical substitutions.

Remember when we placed a neutral conductor inside an electric field? Free charges moved with the field to create an electric field inside which opposed the electric field outside. In this state we said that the conductor was polarized by the electric field. In a conductor this was allowed because free electrons can easily move around. Insulating materials do not exhibit this behavior due to all their charge being bound.

We now introduce a third, intermediary class of materials called dielectrics. In dielectric materials charges are free to move but are still in bound states, kind of like being attached to a spring. In the presence of electric fields dielectrics polarize like a conductor but unlike a conductor the dielectric does not fully polarize because the dielectric material holds charges close their “rest” positions.

Many different types of dielectrics exist but for this book we will limit ourselves to what are called linear dielectrics. In a linear dielectric the polarization and hence the electric field within the material varies linearly with the electric field outside.

Since a linear dielectric is much like a conductor when it is placed in an electric field a polarization occurs. This polarization if you recall causes an electric field to be produced which opposes the outside electric field. Since a dielectric will not completely polarize instead of completely counteracting the outside electric field the inside electric field will only reduce the strength of the outside field inside the dielectric. This decrease in electric field can be modeled with a constant. This means that the voltage will also be reduced by the same constant value. If we let this reduction be represented by

Then for a capacitor with vacuum capacitance C and voltage V we can get an expression for the capacitance of a capacitor using a linear dielectric.

Interestingly despite a decrease in the field between the plates the capacitance of the capacitor changes by a multiple of the dielectric constant K.

Example 3.3.2

If a capacitor is measured to have capacitance 1 F in vacuum and after a linear dielectric is placed inside the capacitance of the capacitor is measured as 2 F what is the dielectric constant of the material inserted into the capacitor?

To solve this problem all we must remember is the relationship between a capacitor with a dielectric and the capacitance of a capacitor

Solving for K we have that

Now all we have to do is plug in a few numbers.

Section 3

Stored Energy and Dielectrics

To get an expression for this energy we must consider the capacitance of the capacitor.

Since the capacitance is a property of the geometry of the capacitor and not the charge and voltage this ratio holds for any amount of charge and voltage on the capacitor so that at any point in time

Recall that potential energy and voltage are related by the fact that

If we move charges onto the capacitor we are then creating potential energy. If we solve the capacitance equation for v we then get that

If we were to then add an infinitesimal amount of charge dq to the capacitor plate we would be increasing the potential energy by an infinitesimal amount du

Finally when we completely charge the capacitor we then creating a potential energy

Example 3.3.1

A 2200 μF capacitor is charged to 25 V. Find its stored energy.

Authored clarification: Use ideal lumped capacitance; μF=10⁻⁶ F.

Reviewed worked solution:

Choose the energy formula.
$U=\tfrac12CV^2$
Convert capacitance.
$C=2200\times10^{-6}\,\mathrm F=0.0022\,\mathrm F$
Substitute.
$U=\tfrac12(0.0022)(25)^2=0.6875\,\mathrm J$

Answer: Stored energy is 0.6875 J.

Check: C·V² has units joules; the source’s 1100 J is incorrect.

Gallery 1.2 Stored Energy

Scroll within this figure to read its labels. Open the full-size reviewed figure.

Original frame 2

In the ideal wide parallel-plate model, the interior electric field points from the positive right plate toward the negative left plate. Exterior field is negligible away from the edges. The corrected illustration shows this direction.

Reviewed drawing: In the ideal wide-plate model, electric field points left from the positive right plate toward the negative left plate. The exterior field is negligible away from the edges.

Original image

Archived original widget · contains the source error

Archived uncorrected source figure — When the plates are uncharged there is no force on an approaching charge.

A neutral conductor polarizes through redistribution of free electrons opposite the applied field. Insulating materials can also polarize: their bound charges shift slightly or molecular dipoles align, without free conduction across the material. Both processes conserve total charge.

A dielectric is an insulating material whose bound charges can polarize. It is not a third class between conductors and insulators. For a linear dielectric, the polarization is proportional to the local electric field; bound charges remain tied to their atoms or molecules.

For a capacitor completely filled by a linear dielectric of relative permittivity K, $C=KC_0$ . At fixed free charge Q (an isolated capacitor), $E=E_0/K,\quad V=V_0/K,\quad U=U_0/K$ . At fixed voltage V (a connected ideal supply), the plate separation fixes E=V/d, while Q and stored energy U increase by K. The boundary condition must be stated.

Then for a capacitor with vacuum capacitance C and voltage V we can get an expression for the capacitance of a capacitor using a linear dielectric.

Interestingly despite a decrease in the field between the plates the capacitance of the capacitor changes by a multiple of the dielectric constant K.

Example 3.3.2

An air-filled capacitor has capacitance 1 F; complete filling by a linear dielectric makes it 2 F. Find K.

Authored clarification: Geometry is unchanged and the dielectric fully fills the field region.

To solve this problem all we must remember is the relationship between a capacitor with a dielectric and the capacitance of a capacitor

Solving for K we have that

Now all we have to do is plug in a few numbers.

Review 1.3 Stored Energy and Dielectrics

This review preserves the author’s recorded prompts, choices, and answer states. Corrected items have separate reviewed explanations and keys; unchanged draft keys are marked unreviewed. Unfinished items are not scored.

Original question 1

An ideal vacuum parallel-plate capacitor has area A, gap d and free plate charge densities ±σ. What is its energy?

$U={\sigma^2Ad\over2\epsilon_0}$
$U={\sigma^2Ad\over\epsilon_0}$
0

Reviewed answer and reasoning

Use E=σ/ε₀ and electric energy density ε₀E²/2 over volume Ad.

Authored correction or explicit assumptions; this reviewed key is separate from the archived original recorded states.

Archived original question and recorded key · contains the source error

Calculate the energy stored in a parallel-plate capacitor of area A, separation d, and charge distribution σ.

None of the above

Original recorded key: A

Original question 2

A linear dielectric with relative permittivity K completely fills a capacitor while an ideal supply keeps V fixed. By what factor does its stored electric energy change?

$K$
$1/K$
1
$K^2$

Reviewed answer and reasoning

Since C becomes KC₀ and $U=CV^2/2$ , the energy becomes KU₀. An isolated fixed-Q capacitor instead has U₀/K.

Authored correction or explicit assumptions; this reviewed key is separate from the archived original recorded states.

Archived original question and recorded key · contains the source error

In real world applications dielectrics are used to increase the energy capacity of a capacitor. By what factor does a dielectric with constant K increase the stored energy?

None of the above

Original recorded key: D

Original question 3

A long vacuum coaxial capacitor has inner radius a and outer radius b>a. The outer radius becomes 2b while length and voltage stay fixed. What is U′/U?

${\ln(b/a)\over\ln(2b/a)}$
${\ln(2b/a)\over\ln(b/a)}$
2
1

Reviewed answer and reasoning

At fixed V, energy is proportional to $C=2\pi\epsilon_0L/\ln(b/a)$ . Both nonzero Q and V cannot remain fixed if C changes.

Authored correction or explicit assumptions; this reviewed key is separate from the archived original recorded states.

Archived original question and recorded key · contains the source error

If charge and voltage are assumed to be constant by what factor does the energy capacity of a cylindrical capacitor increase if the outer cylinder is doubled.

0
2
None of the above

Original recorded key: C

Original question 4

A parallel-plate capacitor’s area doubles with its plate separation and voltage kept fixed. What is U′/U?

Reviewed answer and reasoning

Capacitance doubles because C=εA/d; at fixed V, U=CV²/2 also doubles. Charge is not fixed in this setup.

Authored correction or explicit assumptions; this reviewed key is separate from the archived original recorded states.

Archived original question and recorded key · contains the source error

If charge and voltage are kept constant by what factor does the stored energy of a parallel plate capacitor increase if the area of the plates is doubled?

1
2
3
4
None of the above

Original recorded key: B

Original question 5

A dielectric material polarizes when placed into an electric field.

True
False

Reviewed answer and reasoning

A dielectric polarizes in an applied electric field.

The independent semantic audit confirms this source prompt and recorded key under its stated conditions.

Archived original question and recorded key · contains the source error

A dielectric material polarizes when placed into an electric field.

True
False

Original recorded key: A

Reviewed worked examples

These authored derivations are paired with the original example identities. Added assumptions are stated.

Example 3.3.1 · reviewed solution

A 2200 μF capacitor is charged to 25 V. Find its stored energy.

Authored clarification: Use ideal lumped capacitance; μF=10⁻⁶ F.

Choose the energy formula.
$U=\tfrac12CV^2$
Convert capacitance.
$C=2200\times10^{-6}\,\mathrm F=0.0022\,\mathrm F$
Substitute.
$U=\tfrac12(0.0022)(25)^2=0.6875\,\mathrm J$

Stored energy is 0.6875 J.

Check

C·V² has units joules; the source’s 1100 J is incorrect.

Example 3.3.2 · reviewed solution

An air-filled capacitor has capacitance 1 F; complete filling by a linear dielectric makes it 2 F. Find K.

Authored clarification: Geometry is unchanged and the dielectric fully fills the field region.

Use dielectric capacitance relation.
$C=KC_0$
Solve for relative permittivity.
$K=C/C_0$
Substitute.
$K=2/1=2$

The relative permittivity is K=2.

Check

K is dimensionless.

Glossary and deeper reading · 6 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Dielectric
A dielectric is an insulating material that polarizes in an electric field. In a capacitor it changes the field response and capacitance; charge or voltage changes depend on whether the battery remains connected. For example, inserting a dielectric into an isolated charged capacitor can reduce its voltage while its free plate-charge magnitude remains fixed.
Read more: Dielectric in detail →
Capacitor
A capacitor stores energy by separating charge between conducting regions. In the ideal two-plate model, opposite plate charges create a potential difference related to charge by the capacitance. For example, disconnecting a charged capacitor from a battery leaves energy stored until a discharge path is provided.
Read more: Capacitor in detail →
Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
Read more: Electric charge in detail →
Capacitance
Capacitance is the plate-charge magnitude divided by the potential difference for an ideal capacitor. It is measured in farads and depends on geometry and the material between the conductors. For example, increasing the overlap area of ideal parallel plates increases their capacitance when separation and dielectric stay fixed.
Read more: Capacitance in detail →
Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
Read more: Potential difference in detail →
Potential energy
Potential energy describes energy associated with the configuration of interacting parts of a system. Its numerical zero is a chosen reference, while differences in potential energy determine physical energy changes. For example, separating Earth and a ball changes their gravitational potential energy even though the energy belongs to their interaction.
Read more: Potential energy in detail →

Gallery 1.2 Stored Energy

Gallery 1.3 Dielectric Media

Review 1.3 Stored Energy and Dielectrics

Original question 1

Original question 2

Original question 3

Original question 4

Original question 5

Reviewed worked examples

Dielectric

Capacitor

Electric charge

Capacitance

Potential difference

Potential energy

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