Capacitors and Energy Storage

What Is a Capacitor?

A capacitor is an electrical component designed to store separated electric charge and energy in an electric field.

A simple capacitor consists of:

• two conducting plates
• an insulating material or gap between the plates
• connections allowing the plates to be connected to a circuit

The insulating region prevents charge from flowing directly from one plate to the other.

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Structure of a Parallel-Plate Capacitor

The simplest model is the parallel-plate capacitor.

It consists of two conducting plates separated by a distance:

d

When connected to a voltage source:

• electrons are transferred onto one plate
• electrons are removed from the other plate
• one plate becomes negatively charged
• the other becomes positively charged

If one plate has:

+Q

the other has:

−Q

The capacitor as a whole can remain electrically neutral because the charges are separated rather than created.

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The Electric Field Between the Plates

The separated charges create an electric field between the plates.

For an ideal parallel-plate capacitor, away from the edges, this field is approximately:

• uniform
• perpendicular to the plates
• directed from the positive plate toward the negative plate

Energy is stored in this electric field.

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What Does a Capacitor Store?

It is common to say that a capacitor "stores charge."

More precisely, a capacitor stores:

separated charge and energy in its electric field

The two plates carry equal amounts of opposite charge:

+Q and −Q

The important feature is the separation of charge.

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Capacitance

Capacitance measures how much charge a capacitor can store for each volt of potential difference.

It is represented by:

C

and defined by:

C = Q/V

where:

• C = capacitance (F)
• Q = magnitude of charge on either plate (C)
• V = potential difference across the capacitor (V)

Rearranging:

Q = CV

and:

V = Q/C

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The Farad

The SI unit of capacitance is the:

farad (F)

From:

C = Q/V

we obtain:

1 F = 1 C/V

A capacitor with capacitance:

1 F

stores:

1 C

of charge on each plate when the potential difference is:

1 V

One farad is quite large for many ordinary electronic circuits.

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Common Capacitance Units

Smaller units are commonly used:

1 mF = 10⁻³ F

1 μF = 10⁻⁶ F

1 nF = 10⁻⁹ F

1 pF = 10⁻¹² F

For example:

470 μF = 470 × 10⁻⁶ F

= 4.70 × 10⁻⁴ F

Unit conversion is very important in capacitor calculations.

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Example: Finding Capacitance

A capacitor stores:

Q = 0.003 C

when connected across:

V = 12 V

Use:

C = Q/V

C = 0.003/12

C = 0.00025 F

Therefore:

C = 250 μF

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Example: Finding Charge

A:

200 μF

capacitor is connected across:

12 V.

Convert:

C = 200 × 10⁻⁶ F

Use:

Q = CV

Q = (200 × 10⁻⁶)(12)

Q = 2.4 × 10⁻³ C

Therefore:

Q = 2.4 mC

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What Determines Capacitance?

For a parallel-plate capacitor:

C = εA/d

where:

• C = capacitance
• ε = permittivity of the material between the plates
• A = overlapping plate area
• d = plate separation

This equation shows that capacitance depends on the capacitor's physical construction.

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Increasing Plate Area

From:

C = εA/d

we see:

C ∝ A

Increasing plate area increases capacitance.

Larger plates can hold more separated charge for the same potential difference.

Therefore:

larger area → larger capacitance

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Changing Plate Separation

Capacitance is inversely proportional to plate separation:

C ∝ 1/d

Therefore:

smaller separation → larger capacitance

If the plate separation is doubled:

capacitance is halved

provided everything else remains constant.

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Dielectrics

The insulating material between capacitor plates is called a:

dielectric

Possible dielectric materials include:

• air
• ceramic
• plastic
• paper
• mica
• oxide layers

A dielectric increases capacitance compared with a vacuum for the same geometry.

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Why a Dielectric Increases Capacitance

The molecules or atoms in a dielectric become polarized by the capacitor's electric field.

This polarization partially opposes the field produced by the free charges on the plates.

As a result, the capacitor can store more free charge for a given potential difference.

Therefore:

dielectric → greater capacitance

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Relative Permittivity

The capacitance may also be written:

C = ε₀εᵣA/d

where:

• ε₀ = permittivity of free space
• εᵣ = relative permittivity of the dielectric

A material with a larger εᵣ generally produces a larger capacitance for the same plate geometry.

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Charging a Capacitor

A capacitor can be charged by connecting it to a voltage source through a conducting path.

Initially, an uncharged capacitor has:

Q = 0

and:

V_C = 0

When connected to a battery, charge begins moving through the circuit.

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What Happens During Charging?

As the capacitor charges:

• one plate gains electrons
• the other loses electrons
• Q increases
• V_C increases
• the electric field becomes stronger
• stored energy increases
• charging current gradually decreases

Eventually, in an ideal DC circuit:

V_C ≈ V_supply

and:

I ≈ 0

The capacitor is then considered fully charged.

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Why Does the Current Decrease?

At the beginning:

V_C = 0

so the battery produces the greatest potential difference across the resistor.

The current is therefore largest.

As the capacitor charges:

V_C increases

This leaves less voltage across the resistor.

Since:

I = V_R/R

the current decreases.

Eventually:

V_C ≈ V_supply

V_R ≈ 0

and:

I ≈ 0

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Capacitor Voltage During Charging

The voltage across a charging capacitor rises exponentially.

The exact relationship is:

V_C = V₀(1 − e^(−t/RC))

where:

• V_C = capacitor voltage at time t
• V₀ = supply voltage
• R = resistance
• C = capacitance
• t = time

The voltage approaches the supply voltage but, mathematically, never reaches it exactly.

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Charge During Charging

Since:

Q = CV

the charge follows the same general pattern:

Q = Q_max(1 − e^(−t/RC))

where:

Q_max = CV₀

Charge rises rapidly at first and then more slowly.

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Current During Charging

Charging current decreases exponentially:

I = I₀e^(−t/RC)

where:

I₀ = V₀/R

The current is greatest immediately after the circuit is connected.

It then decreases toward zero.

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The RC Time Constant

The rate at which a capacitor charges or discharges depends on:

R

and:

C

The time constant is:

τ = RC

where:

• τ = time constant (s)
• R = resistance (Ω)
• C = capacitance (F)

Because:

Ω × F = s

the unit of RC is seconds.

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Meaning of One Time Constant

After one time constant during charging:

V_C ≈ 63% of its final value

and:

Q ≈ 63% of its final value

The current has fallen to approximately:

37% of its initial value

After about:

5τ

the capacitor is usually considered essentially fully charged.

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Example: Time Constant

A circuit contains:

R = 20 kΩ

C = 100 μF

Convert:

R = 20,000 Ω

C = 100 × 10⁻⁶ F

Then:

τ = RC

τ = (20,000)(100 × 10⁻⁶)

τ = 2.0 s

After approximately:

2 s

the capacitor has reached about 63% of its final voltage.

After approximately:

10 s

or 5τ, it is essentially fully charged.

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Effect of Resistance

If capacitance remains constant:

larger R → larger τ

Therefore, increasing resistance causes the capacitor to:

• charge more slowly
• discharge more slowly

Resistance limits the current flowing into or out of the capacitor.

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Effect of Capacitance

If resistance remains constant:

larger C → larger τ

A larger-capacitance capacitor takes longer to charge or discharge through the same resistance.

It can also store more charge at the same voltage because:

Q = CV

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Discharging a Capacitor

A charged capacitor can release its stored energy by connecting its plates through a conducting path.

During discharge:

• charge decreases
• capacitor voltage decreases
• electric field weakens
• stored energy decreases
• current decreases in magnitude

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Discharge Equation

For an ideal RC discharge:

V_C = V₀e^(−t/RC)

Similarly:

Q = Q₀e^(−t/RC)

The magnitude of the current also decreases exponentially.

After one time constant:

V_C ≈ 37% of its initial value

and:

Q ≈ 37% of its initial value

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Charging vs Discharging

Both processes are exponential rather than linear.

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Energy Stored in a Capacitor

A capacitor stores energy in its electric field.

The stored energy is:

E = ½CV²

where:

• E = energy stored (J)
• C = capacitance (F)
• V = potential difference (V)

Alternative forms are:

E = ½QV

and:

E = Q²/(2C)

These equations are equivalent because:

Q = CV

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Why Is There a Factor of ½?

While a capacitor is charging, its voltage does not remain constant.

It begins at:

0 V

and rises to:

V

For an ideal capacitor, the average voltage during the charging process is:

V/2

Therefore:

energy = charge × average voltage

E = Q(V/2)

so:

E = ½QV

Since:

Q = CV

we obtain:

E = ½CV²

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Example: Energy Stored

A capacitor has:

C = 500 μF

and is charged to:

V = 12 V

Convert:

C = 500 × 10⁻⁶ F

Use:

E = ½CV²

E = ½(500 × 10⁻⁶)(12²)

E = 0.036 J

Therefore:

E = 0.036 J

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Voltage Has a Strong Effect on Stored Energy

Since:

E = ½CV²

energy depends on the square of voltage.

If voltage doubles:

V → 2V

then:

E → ½C(2V)²

E → 4(½CV²)

Therefore:

doubling voltage quadruples stored energy

for a capacitor with constant capacitance.

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Example: Doubling Voltage

Suppose a capacitor stores:

5 J

at:

100 V

If the voltage increases to:

200 V

then:

E_new/E_old = (200/100)²

= 4

Therefore:

E_new = 20 J

The voltage doubled, but the energy quadrupled.

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Energy During Discharge

When a capacitor discharges, its stored electric-field energy is transferred into other forms.

If discharged through a resistor:

electric-field energy → thermal energy

If discharged through a lamp:

electric-field energy → light + thermal energy

If discharged through a motor:

electric-field energy → kinetic + thermal + sound energy

Energy is conserved.

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Energy Does Not Decrease Linearly

Because:

E = ½CV²

and voltage decreases exponentially during discharge, stored energy also decreases nonlinearly.

If capacitor voltage falls to half its original value:

V_new = V/2

then:

E_new = ½C(V/2)²

Therefore:

E_new = E_original/4

Half the voltage means only one-quarter of the original stored energy remains.

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Capacitors in Parallel

Capacitors connected in parallel have the same voltage across them.

Their total capacitance is:

C_total = C₁ + C₂ + C₃ + ...

For example:

C₁ = 10 μF

C₂ = 20 μF

C₃ = 30 μF

Then:

C_total = 10 + 20 + 30

C_total = 60 μF

Adding capacitors in parallel increases total capacitance.

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Capacitors in Series

For capacitors in series:

1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + ...

For two capacitors:

C_total = C₁C₂/(C₁ + C₂)

The total capacitance of capacitors in series is less than the smallest individual capacitance.

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Example: Capacitors in Series

Suppose:

C₁ = 6 μF

C₂ = 3 μF

Then:

C_total = (6 × 3)/(6 + 3)

C_total = 18/9

C_total = 2 μF

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Capacitors vs Batteries

Capacitors and batteries can both store energy, but they operate differently.

Capacitors are particularly useful when energy must be stored or released quickly.

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Camera Flash

A camera flash is a classic capacitor application.

The capacitor:

1. charges gradually from the battery
2. stores electrical energy
3. releases the energy very rapidly
4. produces a short, intense flash of light

The battery alone may not be able to deliver the required power as quickly.

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Power-Supply Smoothing

Capacitors are widely used in electronic power supplies.

After AC has been rectified, the voltage may fluctuate.

A capacitor can:

• charge when voltage rises
• discharge when voltage falls

This helps produce a smoother output voltage.

This process is called:

smoothing or filtering

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Timing Circuits

The predictable charging and discharging of capacitors makes them useful for timing.

Because:

τ = RC

designers can choose R and C to control how quickly voltage changes.

Applications include:

• flashing lights
• delay circuits
• timers
• oscillators
• control circuits

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Capacitors in Electronic Devices

Capacitors are found throughout modern electronics.

Applications include:

• computers
• smartphones
• televisions
• audio equipment
• power supplies
• communication systems
• vehicles
• sensors
• motor circuits
• medical equipment

A single electronic device may contain many different types of capacitors.

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Decoupling Capacitors

Digital circuits can produce rapid changes in current.

Small capacitors placed close to integrated circuits can provide charge during these brief changes.

They also help reduce unwanted voltage fluctuations and electrical noise.

These are often called:

decoupling capacitors

or:

bypass capacitors

They are extremely common on computer circuit boards.

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Capacitors in Audio Systems

Capacitors can help separate signals of different frequencies.

They may be used in:

• filters
• amplifiers
• speaker crossover networks
• tone-control circuits

Their response depends on factors including:

• capacitance
• signal frequency
• surrounding resistance

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Capacitors in Motors

Some AC motors use capacitors to help create appropriate phase relationships between currents.

Applications include:

• fans
• pumps
• compressors
• air-conditioning systems

These may use:

• start capacitors
• run capacitors

depending on the motor design.

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Supercapacitors

A supercapacitor is designed to provide much larger capacitance than many conventional capacitors.

Supercapacitors can:

• charge quickly
• discharge quickly
• survive many cycles
• provide large bursts of power

Applications can include:

• regenerative braking
• backup power
• energy recovery
• industrial systems
• short-term energy storage

They generally store less energy per unit mass than many batteries but can deliver power much more rapidly.

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Regenerative Braking

Some transportation systems recover kinetic energy during braking.

Instead of converting all kinetic energy into thermal energy at the brakes:

kinetic energy → electrical energy → stored energy

Capacitors or supercapacitors can sometimes store this recovered energy rapidly.

The energy can later help accelerate the vehicle again.

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Capacitor Voltage Ratings

Real capacitors have a maximum safe operating voltage called the:

voltage rating

For example, a capacitor might be marked:

470 μF, 25 V

This means:

• capacitance = 470 μF
• maximum intended operating voltage is approximately 25 V according to its specification

Operating above the voltage rating can damage the dielectric and may cause capacitor failure.

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Polarized Capacitors

Some capacitors, especially many electrolytic capacitors, are polarized.

They have designated:

• positive terminal
• negative terminal

Connecting a polarized capacitor incorrectly can damage it and may create a safety hazard.

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Capacitor Safety

A capacitor can remain charged even after a circuit has been switched off.

Therefore:

disconnected does not necessarily mean discharged

Large or high-voltage capacitors can store dangerous amounts of energy.

They should be handled using appropriate electrical safety procedures.

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High-Voltage Energy Example

Suppose:

C = 1000 μF

V = 400 V

Convert:

C = 0.001 F

Stored energy:

E = ½CV²

E = ½(0.001)(400²)

E = 80 J

Therefore:

E = 80 J

This demonstrates why large high-voltage capacitors can remain hazardous even after the power supply is disconnected.

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Analyzing a Capacitor Problem

A useful approach is:

Identify what is known

For example:

C = 200 μF

V = 10 V

R = 50 kΩ

Convert units

C = 200 × 10⁻⁶ F

R = 50,000 Ω

Choose the required equation

For charge:

Q = CV

For energy:

E = ½CV²

For charging time:

τ = RC

Calculate carefully

Then check that the answer has a reasonable unit and magnitude.

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Multi-Step Example

A capacitor has:

C = 400 μF

It is connected to:

V = 20 V

through:

R = 10 kΩ

Maximum charge

Q = CV

Q = (400 × 10⁻⁶)(20)

Q = 8.0 × 10⁻³ C

or:

Q = 8.0 mC

Stored energy

E = ½CV²

E = ½(400 × 10⁻⁶)(20²)

E = 0.080 J

Time constant

τ = RC

τ = (10,000)(400 × 10⁻⁶)

τ = 4.0 s

After approximately:

4 s

the capacitor has reached about 63% of its final voltage and charge.

After approximately:

20 s

it is essentially fully charged.

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Choosing the Correct Equation

Use:

C = Q/V

when relating capacitance, charge and voltage.

Use:

Q = CV

when calculating stored charge.

Use:

C = εA/d

for an ideal parallel-plate capacitor.

Use:

E = ½CV²

when calculating stored energy from capacitance and voltage.

Use:

E = ½QV

when charge and voltage are known.

Use:

E = Q²/(2C)

when charge and capacitance are known.

Use:

τ = RC

when analyzing charging or discharging time.

Use:

V_C = V₀(1 − e^(−t/RC))

for charging voltage.

Use:

V_C = V₀e^(−t/RC)

for discharging voltage.

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Common Mistakes

Mistake: Saying a capacitor creates charge.
It separates existing charge between its plates.

Mistake: Saying energy is stored only "on the plates."
The stored energy is associated with the capacitor's electric field.

Mistake: Confusing capacitance with charge.
Capacitance measures the ability to store charge per volt.

Mistake: Using C = QV.
The correct relationship is C = Q/V.

Mistake: Forgetting to convert μF into F.
1 μF = 10⁻⁶ F.

Mistake: Assuming capacitor voltage changes instantly during ordinary RC charging.
Voltage changes progressively according to the charging curve.

Mistake: Assuming charging current remains constant.
It decreases as the capacitor charges.

Mistake: Saying a fully charged ideal capacitor continues carrying steady DC current.
In the ideal steady state, current approaches zero.

Mistake: Assuming 1τ means fully charged.
After 1τ, the capacitor is only about 63% charged.

Mistake: Assuming half the voltage means half the stored energy.
Because E ∝ V², half the voltage means one-quarter of the energy.

Mistake: Assuming a disconnected capacitor is automatically safe.
It may remain charged.

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Key Terms

Capacitor — component that stores separated electric charge and energy in an electric field.

Capacitance — amount of charge stored per unit potential difference.

Farad (F) — SI unit of capacitance; 1 F = 1 C/V.

Plate — conducting surface of a capacitor on which charge accumulates.

Dielectric — insulating material between capacitor plates.

Permittivity — property describing how a material responds to an electric field.

Charging — process of increasing separated charge and stored energy in a capacitor.

Discharging — process of releasing stored electrical energy from a capacitor.

Time constant — characteristic charging or discharging time given by τ = RC.

RC circuit — circuit containing resistance and capacitance.

Stored energy — energy associated with the electric field of a charged capacitor.

Voltage rating — maximum intended operating potential difference across a capacitor.

Polarized capacitor — capacitor designed to be connected with a particular polarity.

Supercapacitor — very-high-capacitance device capable of rapid charge and discharge.

Decoupling capacitor — capacitor used to reduce voltage fluctuations and electrical noise near electronic components.

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Key Takeaways

• A capacitor consists essentially of two conductors separated by an insulator.
• Capacitors store separated charge and energy in an electric field.
• The plates carry equal and opposite charges.
• Capacitance is defined by C = Q/V.
• The unit of capacitance is the farad.
• Greater plate area increases capacitance.
• Smaller plate separation increases capacitance.
• A dielectric can increase capacitance.
• For parallel plates, C = εA/d.
• During charging, charge and capacitor voltage increase while current decreases.
• During discharging, charge and voltage decrease.
• Capacitor charging and discharging are exponential processes.
• The RC time constant is τ = RC.
• After 1τ, a charging capacitor reaches about 63% of its final voltage.
• After 1τ of discharge, about 37% of the original voltage remains.
• Capacitor energy can be calculated using E = ½CV².
• Stored energy depends on the square of voltage.
• Capacitors in parallel add directly.
• Capacitors in series combine reciprocally.
• Capacitors are used for energy storage, filtering, timing, smoothing, signal processing and rapid power delivery.
• Real capacitors have voltage, polarity and safety limitations.

Check Your Understanding

1. Describe the basic structure of a capacitor.
2. Explain what happens to charge when a capacitor is connected to a battery.
3. Where is the energy of a charged capacitor stored?
4. Define capacitance.
5. What does a capacitance of 1 F mean?
6. A capacitor stores 0.006 C at 12 V. Calculate its capacitance.
7. Calculate the charge stored by a 300 μF capacitor connected across 20 V.
8. State how increasing plate area affects capacitance.
9. State how increasing plate separation affects capacitance.
10. Explain why inserting a dielectric increases capacitance.
11. Describe how capacitor voltage changes during charging.
12. Describe how current changes during charging.
13. Define the RC time constant.
14. Calculate τ for R = 25 kΩ and C = 200 μF.
15. Approximately what percentage of its final voltage has a capacitor reached after one time constant?
16. Approximately what percentage of its original voltage remains after one time constant of discharge?
17. Calculate the energy stored by a 500 μF capacitor charged to 20 V.
18. If the voltage across a capacitor doubles, by what factor does its stored energy change?
19. Explain how a capacitor can be used in a camera flash or electronic power supply.
20. Explain why a high-voltage capacitor may remain dangerous even after the power supply has been disconnected.