Capacitors

What Is a Capacitor?

A capacitor is an electrical component that stores 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 to an electrical circuit

When connected to a voltage source, electrons are moved from one plate to the other.

One plate becomes:

negatively charged

while the other becomes:

positively charged

The plates carry equal amounts of opposite charge.

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How a Capacitor Stores Charge

Suppose an uncharged capacitor is connected to a battery.

Electrons move:

• away from one plate
• toward the other plate

This creates a separation of charge.

If one plate has charge:

+Q

the other has:

−Q

The capacitor is described as storing charge Q, meaning the magnitude of charge on either plate.

The total charge of the complete capacitor remains zero because the positive and negative charges balance.

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Charging a Capacitor

When a capacitor is first connected to a battery, charge begins to move through the circuit.

As charge accumulates on the plates:

• the electric field between the plates becomes stronger
• the voltage across the capacitor increases
• further charging becomes increasingly difficult

Eventually, the capacitor voltage becomes equal to the supply voltage.

For an ideal capacitor connected to a DC source, the charging current then falls to zero.

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The Electric Field

The separated charges create an electric field between the capacitor plates.

Energy is stored in this electric field.

For a simple parallel-plate capacitor, the electric field between the plates is approximately uniform away from the edges.

The field points from:

positive plate → negative plate

The electric field becomes stronger as the amount of separated charge increases.

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What Is Capacitance?

Capacitance describes a capacitor's ability to store charge for a given potential difference.

It is represented by:

C

and defined by:

C = Q/V

where:

• C = capacitance (F)
• Q = charge (C)
• V = potential difference (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 has a capacitance of 1 F if a charge of 1 C produces a potential difference of 1 V.

One farad is quite large for many electronic applications.

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Common Capacitance Units

Capacitance is often measured using smaller units.

1 millifarad (mF) = 10⁻³ F

1 microfarad (μF) = 10⁻⁶ F

1 nanofarad (nF) = 10⁻⁹ F

1 picofarad (pF) = 10⁻¹² F

For example:

100 μF = 100 × 10⁻⁶ F

= 1.0 × 10⁻⁴ F

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Calculating Stored Charge

Suppose a capacitor has:

C = 200 μF

and is connected to:

V = 12 V

Convert capacitance:

200 μF = 200 × 10⁻⁶ F

Use:

Q = CV

Q = (200 × 10⁻⁶)(12)

Q = 2.4 × 10⁻³ C

or:

Q = 2.4 mC

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Comparing Capacitors

From:

Q = CV

at the same voltage, a capacitor with greater capacitance stores more separated charge.

For example:

Capacitor A:

C = 100 μF

Capacitor B:

C = 500 μF

If both are connected to 10 V:

Capacitor A

Q = CV

Q = (100 × 10⁻⁶)(10)

Q = 1.0 × 10⁻³ C

Capacitor B

Q = (500 × 10⁻⁶)(10)

Q = 5.0 × 10⁻³ C

Capacitor B stores five times as much separated charge at the same voltage.

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What Determines Capacitance?

For a simple parallel-plate capacitor, capacitance depends on:

• plate area
• distance between the plates
• material between the plates

A larger plate area generally produces:

greater capacitance

A smaller separation generally produces:

greater capacitance

A suitable insulating material between the plates can also increase capacitance.

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Parallel-Plate Capacitor

For an ideal parallel-plate capacitor:

C = εA/d

where:

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

Therefore:

C ∝ A

and:

C ∝ 1/d

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Effect of Plate Area

Increasing plate area allows more charge to be stored for the same voltage.

Therefore:

larger plate area → greater capacitance

For example, doubling the plate area approximately doubles the capacitance if all other factors remain unchanged.

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Effect of Plate Separation

Moving the plates closer together increases capacitance.

Therefore:

smaller separation → greater capacitance

For example, halving the plate separation approximately doubles the capacitance in the ideal parallel-plate model.

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Dielectrics

The insulating material between capacitor plates is called a dielectric.

Examples include:

• air
• ceramic
• plastic
• glass
• paper
• oxide layers

A dielectric can increase capacitance by responding to the electric field inside the capacitor.

Different dielectric materials have different electrical properties.

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Energy Stored in a Capacitor

Charging a capacitor requires work.

As charge accumulates, energy is stored in the electric field.

The energy stored can be calculated using:

E = ½CV²

where:

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

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Why Voltage Is Squared

Notice:

E = ½CV²

Therefore, for constant capacitance:

E ∝ V²

If voltage doubles:

V → 2V

then:

E → ½C(2V)²

E → 4(½CV²)

Therefore:

doubling the voltage produces four times the stored energy

if capacitance remains constant.

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Energy Calculation

A capacitor has:

C = 500 μF

and:

V = 20 V

Convert:

500 μF = 500 × 10⁻⁶ F

Energy:

E = ½CV²

E = ½(500 × 10⁻⁶)(20²)

E = ½(500 × 10⁻⁶)(400)

E = 0.10 J

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Other Capacitor Energy Equations

Since:

Q = CV

the energy equation can also be written as:

E = ½QV

or:

E = Q²/(2C)

Therefore:

E = ½CV²

E = ½QV

E = Q²/(2C)

Choose the equation that matches the quantities given.

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Example Using E = ½QV

A capacitor stores:

Q = 0.020 C

at:

V = 50 V

Energy:

E = ½QV

E = ½(0.020)(50)

E = 0.50 J

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Why Is There a ½ in the Energy Equation?

When an initially uncharged capacitor begins charging:

V = 0

As charge accumulates, the voltage gradually increases until it reaches the final value V.

The average voltage during the charging process is therefore:

V_average = V/2

Since:

E = QV_average

we obtain:

E = Q(V/2)

Therefore:

E = ½QV

Using Q = CV:

E = ½CV²

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Discharging a Capacitor

A charged capacitor can release its stored energy into a circuit.

If its terminals are connected through a component such as a resistor:

• charge begins moving
• current flows
• stored energy decreases
• capacitor voltage decreases

Eventually, an ideal capacitor becomes discharged.

The capacitor therefore acts as a temporary energy-storage device.

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Capacitor Voltage During Discharge

As the capacitor loses charge:

Q decreases

Since:

V = Q/C

and C remains approximately constant:

V also decreases

This means a capacitor does not usually maintain a constant voltage while discharging.

This is an important difference between a capacitor and an idealized battery.

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Charging Current

When an uncharged capacitor is first connected to a DC circuit, the charging current may initially be relatively large.

As the capacitor charges:

• capacitor voltage increases
• the driving voltage across the rest of the charging circuit decreases
• current decreases

Eventually:

current → 0

for an ideal fully charged capacitor in a simple DC circuit.

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Capacitors and Resistors

A resistor can control how quickly a capacitor charges or discharges.

A circuit containing a resistor and capacitor is called an:

RC circuit

The characteristic time is described by the time constant:

τ = RC

where:

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

A larger R or C produces slower charging and discharging.

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Understanding the Time Constant

After one time constant during charging, a capacitor reaches approximately:

63% of its final voltage

During discharge, after one time constant, the voltage falls to approximately:

37% of its initial value

After several time constants, charging or discharging is nearly complete.

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Example Time Constant

Suppose:

R = 10 kΩ

C = 100 μF

Convert:

R = 10,000 Ω

C = 100 × 10⁻⁶ F

Then:

τ = RC

τ = 10,000 × 100 × 10⁻⁶

τ = 1.0 s

The capacitor charges and discharges over a timescale of several seconds.

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Capacitors and Batteries

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

Capacitor

Stores energy primarily in an electric field.

Battery

Stores energy primarily in chemical form.

These differences give them different advantages and applications.

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Capacitor vs Battery

The exact comparison depends on the capacitor and battery technologies involved.

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Energy vs Power

Capacitors are especially useful when energy needs to be delivered quickly.

A battery may store much more total energy.

A capacitor may be able to release its smaller amount of stored energy at a very high rate.

This means:

battery → often better for energy capacity

capacitor → often better for rapid power delivery

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Camera Flash

One classic capacitor application is a camera flash.

The battery gradually charges a capacitor.

The capacitor then releases its stored energy very quickly through the flash lamp.

This produces a short, intense burst of light.

The sequence is:

battery → capacitor → rapid discharge → flash

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Capacitors in Power Supplies

Electronic devices often require relatively smooth DC voltage.

After AC has been converted into pulsating DC, capacitors can help smooth the voltage.

The capacitor:

• charges when voltage rises
• discharges when voltage falls

This reduces fluctuations in the output.

Capacitors used this way are often called smoothing or filter capacitors.

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Capacitors in Electronic Circuits

Capacitors are extremely common in electronic circuits.

Applications include:

• filtering
• timing
• smoothing voltage
• coupling signals
• separating AC and DC components
• reducing electrical noise
• storing small amounts of energy

A computer, television, phone charger, or other electronic device may contain many capacitors.

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Timing Circuits

Because capacitors take time to charge and discharge through resistors, they can be used in timing circuits.

Examples include:

• flashing lights
• delay circuits
• oscillators
• sensor circuits
• electronic timing systems

Changing R or C changes:

τ = RC

and therefore changes the timing behavior.

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Capacitors and Motors

Some AC motors use capacitors to help create the electrical conditions required for starting or running.

These are commonly found in equipment such as:

• fans
• pumps
• compressors
• air-conditioning systems

The capacitor's role depends on the type of motor and circuit.

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Capacitors for Energy Bursts

Some applications require a short burst of relatively high power.

Capacitors can be useful because they can discharge quickly.

Examples include:

• camera flashes
• pulsed electronics
• emergency power systems
• some medical equipment
• energy-recovery systems

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Supercapacitors

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

Supercapacitors can:

• charge rapidly
• discharge rapidly
• survive many charge/discharge cycles
• store more energy than conventional capacitors

However, batteries generally still provide much greater energy storage for a given mass or volume in many applications.

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Regenerative Braking

Some transportation systems recover energy during braking.

Instead of transferring all kinetic energy into unwanted thermal energy in friction brakes, part of the kinetic energy can be converted into electrical energy.

That energy may be stored temporarily in:

• batteries
• supercapacitors

Supercapacitors are useful when energy must be absorbed and released rapidly.

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Capacitors in Parallel

When capacitors are connected in parallel:

C_total = C₁ + C₂ + C₃ + ...

The total capacitance increases.

For example:

C₁ = 20 μF

C₂ = 30 μF

Then:

C_total = 20 + 30

C_total = 50 μF

Parallel capacitors experience the same voltage.

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Capacitors in Series

For capacitors in series:

1/C_total = 1/C₁ + 1/C₂ + ...

For two capacitors:

C_total = C₁C₂/(C₁ + C₂)

The total capacitance is smaller than either individual capacitance.

This is opposite to the familiar behavior of resistors in series.

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Series Capacitor Example

Suppose:

C₁ = 6 μF

C₂ = 3 μF

Then:

C_total = (6 × 3)/(6 + 3)

C_total = 18/9

C_total = 2 μF

Notice that 2 μF is smaller than both 6 μF and 3 μF.

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Capacitor Voltage Ratings

Capacitors have maximum voltage ratings.

For example, a capacitor might be labelled:

100 μF, 25 V

This means:

• capacitance = 100 μF
• maximum intended operating voltage is approximately 25 V according to its rating

Applying excessive voltage can damage the dielectric and may cause capacitor failure.

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Polarized Capacitors

Some capacitors, particularly many electrolytic capacitors, are polarized.

They must be connected with the correct polarity.

Incorrect connection can damage the component and may be hazardous.

Other capacitor types are non-polarized and can operate with voltage in either direction within their specified ratings.

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Capacitor Safety

A capacitor can remain charged even after a circuit has been disconnected from its power source.

Therefore, a disconnected circuit is not automatically energy-free.

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

For example:

C = 1000 μF

V = 400 V

Convert:

C = 0.001 F

Energy:

E = ½CV²

E = ½(0.001)(400²)

E = 80 J

This is a substantial amount of stored energy.

High-voltage capacitors should only be handled using appropriate electrical safety procedures.

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Capacitor Energy and Voltage

Because:

E = ½CV²

voltage has a strong effect on stored energy.

Suppose the same capacitor operates first at:

100 V

and then at:

200 V

Since voltage doubles:

stored energy becomes four times greater

This is one reason high-voltage capacitors require careful handling.

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Comparing Two Capacitors

Capacitor A:

C = 100 μF

V = 10 V

Capacitor B:

C = 50 μF

V = 20 V

Capacitor A

E = ½CV²

E = ½(100 × 10⁻⁶)(10²)

E = 0.005 J

Capacitor B

E = ½(50 × 10⁻⁶)(20²)

E = 0.010 J

Even though Capacitor B has lower capacitance, it stores more energy because it operates at a higher voltage.

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Capacitance vs Stored Energy

Capacitance and stored energy are not the same thing.

Capacitance describes how much charge is stored per volt:

C = Q/V

Energy depends on both capacitance and voltage:

E = ½CV²

Therefore, knowing capacitance alone is not enough to determine stored energy.

The voltage must also be known.

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Common Mistakes

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

Mistake: Saying the entire capacitor has a net charge Q.
The plates normally carry +Q and −Q, so the capacitor as a whole remains approximately neutral.

Mistake: Confusing capacitance with charge.
Capacitance measures charge stored per volt.

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

Mistake: Forgetting to square voltage in E = ½CV².

Mistake: Saying a capacitor stores energy "as charge."
The energy is stored in the electric field associated with the separated charges.

Mistake: Assuming a charged capacitor maintains constant voltage while discharging.
Its voltage normally decreases as charge leaves.

Mistake: Assuming capacitors and batteries work in the same way.
Their energy-storage mechanisms and discharge behavior are different.

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

Mistake: Ignoring capacitor voltage and polarity ratings.
Exceeding ratings or incorrectly connecting polarized capacitors can cause failure.

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Key Terms

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

Capacitance — charge stored per unit potential difference.

Farad (F) — SI unit of capacitance.

Microfarad (μF) — 10⁻⁶ F.

Charge (Q) — quantity of electric charge, measured in coulombs.

Dielectric — insulating material between capacitor conductors.

Parallel-plate capacitor — simple capacitor consisting of two conducting plates separated by an insulator or gap.

Charging — process of increasing the separated charge and voltage of a capacitor.

Discharging — process of releasing stored charge and energy into a circuit.

RC circuit — circuit containing resistance and capacitance.

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

Electric field — region in which electric charges experience electrical forces.

Stored energy — energy held in the capacitor's electric field.

Supercapacitor — capacitor designed to provide extremely large capacitance compared with conventional capacitors.

Polarized capacitor — capacitor that must be connected with the correct polarity.

Voltage rating — maximum voltage for which a component is designed.

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Key Takeaways

• A capacitor consists of conductors separated by an insulating region.
• Capacitors store separated electric charge.
• The two plates carry equal and opposite charges, +Q and −Q.
• Energy is stored in the electric field between the charged conductors.
• Capacitance describes the amount of charge stored per unit voltage.
• C = Q/V.
• The SI unit of capacitance is the farad.
• Q = CV.
• Larger capacitance means more charge can be stored at a given voltage.
• Capacitance increases with plate area and decreases with plate separation.
• Dielectric materials affect capacitance.
• Capacitor energy is E = ½CV².
• Energy can also be calculated using E = ½QV or E = Q²/(2C).
• Doubling voltage produces four times the stored energy for constant capacitance.
• Capacitors charge and discharge over time when resistance is present.
• The RC time constant is τ = RC.
• Capacitors can charge and discharge much more rapidly than many batteries.
• Batteries generally store more total energy, while capacitors are useful for rapid energy transfer.
• Capacitors are used in filtering, timing, smoothing, flashes, motors, power supplies, and energy-storage systems.
• Capacitors can remain charged after power is disconnected and must be treated appropriately.

Check Your Understanding

1. What is a capacitor?
2. Describe how a capacitor stores separated charge.
3. If one plate has +5 mC, what charge is on the other plate?
4. Explain why the capacitor as a whole can remain electrically neutral even though its plates are charged.
5. Define capacitance.
6. Write the equation relating capacitance, charge, and voltage.
7. What is the SI unit of capacitance?
8. A 200 μF capacitor is connected to 10 V. Calculate the stored charge.
9. Explain how increasing plate area affects capacitance.
10. Explain how decreasing plate separation affects capacitance.
11. What is the purpose of a dielectric?
12. Write the equation for energy stored in a capacitor using capacitance and voltage.
13. A 500 μF capacitor is charged to 12 V. Calculate its stored energy.
14. If the voltage across a capacitor doubles, by what factor does its stored energy change?
15. Explain what happens to capacitor voltage as it discharges.
16. A 20 kΩ resistor is connected to a 50 μF capacitor. Calculate the RC time constant.
17. Compare how capacitors and batteries store energy.
18. Give four applications of capacitors.
19. Explain why capacitors are useful when a short burst of high power is required.
20. Explain why a large capacitor may remain an electrical hazard even after the power supply has been disconnected.