Charging and Discharging Capacitors

Capacitors in Circuits

A capacitor does not normally charge or discharge instantaneously when resistance is present in a circuit.

Instead, the:

• charge on the capacitor
• voltage across the capacitor
• current in the circuit

change continuously with time.

A circuit containing a resistor (R) and capacitor (C) is called an RC circuit.

RC circuits are important because they can control how quickly electrical conditions change.

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Charging a Capacitor

Consider an initially uncharged capacitor connected to a battery through a resistor.

When the switch is closed:

• current begins flowing
• electrons accumulate on one capacitor plate
• electrons leave the other plate
• charge separation increases
• capacitor voltage increases
• current gradually decreases

Eventually, the capacitor becomes essentially fully charged.

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At the Moment Charging Begins

Initially:

Q = 0

and therefore:

V_C = 0

because:

Q = CV_C

The capacitor initially has no voltage across it.

This means the resistor initially experiences approximately the full supply voltage.

Therefore, the charging current is greatest immediately after the switch is closed.

Using Ohm's Law:

I₀ = V_supply/R

where I₀ is the initial current.

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Initial Charging Current Example

Suppose:

V_supply = 12 V

R = 2000 Ω

Initially:

I₀ = V/R

I₀ = 12/2000

I₀ = 0.006 A

or:

I₀ = 6 mA

This is the maximum charging current.

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What Happens During Charging?

As charge accumulates:

Q increases

Since:

V_C = Q/C

the voltage across the capacitor increases.

Kirchhoff's Voltage Law gives:

V_supply = V_R + V_C

As V_C increases:

V_R decreases

Since:

I = V_R/R

the current also decreases.

Therefore:

capacitor charges → V_C increases → V_R decreases → I decreases

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When the Capacitor Is Fully Charged

After sufficient time:

V_C ≈ V_supply

Therefore:

V_R ≈ 0

and:

I ≈ 0

In an ideal DC circuit, a fully charged capacitor behaves approximately like an open circuit.

No steady current continues through the capacitor.

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Maximum Stored Charge

Since:

Q = CV

when the capacitor is fully charged:

Q_max = CV_supply

Suppose:

C = 500 μF

V_supply = 12 V

Convert:

500 μF = 500 × 10⁻⁶ F

Then:

Q_max = CV

Q_max = (500 × 10⁻⁶)(12)

Q_max = 0.006 C

or:

6 mC

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Charging Is Not Linear

A capacitor does not gain equal amounts of charge during equal time intervals.

Initially, charging is rapid.

Later, charging becomes progressively slower.

The capacitor approaches its final charge gradually.

The same is true for capacitor voltage.

The charging curve rises quickly at first and then gradually levels off.

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Charging Voltage Graph

A graph of capacitor voltage against time begins at:

V_C = 0

and approaches:

V_supply

The curve:

• rises steeply initially
• becomes progressively less steep
• approaches the supply voltage
• does not suddenly jump to its final value

The horizontal leveling represents the capacitor approaching full charge.

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Charging Charge Graph

Because:

Q = CV_C

charge follows the same general shape as capacitor voltage.

It begins at:

Q = 0

and approaches:

Q_max = CV_supply

Therefore:

• V_C and Q both increase exponentially toward maximum values
• their graphs have the same basic shape

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Charging Current Graph

Current behaves differently.

At the moment charging begins:

I = I₀

where:

I₀ = V_supply/R

The current then decreases toward:

0 A

Therefore, during charging:

V_C increases

Q increases

I decreases

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Why Does Charging Slow Down?

At the beginning:

V_C = 0

so almost the entire supply voltage appears across the resistor.

This produces the maximum current.

As the capacitor charges:

V_C increases.

Therefore:

V_R = V_supply − V_C

becomes smaller.

This produces a smaller current:

I = V_R/R

Because less current flows, charge is added to the capacitor more slowly.

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The RC Time Constant

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

• resistance R
• capacitance C

The characteristic timescale is called the time constant.

It is represented by:

τ

and calculated using:

τ = RC

where:

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

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Units of the Time Constant

The units work out as:

Ω × F = s

Therefore:

RC has units of seconds

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Time Constant Example

Suppose:

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

The circuit therefore has a time constant of 2 seconds.

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What Does One Time Constant Mean?

During charging, after one time constant:

V_C ≈ 63% of its final value

and:

Q ≈ 63% of its final value

At the same time, the current has fallen to approximately:

37% of its initial value

These percentages are useful for interpreting RC graphs.

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Charging Over Several Time Constants

Approximately:

After about:

5τ

the capacitor is normally treated as essentially fully charged.

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Example Using the Time Constant

Suppose:

V_supply = 10 V

τ = 4 s

After one time constant:

t = 4 s

Capacitor voltage is approximately:

V_C ≈ 0.63 × 10

V_C ≈ 6.3 V

After about:

5τ = 20 s

the capacitor is essentially fully charged:

V_C ≈ 10 V

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Effect of Resistance

Since:

τ = RC

increasing resistance increases the time constant.

Therefore:

larger R → slower charging

Why?

A larger resistance reduces current.

Since:

I = ΔQ/Δt

less current means charge moves onto the capacitor more slowly.

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Effect of Capacitance

Increasing capacitance also increases the time constant.

Therefore:

larger C → slower charging

A larger capacitor requires more charge to reach a given voltage because:

Q = CV

So more charge must be transferred before the capacitor reaches its final voltage.

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Predicting Changes

If resistance doubles:

τ doubles

If capacitance doubles:

τ doubles

If both resistance and capacitance double:

τ_new = (2R)(2C)

τ_new = 4RC

Therefore:

the time constant becomes four times larger

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Comparing RC Circuits

Consider two circuits.

Circuit A

R = 10 kΩ

C = 100 μF

τ = RC

τ = 10,000 × 100 × 10⁻⁶

τ = 1 s

Circuit B

R = 20 kΩ

C = 200 μF

τ = 20,000 × 200 × 10⁻⁶

τ = 4 s

Circuit B charges and discharges much more slowly.

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The Charging Equation

For a capacitor charging from an initially uncharged state:

V_C = V₀(1 − e^(−t/RC))

where:

• V_C = capacitor voltage at time t
• V₀ = supply voltage
• t = time
• R = resistance
• C = capacitance
• e ≈ 2.718

Similarly:

Q = Q_max(1 − e^(−t/RC))

These equations describe the exponential shape of the charging curves.

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Charging Current Equation

During charging:

I = I₀e^(−t/RC)

where:

I₀ = V₀/R

The current therefore decreases exponentially.

At:

t = RC = τ

I = I₀e⁻¹

I ≈ 0.37I₀

So after one time constant, approximately 37% of the initial charging current remains.

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Discharging a Capacitor

Suppose a charged capacitor is disconnected from the battery and connected across a resistor.

The capacitor now acts temporarily as the energy source.

Charge flows from one plate through the resistor toward the other plate.

As this happens:

• stored charge decreases
• capacitor voltage decreases
• current decreases
• stored electric-field energy decreases

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At the Beginning of Discharge

Suppose the capacitor initially has voltage:

V₀

The initial current magnitude is:

I₀ = V₀/R

The current is largest immediately after discharge begins.

As capacitor voltage decreases, the current also decreases.

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Discharging Voltage Graph

During discharge:

V_C decreases exponentially

The voltage starts at:

V₀

and approaches:

0 V

The curve:

• falls rapidly at first
• falls more slowly later
• approaches zero

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Discharging Charge Graph

Since:

Q = CV

the charge also decreases exponentially.

It begins at:

Q₀

and approaches:

0 C

Therefore, charge and voltage have the same general decay shape.

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Discharging Current

The magnitude of current is initially greatest and then decreases toward zero.

If direction is included using a sign convention, discharge current may be represented as negative because it flows opposite to the charging direction.

The important physical idea is:

the magnitude of discharge current decreases with time

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Discharge Equation

During discharge:

V_C = V₀e^(−t/RC)

Similarly:

Q = Q₀e^(−t/RC)

The current magnitude also follows exponential decay:

I = I₀e^(−t/RC)

The same time constant:

τ = RC

controls the rate of discharge.

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Discharging After One Time Constant

After:

t = τ

the capacitor retains approximately:

37%

of its initial:

• voltage
• charge

Therefore:

V_C ≈ 0.37V₀

and:

Q ≈ 0.37Q₀

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Discharge Over Several Time Constants

Approximately:

After approximately:

5τ

the capacitor is usually treated as essentially discharged.

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Charging vs Discharging Graphs

Charging and discharging produce opposite-looking voltage curves.

Charging

Starts at:

0

and approaches:

maximum.

Discharging

Starts at:

maximum

and approaches:

0.  

Both are exponential processes.

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Interpreting the Shape of an RC Graph

When looking at an RC graph, first determine:

• what quantity is on the vertical axis
• whether the graph starts at zero or a maximum
• whether it increases or decreases
• its final value
• how quickly it approaches that value

A curve rising from zero usually represents:

charging voltage or charging charge

A curve falling from a maximum toward zero could represent:

• discharge voltage
• discharge charge
• charging current magnitude
• discharge current magnitude

The axis label and circuit context are therefore essential.

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Finding the Time Constant from a Charging Graph

For a charging voltage graph:

1. Identify the final voltage.
2. Calculate 63% of that value.
3. Find where the curve reaches this voltage.
4. Read the corresponding time.

That time is approximately:

τ

Example:

Final voltage = 12 V

63%:

0.63 × 12

= 7.56 V

If the graph reaches 7.56 V at 3 seconds:

τ ≈ 3 s

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Finding the Time Constant from a Discharge Graph

For a discharge graph:

1. Identify the initial voltage.
2. Calculate 37% of that value.
3. Find where the graph reaches this voltage.
4. Read the corresponding time.

Example:

Initial voltage = 20 V

37%:

0.37 × 20

= 7.4 V

If the graph reaches 7.4 V at 5 seconds:

τ ≈ 5 s

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Using the Initial Gradient

At the beginning of charging or discharging, the graph changes most rapidly.

Later, the graph becomes flatter.

This occurs because the current decreases.

Since:

I = ΔQ/Δt

the current represents the rate at which charge changes.

Large current:

charge changes rapidly

Small current:

charge changes slowly

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Voltage Across the Resistor During Charging

During charging:

V_supply = V_R + V_C

Initially:

V_C = 0

so:

V_R ≈ V_supply

As charging continues:

V_C increases

and:

V_R decreases.

Eventually:

V_C ≈ V_supply

and:

V_R ≈ 0

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Current and Resistor Voltage

Because:

V_R = IR

the voltage across the resistor follows the same general pattern as current.

During charging:

• V_R starts at its maximum
• V_R decreases exponentially
• V_R approaches zero

At the same time:

• V_C starts at zero
• V_C increases
• V_C approaches the supply voltage

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Kirchhoff's Voltage Law in an RC Circuit

During charging:

V_supply = V_R + V_C

Using:

V_R = IR

and:

V_C = Q/C

we obtain:

V_supply = IR + Q/C

This equation shows how the supply voltage is shared between the resistor and capacitor at any moment.

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Example at One Time Constant

Suppose:

V_supply = 10 V

After one time constant:

V_C ≈ 6.3 V

Therefore:

V_R = V_supply − V_C

V_R = 10 − 6.3

V_R ≈ 3.7 V

This matches the fact that current has fallen to approximately 37% of its initial value.

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Energy During Charging

Energy stored in the capacitor is:

E = ½CV²

As capacitor voltage increases, stored energy increases.

For example, if:

C = 1000 μF

V = 10 V

then:

C = 0.001 F

E = ½(0.001)(10²)

E = 0.050 J

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Energy During Discharge

As the capacitor discharges:

V decreases.

Since:

E = ½CV²

the stored energy decreases rapidly.

The energy is transferred into other forms in the circuit.

If discharge occurs through a resistor, the stored electrical energy is mainly transferred into:

thermal energy

in the resistor.

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Energy Is Conserved

During discharge:

energy stored in capacitor → thermal energy in resistor

Energy does not disappear.

During charging, some energy supplied by the source becomes stored in the capacitor's electric field, while some may be transferred as thermal energy in the resistance of the charging circuit.

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RC Circuit Example

A capacitor has:

C = 200 μF

and is charged through:

R = 15 kΩ

from a:

12 V

supply.

Time constant

Convert:

C = 200 × 10⁻⁶ F

R = 15,000 Ω

τ = RC

τ = 15,000 × 200 × 10⁻⁶

τ = 3.0 s

Maximum charge

Q_max = CV

Q_max = (200 × 10⁻⁶)(12)

Q_max = 2.4 × 10⁻³ C

Initial current

I₀ = V/R

I₀ = 12/15,000

I₀ = 8.0 × 10⁻⁴ A

or:

0.80 mA

After approximately:

5τ = 15 s

the capacitor is essentially fully charged.

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Predicting the Effect of Increasing Resistance

Suppose the previous resistance increases from:

15 kΩ → 30 kΩ

Capacitance remains:

200 μF

New time constant:

τ = RC

τ = 30,000 × 200 × 10⁻⁶

τ = 6.0 s

The charging process becomes slower.

The initial current also decreases:

I₀ = 12/30,000

I₀ = 0.40 mA

So increasing resistance:

• increases τ
• decreases initial current
• makes the capacitor charge more slowly

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Predicting the Effect of Increasing Capacitance

Suppose resistance stays constant while capacitance doubles.

Since:

τ = RC

the time constant doubles.

The capacitor also stores more charge at the same final voltage because:

Q = CV

Therefore, a larger capacitor:

• requires more charge
• stores more charge
• charges more slowly through the same resistance
• stores more energy at the same voltage

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Supply Voltage and Time Constant

An important point is that:

τ = RC

does not directly contain supply voltage.

Therefore, changing the battery voltage does not change the ideal RC time constant if R and C remain constant.

However, increasing supply voltage does increase:

• final capacitor voltage
• maximum charge
• initial current
• final stored energy

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Example: Doubling Supply Voltage

Suppose R and C remain unchanged.

Voltage increases:

5 V → 10 V

Then:

Time constant

unchanged

Maximum charge

Q = CV

Therefore:

doubles

Initial current

I₀ = V/R

Therefore:

doubles

Maximum stored energy

E = ½CV²

Therefore:

quadruples

This is an important distinction.

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Applications of Charging and Discharging

RC circuits are useful whenever a circuit requires a controlled change over time.

Applications include:

• timing circuits
• flashing lights
• delay circuits
• sensor circuits
• power-supply smoothing
• signal filtering
• camera flashes
• electronic control systems

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Capacitor Smoothing

In a power supply, a capacitor can help reduce fluctuations in voltage.

When supply voltage rises:

capacitor charges

When supply voltage falls:

capacitor discharges

The capacitor therefore helps maintain a smoother output voltage.

A larger RC time constant generally allows the capacitor voltage to change more slowly.

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Timing Applications

Suppose a circuit needs a delay before another component activates.

An RC circuit can be designed so that capacitor voltage gradually rises.

When the voltage reaches a particular threshold, another part of the circuit can switch.

Changing R or C changes the delay.

Therefore:

larger RC → longer delay

smaller RC → shorter delay

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Experimental Investigation

A capacitor charging experiment can use:

• capacitor
• resistor
• DC power supply
• switch
• voltmeter or data logger
• connecting wires

Voltage across the capacitor can be measured at regular time intervals.

A graph of:

capacitor voltage vs time

can then be plotted.

The resulting curve should show exponential charging.

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Investigating Discharge

First charge the capacitor.

Then disconnect the supply and allow the capacitor to discharge through a resistor.

Measure capacitor voltage over time.

The graph should show:

exponential decay

The time constant can be estimated from the point where voltage reaches approximately:

37% of its initial value

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Capacitor Safety

Capacitors can remain charged after a power source has been disconnected.

This is particularly important for:

• high-voltage capacitors
• large-capacitance devices
• power supplies
• camera flash circuits
• electrical equipment

A disconnected circuit should not automatically be assumed to contain no stored electrical energy.

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Common Mistakes

Mistake: Saying a capacitor charges at a constant rate.
Charging is exponential, not linear.

Mistake: Saying charging current increases with time.
Charging current is greatest initially and decreases toward zero.

Mistake: Saying capacitor voltage decreases while charging.
It increases toward the supply voltage.

Mistake: Saying current continues indefinitely through an ideal fully charged capacitor connected to DC.
The steady current approaches zero.

Mistake: Confusing the 63% and 37% values.
After one time constant:

charging voltage/charge ≈ 63%

discharging voltage/charge ≈ 37%

Mistake: Assuming a capacitor is completely charged after exactly one time constant.
It is only about 63% charged.

Mistake: Saying τ = R/C.
The correct equation is τ = RC.

Mistake: Forgetting to convert kΩ and μF before calculating τ.

Mistake: Assuming a larger resistor makes charging faster.
Greater resistance reduces current and increases the charging time.

Mistake: Assuming increasing supply voltage increases the time constant.
The ideal time constant depends on R and C, not supply voltage.

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Key Terms

RC circuit — circuit containing resistance and capacitance.

Charging — process in which separated charge and stored energy increase on a capacitor.

Discharging — process in which a capacitor releases stored charge and energy.

Time constant (τ) — characteristic timescale for charging or discharging, equal to RC.

Exponential growth — change that approaches a maximum increasingly slowly, as seen in capacitor charging.

Exponential decay — decrease that becomes progressively slower, as seen during capacitor discharge.

Initial current — maximum current immediately after charging or discharging begins.

Final voltage — voltage approached by a capacitor after a long charging time.

Maximum charge — final stored charge given by Q_max = CV_supply.

Open circuit — circuit condition in which no steady current flows; an ideal fully charged capacitor behaves approximately this way in a DC circuit.

RC time constant — product of resistance and capacitance that determines the charging/discharging timescale.

Smoothing capacitor — capacitor used to reduce voltage fluctuations in a power supply.

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Key Takeaways

• Capacitors do not charge or discharge instantaneously when resistance is present.
• A resistor and capacitor together form an RC circuit.
• During charging, capacitor charge and voltage increase.
• Charging current decreases toward zero.
• Initially, an uncharged capacitor has approximately zero voltage across it.
• The initial charging current is approximately I₀ = V/R.
• A fully charged ideal capacitor connected to DC behaves approximately like an open circuit.
• Maximum capacitor charge is Q_max = CV.
• Charging and discharging are exponential processes.
• The RC time constant is τ = RC.
• After one time constant, a charging capacitor reaches approximately 63% of its final charge and voltage.
• After one time constant of discharge, approximately 37% of the initial charge and voltage remains.
• After approximately five time constants, charging or discharging is usually considered essentially complete.
• Increasing resistance increases the time constant.
• Increasing capacitance increases the time constant.
• Changing supply voltage does not change the ideal time constant if R and C remain constant.
• During discharge, capacitor voltage, charge, and current magnitude decrease toward zero.
• RC graphs can be used to estimate the time constant.
• Capacitors store energy according to E = ½CV².
• RC circuits are widely used for timing, filtering, smoothing, delays, and electronic control.

Check Your Understanding

1. Describe what happens when an initially uncharged capacitor is connected to a battery through a resistor.
2. Why is the charging current greatest immediately after the switch is closed?
3. What happens to capacitor voltage during charging?
4. What happens to current during charging?
5. Explain why the charging rate decreases with time.
6. Write the equation for the RC time constant.
7. Calculate τ for R = 50 kΩ and C = 100 μF.
8. What percentage of its final voltage has a charging capacitor reached after one time constant?
9. What percentage of its initial voltage remains after one time constant of discharge?
10. Approximately how many time constants are required for a capacitor to be considered essentially fully charged?
11. Sketch and describe the shape of a capacitor charging voltage graph.
12. Sketch and describe the shape of a capacitor discharge voltage graph.
13. Explain how a charging current graph differs from a charging voltage graph.
14. A 20 μF capacitor is charged from a 10 V supply through a 100 kΩ resistor. Calculate the time constant.
15. For Question 14, calculate the maximum charge stored on the capacitor.
16. For Question 14, calculate the initial charging current.
17. Predict what happens to the charging time if resistance is doubled.
18. Predict what happens to the time constant, maximum charge, initial current, and maximum stored energy if supply voltage is doubled while R and C remain unchanged.
19. Explain how an RC circuit can be used as a timing device.
20. Explain how you could determine the time constant experimentally from a capacitor discharge graph.