Charging capacitor Equation, Derivation & Rearrangements

Core idea

Overview

This equation models the exponential rise of voltage across a capacitor as it accumulates charge through a resistor. It defines the transient behavior of an RC circuit, where the charging rate is determined by the time constant product of resistance and capacitance.

When to use: Apply this formula when analyzing a series RC circuit connected to a constant DC voltage source. It assumes the capacitor starts with zero initial charge and that components behave ideally throughout the charging process.

Why it matters: Understanding capacitor charging is critical for designing timing circuits, signal filters, and power-on reset mechanisms. It allows engineers to predict how long a system takes to reach a specific logic level or operating threshold.

Symbols

Variables

$V_{0}$ = Supply Voltage, t = Time, R = Resistance, C = Capacitance, V(t) = Capacitor Voltage

V_{0}

Supply Voltage

V

t

Time

s

R

Resistance

Ω

C

Capacitance

F

V(t)

Capacitor Voltage

V

Walkthrough

Derivation

Derivation of Capacitor Charging Equation

In a series RC circuit with a constant supply, capacitor voltage rises exponentially towards the supply voltage.

Capacitor is uncharged at t=0.
Supply voltage $V_{0}$ is constant.
R and C are constant (ideal components).

1

Apply Kirchhoff’s Voltage Law:

Supply voltage equals the sum of resistor and capacitor voltages.

V_{0} = V_{R} + V_{C} = I R + V_{C}

2

Write Current in Terms of Capacitor Voltage:

Since $Q = C V$ , differentiating gives current in terms of the rate of change of capacitor voltage.

I = \frac{d Q}{d t} = C \frac{d V _{C}}{d t}

3

Form the Differential Equation:

Substitute the current expression into the KVL equation.

V_{0} = R C \frac{d V _{C}}{d t} + V_{C}

4

State the Solution:

Solving with $V_{C} (0) = 0$ gives exponential charging towards $V_{0}$ .

V_{C} (t) = V_{0} (1 - e^{- \frac{t}{R C}})

Result

V_{C} (t) = V_{0} (1 - e^{- \frac{t}{R C}})

Source: OCR A-Level Physics — Capacitance

Free formulas

Rearrangements

Solve for $V_{0}$

Make V0 the subject

V_{0} = \frac{V ( t ) e ^{\frac{t}{R C}}}{e ^{\frac{t}{R C}} - 1}

Exact symbolic rearrangement generated deterministically for V0.

Difficulty: 3/5

Solve for $t$

Make t the subject

t = R C \ln\left(\frac{V_0}{V_0 - V(t)} \right)}

Exact symbolic rearrangement generated deterministically for t.

Difficulty: 3/5

Solve for $R$

Make R the subject

R = \frac{t}{C \ln\left(\frac{V_0}{V_0 - V(t)} \right)}}

Exact symbolic rearrangement generated deterministically for R.

Difficulty: 3/5

Solve for $C$

Make C the subject

C = \frac{t}{R \ln\left(\frac{V_0}{V_0 - V(t)} \right)}}

Exact symbolic rearrangement generated deterministically for C.

Difficulty: 3/5

Solve for V(t)

Make Vt the subject

V (t) = V_{0} - V_{0} e^{- \frac{t}{R C}}

Exact symbolic rearrangement generated deterministically for Vt.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph shows an exponential growth curve starting at the origin (0,0) and rising towards a horizontal asymptote at V_0. As the independent variable increases, the rate of change in Voltage (Vt) decreases, reflecting the inverse relationship between the charging rate and the remaining potential difference.

Graph type: exponential

Why it behaves this way

Intuition

Imagine a bathtub filling with water, but the water flow rate decreases as the water level rises because the pressure difference between the tap and the water surface diminishes.

V(t)

The instantaneous voltage across the capacitor plates at time t.

This is the measured voltage across the capacitor at any given moment, showing how 'full' it is.

V_{0}

The constant DC voltage supplied by the power source.

This represents the maximum voltage the capacitor can eventually charge to, acting as the 'target' voltage.

t

The elapsed time since the charging process began.

This is how long the capacitor has been charging.

R

The electrical resistance in series with the capacitor.

A larger resistor acts like a narrower pipe, slowing down how quickly charge can flow onto the capacitor.

C

The capacitance of the capacitor, a measure of its ability to store electric charge per unit voltage.

A larger capacitor is like a bigger tank; it can hold more charge and thus takes longer to fill to a given voltage.

RC

The time constant (τ) of the RC circuit, representing the characteristic time for the capacitor to charge or discharge.

This product dictates the 'speed' of the charging process. A larger RC means a slower charge.

e^{- t / R C}

An exponential decay term that represents the fraction of the remaining voltage difference (V_0 - V(t)) that has not yet been charged.

This term shows that the rate of charging slows down as the capacitor approaches its full voltage, because the driving force (voltage difference) diminishes.

Signs and relationships

-t/RC: The negative sign in the exponent ` $e^{- t / R C}$ ` indicates an exponential decay of the term itself. This decay models how the *difference* between the source voltage and the capacitor voltage ( $V_{0}$ - V(t))
1 - e^{-t/RC}: The `1 -` operation transforms the decaying exponential ` $e^{- t / R C}$ ` (which goes from 1 to 0) into a rising function `(1 - $e^{- t / R C}$ )` (which goes from 0 to 1).

Free study cues

Insight

Canonical usage

The equation is typically used with SI units, where voltage is in volts, time in seconds, resistance in ohms, and capacitance in farads, ensuring the exponent is dimensionless.

Common confusion

A common mistake is using inconsistent units, such as milliseconds for time with microfarads for capacitance, without proper conversion to seconds and farads, leading to incorrect time constant calculations.

Dimension note

The exponent `t/RC` must be dimensionless for the exponential function to be physically meaningful. This implies that the product `RC` has the dimension of time.

Unit systems

V(t)V · Voltage across the capacitor at time t.

$V_{0}$ V · Source voltage or initial voltage across the capacitor (if charging from a source).

$t$ s · Time elapsed since charging began.

$R$ Ω · Electrical resistance.

$C$ F · Electrical capacitance.

RCs · The time constant of the RC circuit. The product RC must have units of time.

t/RCdimensionless · The argument of the exponential function must be dimensionless.

Ballpark figures

Quantity:

One free problem

Practice Problem

A 12V battery is connected to a 100 µF capacitor through a 10 kΩ resistor. Calculate the voltage across the capacitor after 1 second of charging.

Supply Voltage12 V

Resistance10000 \Omega

Capacitance0.0001 F

Time1 s

Solve for: Vt

Hint: Calculate the time constant (RC) first, then plug it into the exponential part of the formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When predicting LED fade⁻in time with an RC circuit, Charging capacitor is used to calculate Voltage from Supply Voltage, Time, and Resistance. The result matters because it helps check whether a circuit component is operating within the required voltage, current, power, or resistance range.

Study smarter

Tips

The product R × C is the time constant τ, measured in seconds.
The capacitor reaches approximately 63.2% of its maximum voltage after one time constant.
Use base SI units: Ohms for R, Farads for C, and Volts for V.
After 5 time constants, the capacitor is considered fully charged for most practical purposes.

Avoid these traps

Common Mistakes

Using base⁻10 exponent instead of e.
Using t in ms while RC in s.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

In a series RC circuit with a constant supply, capacitor voltage rises exponentially towards the supply voltage.

Apply this formula when analyzing a series RC circuit connected to a constant DC voltage source. It assumes the capacitor starts with zero initial charge and that components behave ideally throughout the charging process.

Understanding capacitor charging is critical for designing timing circuits, signal filters, and power-on reset mechanisms. It allows engineers to predict how long a system takes to reach a specific logic level or operating threshold.

Using base⁻10 exponent instead of e. Using t in ms while RC in s.

When predicting LED fade⁻in time with an RC circuit, Charging capacitor is used to calculate Voltage from Supply Voltage, Time, and Resistance. The result matters because it helps check whether a circuit component is operating within the required voltage, current, power, or resistance range.

The product R × C is the time constant τ, measured in seconds. The capacitor reaches approximately 63.2% of its maximum voltage after one time constant. Use base SI units: Ohms for R, Farads for C, and Volts for V. After 5 time constants, the capacitor is considered fully charged for most practical purposes.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics, 10th Edition
Wikipedia: RC circuit
Halliday, Resnick, Walker, Fundamentals of Physics, Extended
IUPAC Gold Book
Halliday, Resnick, and Walker Fundamentals of Physics
Nilsson and Riedel Electric Circuits
OCR A-Level Physics — Capacitance

Charging capacitor

Overview

Variables

Derivation

Apply Kirchhoff’s Voltage Law:

Write Current in Terms of Capacitor Voltage:

Form the Differential Equation:

State the Solution:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

RC time constant

Discharging capacitor

Frequently Asked Questions

Sources