Discharging capacitor Equation, Derivation & Rearrangements

Core idea

Overview

This equation describes the exponential decay of voltage across a capacitor as it discharges its stored energy through a resistor. It is a fundamental model for first-order transient circuits, illustrating how the electric field between capacitor plates weakens over time when the charging source is removed.

When to use: Use this formula when analyzing a passive circuit where a previously charged capacitor is allowed to bleed charge through a resistive path. It assumes the circuit has no active voltage sources during the discharge phase and that components behave ideally.

Why it matters: Understanding discharge rates is critical for designing safety circuits that bleed off high voltage, timing mechanisms in electronics, and filtering stages in power supplies. It determines how long a device can maintain operation during a power loss or how quickly a camera flash can reset.

Symbols

Variables

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

V_{0}

Initial Voltage

V

t

Time

s

R

Resistance

Ω

C

Capacitance

F

V(t)

Capacitor Voltage

V

Walkthrough

Derivation

Derivation of Capacitor Discharging Equation

When a charged capacitor discharges through a resistor, the capacitor voltage decays exponentially.

Initial capacitor voltage at t=0 is $V_{0}$ .
No external EMF in the discharge loop.
R and C are constant (ideal components).

1

Apply Kirchhoff’s Voltage Law:

With no supply, resistor and capacitor voltages sum to zero.

V_{R} + V_{C} = 0 ⟹ I R + V_{C} = 0

2

Form and Separate the Differential Equation:

Use $I = C \frac{d V _{C}}{d t}$ and separate variables.

R C \frac{d V _{C}}{d t} + V_{C} = 0 ⟹ \frac{d V _{C}}{V _{C}} = - \frac{1}{R C} d t

3

Integrate:

Integrate both sides; K is a constant set by the initial condition.

ln (V_{C}) = - \frac{t}{R C} + K

4

Apply $V_C(0)=V_0$ and Solve:

Exponentiate and substitute the initial condition to get the standard decay law.

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

Result

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

Source: OCR A-Level Physics — Capacitance

Free formulas

Rearrangements

Solve for $V_{0}$

Make V0 the subject

V_{0} = V t e^{t / R C}

Rearrange the formula to solve for the initial voltage V0.

Difficulty: 2/5

Solve for $t$

Make t the subject

t = - R C ln (\frac{V t}{V _{0}})

Rearrange the formula to solve for the time t.

Difficulty: 3/5

Solve for $R$

Make R the subject

R = \frac{t}{- C ln ( \frac{V t}{V _{0}} )}

Rearrange the formula to solve for the resistance R.

Difficulty: 3/5

Solve for $C$

Make C the subject

C = \frac{t}{- R ln ( \frac{V t}{V _{0}} )}

Rearrange the formula to solve for the capacitance C.

Difficulty: 3/5

Solve for V(t)

Make Vt the subject

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

The formula is already in the form where Vt is the subject.

Difficulty: 1/5

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

Visual intuition

Graph

The graph shows an exponential decay curve where voltage (Vt) starts at V0 when time is zero and decreases as the independent variable increases. The curve approaches an asymptote at the x-axis because the negative exponent causes the voltage to decrease at a rate proportional to its current value.

Graph type: exponential

Why it behaves this way

Intuition

A smooth curve starting at the initial voltage $V_{0}$ and gradually bending downwards, asymptotically approaching zero voltage over time, illustrating the fading stored energy.

V(t)

The instantaneous voltage across the capacitor at time t.

Represents the remaining electrical 'pressure' or potential energy stored in the capacitor at any given moment.

V_{0}

The initial voltage across the capacitor at the start of the discharge (t=0).

The maximum voltage the capacitor holds before it begins to lose charge.

t

The time elapsed since the capacitor began discharging.

Measures how long the discharge process has been occurring.

R

The electrical resistance of the discharge path.

Dictates how much the resistor opposes the flow of current, thereby controlling the discharge rate. Higher R means slower discharge.

C

The capacitance of the capacitor, its ability to store charge.

Determines how much charge the capacitor can hold. Higher C means more charge, leading to a slower discharge for a given R.

RC

The time constant (τ) of the RC circuit, representing the time for the voltage to drop to 1/e (approximately 36.8%) of its initial value.

A direct measure of how quickly the capacitor discharges. A larger RC product means a slower discharge.

e

Euler's number, the base of the natural logarithm (approximately 2.718).

It is the fundamental constant governing all continuous growth and decay processes, indicating that the rate of change is proportional to the current value.

Signs and relationships

-t/RC: The negative sign in the exponent signifies exponential decay. As time 't' increases, the value of e^(-t/RC) decreases, reflecting the continuous reduction of voltage as the capacitor discharges.

Free study cues

Insight

Canonical usage

This equation is typically used with SI units, ensuring that the product RC yields a time constant in seconds, consistent with the time variable 't'.

Common confusion

A common mistake is failing to convert component values with SI prefixes (e.g., kΩ, μF) to base SI units (Ω, F) before calculating the time constant RC, leading to incorrect time units for 't'.

Dimension note

The exponent -t/RC must be dimensionless. This requires the product RC to have units of time, consistent with the unit of 't'.

Unit systems

V(t)V · Voltage across the capacitor at time 't'. Must be in the same unit as V_0.

$V_{0}$ V · Initial voltage across the capacitor at t=0. Must be in the same unit as V(t).

$t$ s · Time elapsed since discharge began. Must be in a unit consistent with the time constant RC.

$R$ Ω · Resistance of the discharge path.

$C$ F · Capacitance of the capacitor.

RCs · The product RC is the time constant (τ) of the circuit. It represents the time required for the voltage to drop to approximately 36.8% (1/e) of its initial value. The exponent -t/RC must be dimensionless.

Ballpark figures

Quantity:

One free problem

Practice Problem

A 100 μF capacitor is initially charged to 12V and then discharged through a 50 kΩ resistor. What is the voltage across the capacitor after exactly 2 seconds of discharging?

Initial Voltage12 V

Resistance50000 \Omega

Capacitance0.0001 F

Time2 s

Solve for: Vt

Hint: Calculate the time constant RC first, then use the exponential decay formula Vt = V0 × e^(-t/RC).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating how long a sensor stays powered after cutoff, Discharging capacitor is used to calculate Voltage from Initial 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 RC is the time constant τ (tau), representing the time to reach approx. 37% of the initial voltage.
A capacitor is practically considered fully discharged after 5 time constants (5τ).
Always convert capacitance to Farads and resistance to Ohms before calculating.

Avoid these traps

Common Mistakes

Using 1 - $e^{- t / R C}$ instead of $e^{- t / R C}$ .
Mixing milliseconds and seconds.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

When a charged capacitor discharges through a resistor, the capacitor voltage decays exponentially.

Use this formula when analyzing a passive circuit where a previously charged capacitor is allowed to bleed charge through a resistive path. It assumes the circuit has no active voltage sources during the discharge phase and that components behave ideally.

Understanding discharge rates is critical for designing safety circuits that bleed off high voltage, timing mechanisms in electronics, and filtering stages in power supplies. It determines how long a device can maintain operation during a power loss or how quickly a camera flash can reset.

Using 1 - e^{-t/RC} instead of e^{-t/RC}. Mixing milliseconds and seconds.

When estimating how long a sensor stays powered after cutoff, Discharging capacitor is used to calculate Voltage from Initial 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 RC is the time constant τ (tau), representing the time to reach approx. 37% of the initial voltage. A capacitor is practically considered fully discharged after 5 time constants (5τ). Always convert capacitance to Farads and resistance to Ohms before calculating.

References

Sources

Halliday, Resnick, and Walker, Fundamentals of Physics
Wikipedia: RC circuit
Halliday, Resnick, Walker - Fundamentals of Physics, 10th ed.
Griffiths - Introduction to Electrodynamics, 4th ed.
NIST Guide for the Use of the International System of Units (SI)
Halliday, Resnick, and Walker Fundamentals of Physics
Horowitz and Hill The Art of Electronics
Alexander and Sadiku Fundamentals of Electric Circuits

Discharging capacitor

Overview

Variables

Derivation

Apply Kirchhoff’s Voltage Law:

Form and Separate the Differential Equation:

Integrate:

Apply \(V_C(0)=V_0\) and Solve:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

RC time constant

Charging capacitor

Frequently Asked Questions

Sources