Capacitor Charging in RC Circuits

When you connect a capacitor to a battery through a resistor, the capacitor does not charge instantly — it fills up gradually, like a bath filling from a tap where the flow slows as the bath gets fuller. This behaviour is described mathematically by an exponential function, and the circuit involved is called an RC circuit (R for resistor, C for capacitor). Understanding how charge builds up over time is essential in electronics, from camera flashes to heart pacemakers and computer memory cells. The key idea is that the rate at which charge flows onto the capacitor decreases as the capacitor's own voltage increases. Initially, the full battery voltage drives a large current through the resistor. As charge accumulates, the capacitor's voltage rises, opposing the battery. The current — and therefore the rate of charging — falls. Eventually, the capacitor's voltage equals the battery's voltage, current drops to zero, and the capacitor is fully charged. This self-limiting process produces the characteristic exponential curve. The time it takes for this process depends on both R and C. Their product, τ = RC, is called the time constant and has units of seconds. After one time constant, the capacitor has reached about 63% of its final charge. This single parameter elegantly governs the entire charging behaviour of any RC circuit.

Key takeaways
  • Setting Up the Circuit Equation
  • Solving the Differential Equation
  • Finding the Current
  • The Time Constant τ
  • Energy Considerations

Setting Up the Circuit Equation

Apply Kirchhoff's voltage law around the loop: the battery EMF ε equals the voltage across the resistor plus the voltage across the capacitor. Since V_R = IR and V_C = Q/C, we get ε = IR + Q/C. Replacing current I with dQ/dt (the rate at which charge accumulates) gives the differential equation: ε = R(dQ/dt) + Q/C.

Solving the Differential Equation

Rearranging: dQ/dt = (ε/R) − Q/(RC). This is a first-order linear ODE. Its solution is Q(t) = Cε(1 − e^(−t/RC)), where Cε = Q_max is the maximum charge the capacitor can hold. The exponential term e^(−t/RC) starts at 1 when t = 0 and decays toward 0 as t → ∞, so Q rises from 0 toward Q_max along a smooth curve.

Finding the Current

Differentiate Q(t) with respect to time to get the current: I(t) = dQ/dt = (ε/R)e^(−t/RC). At t = 0 the current is ε/R — exactly what Ohm's law predicts when the capacitor is uncharged (acting like a wire). As t increases, the current decays exponentially toward zero. The current and charge are complementary: one rises while the other falls, both governed by the same time constant τ = RC.

The Time Constant τ

The time constant τ = RC is the most important single quantity in an RC circuit. At t = τ: Q = Cε(1 − e^−1) ≈ 0.632 Q_max. At t = 2τ: Q ≈ 0.865 Q_max. At t = 5τ: Q ≈ 0.993 Q_max, which is considered fully charged in practice. A large resistance slows the current, extending charging time. A large capacitance means more charge is needed, also extending the time. Both effects combine multiplicatively in τ = RC.

Energy Considerations

The battery supplies total energy εQ_max = Cε² during charging. Half of this (½Cε²) is stored in the capacitor's electric field. The other half is dissipated as heat in the resistor — regardless of the resistor's value. This 50% efficiency is a fundamental result of RC circuit physics, not a flaw in any particular design, and has important consequences for energy storage systems.

The Key Equation

Q(t)=Cε(1−e−t/RC)Q(t) = C\varepsilon\left(1 - e^{-t/RC}\right)
  • Q(t) — Charge at time t (Coulombs (C)): The amount of electric charge stored on the capacitor's plates at a given moment.
  • C — Capacitance (Farads (F)): The capacitor's ability to store charge per unit voltage; determines the maximum charge stored.
  • \varepsilon — EMF of battery (Volts (V)): The electromotive force (voltage) of the battery driving the circuit.
  • e — Euler's number (dimensionless): The mathematical constant ≈ 2.718, base of the natural logarithm; appears naturally in all exponential growth and decay.
  • t — Time (Seconds (s)): The time elapsed since the circuit was closed and charging began.
  • R — Resistance (Ohms (Ω)): The resistance in the circuit that limits the current and slows the charging process.
  • RC — Time constant τ (Seconds (s)): The product of resistance and capacitance; the time for the capacitor to reach 63.2% of its maximum charge.

Worked Example

Problem. A 100 μF capacitor is connected in series with a 47 kΩ resistor and a 9.0 V battery. (a) Calculate the time constant τ. (b) Find the charge on the capacitor after one time constant. (c) Calculate the initial charging current and the current after one time constant.

τ=4.7 s,Q(τ)≈5.69×10−4 C,I0≈1.91×10−4 A,I(τ)≈7.03×10−5 A\tau = 4.7\text{ s},\quad Q(\tau) \approx 5.69 \times 10^{-4}\text{ C},\quad I_0 \approx 1.91 \times 10^{-4}\text{ A},\quad I(\tau) \approx 7.03 \times 10^{-5}\text{ A}

Answer. Time constant τ = 4.7 s; charge after one time constant ≈ 0.569 mC; initial current ≈ 0.191 mA, current after one time constant ≈ 0.0703 mA.

Figure
Diagram
Capacitor Charging: Q vs TimetQQ_max0.63Q_maxQ(t)=Cε(1−e^(−t/RC))τ = RCI falls as Q risesI(t)=(ε/R)e^(−t/RC)2τ3τ4τ5τ0≈99%
Origins · 1 minLord Kelvin (William Thomson) and how this was discovered
LK
Lord Kelvin (William Thomson)
1824–1907 · British (Irish-born)

William Thomson, later Lord Kelvin, made fundamental contributions to the analysis of transient electrical phenomena in circuits, including the mathematical treatment of how signals propagate through telegraph cables — essentially an early study of RC-type charging and discharging. His analysis of the Atlantic telegraph cable in the 1850s required understanding exponential signal decay governed by resistance and capacitance, directly underpinning modern RC circuit theory. Kelvin's analysis of the Atlantic telegraph cable's RC properties correctly predicted signal distortion — advice that was initially ignored, causing the first cable to fail. When engineers finally followed his guidance, transatlantic telegraphy became a reliable reality.

  • Capacitors and Capacitance — Understanding what a capacitor is and how capacitance C is defined is essential before analysing how charge builds up over time in an RC circuit.
  • Capacitor Discharging in RC Circuits — The complementary process to charging: charge decays exponentially as Q(t) = Q₀e^(−t/RC), using the same time constant τ = RC and the same mathematical framework.
  • Kirchhoff's Voltage Law — The charging equation is derived directly by applying Kirchhoff's voltage law around the RC loop, making this law the foundation of the entire analysis.
  • Ohm's Law and Resistance — The resistor in the RC circuit limits current according to Ohm's law (V = IR), which determines the initial charging current and the overall time constant.