AP Physics 2: RC Circuits

Understanding RC (resistor-capacitor) circuits is essential because they form the heartbeat of modern electronics. From the timing mechanism in your camera’s flash to the filtering of noise in audio equipment, the predictable, exponential behavior of charging and discharging capacitors is everywhere. Mastering this topic moves you beyond simple circuits and into the dynamic world where voltage and current change with time, a core concept for both the AP exam and future engineering studies.

Core Components and the Time Constant

An RC circuit is a closed loop containing a resistor ($R$) and a capacitor ($C$) in series, connected to a voltage source (for charging) or just to themselves (for discharging). The resistor controls the rate of charge flow, while the capacitor stores energy in its electric field. When the circuit is completed, the capacitor does not charge or discharge instantly. The speed of this process is quantified by the time constant, represented by the Greek letter tau ($\tau$).

The time constant is defined as the product of resistance and capacitance: $\tau =RC$. Resistance is measured in ohms ($\Omega$) and capacitance in farads (F), so the time constant has units of seconds. Physically, $\tau$ represents the time required for the charge on a charging capacitor to rise to about 63.2% of its maximum value, or for the charge on a discharging capacitor to fall to about 36.8% of its initial value. It is the natural scaling factor for all exponential changes in the circuit.

Exponential Charging of a Capacitor

Consider a simple series RC circuit connected to a battery with emf $\mathcal{E}$ via a switch. Initially, the capacitor is uncharged ($q=0$). At time $t=0$, the switch is closed.

Initially, the full battery voltage appears across the resistor, causing maximum current ($I_0=\mathcal{E}/R$) to flow. As charge builds on the capacitor plates, a voltage develops across it ($V_C=q/C$). This voltage opposes the battery voltage, so the net voltage across the resistor—and thus the current—decreases. The charge asymptotically approaches a maximum value $Q_{max}=C\mathcal{E}$.

The mathematical expressions governing this process are exponential functions of time:

• Charge on the capacitor: $q(t)=Q_{max}(1-e^{-t/\tau })$
• Voltage across the capacitor: $V_C(t)=\mathcal{E}(1-e^{-t/\tau })$
• Current in the circuit: $I(t)=I_0{e}^{-t/\tau }$

The charging curve for charge and voltage rises steeply at first and then levels off, approaching its maximum value. The current starts at a maximum and decays exponentially to zero.

Exponential Discharging of a Capacitor

Now, imagine the fully charged capacitor is disconnected from the battery and connected directly across the resistor. At $t=0$, the switch is moved to complete this new loop.

The capacitor now acts as the voltage source. It drives current through the resistor, which dissipates energy. The charge, voltage, and current all decay exponentially from their initial values to zero.

The governing equations for discharging are:

• Charge on the capacitor: $q(t)=Q_0{e}^{-t/\tau })$
• Voltage across the capacitor: $V_C(t)=V_0{e}^{-t/\tau })$
• Current in the circuit: $I(t)=I_0{e}^{-t/\tau })$

Note that the current during discharging is often defined with a negative sign to indicate direction opposite to the charging current, but its magnitude decays with the same $e^{-t/\tau }$ factor. The time constant $\tau =RC$ is the same as in the charging process.

Calculating Values at Any Time

The power of the exponential model is that it allows you to determine charge, voltage, or current at any moment. You simply need the initial value, the final (asymptotic) value, the elapsed time, and the time constant.

For a charging capacitor, to find the voltage at time $t$:

1. Calculate the time constant: $\tau =RC$.
2. Compute the exponent: $-t/\tau$.
3. Evaluate $1-e^{-t/\tau }$.
4. Multiply by the maximum voltage $\mathcal{E}$.

For a discharging capacitor, to find the remaining charge at time $t$:

1. Calculate the time constant: $\tau =RC$.
2. Compute the exponent: $-t/\tau$.
3. Evaluate $e^{-t/\tau }$.
4. Multiply by the initial charge $Q_0$.

Example: A $2.2\text{ k}\Omega$ resistor and a $100\mu \text{F}$ capacitor are in series with a 9V battery. What is the charge on the capacitor 1 second after the switch is closed?

• First, $\tau =RC=(2200\Omega )(100\times 10^{-6}\text{ F})=0.22\text{ s}$.
• Maximum charge: $Q_{max}=C\mathcal{E}=(100\times 10^{-6}\text{ F})(9\text{ V})=9\times 10^{-4}\text{ C}$.
• At $t=1.0\text{ s}$: $q=Q_{max}(1-e^{-t/\tau })=(9\times 10^{-4}\text{ C})(1-e^{-1.0/0.22})$.
• Since $1.0/0.22\approx 4.55$, $e^{-4.55}$ is very small (~0.0106).
• Therefore, $q\approx (9\times 10^{-4}\text{ C})(1-0.0106)\approx 8.90\times 10^{-4}\text{ C}$.

After about 4.55 time constants, the capacitor is over 99% charged.

Energy Stored and Power Dissipated

A capacitor stores potential energy in its electric field. The energy stored in a capacitor with charge $q$ and capacitance $C$ is given by $U_C=\frac{1}{2}\frac{q^2}{C}$ or equivalently $\frac{1}{2}C{V}_C^2$ or $\frac{1}{2}Q{V}_C$. During charging, this energy increases from zero to a maximum of $U_{max}=\frac{1}{2}C{\mathcal{E}}^2$.

Where does this energy come from? It is supplied by the battery. However, the total energy supplied by the battery during the full charging process is actually $Q_{max}\mathcal{E}=C{\mathcal{E}}^2$, which is twice the energy finally stored in the capacitor. The other half is dissipated as heat by the resistor during the charging process. This 50/50 split between capacitor storage and resistor heating is independent of the resistance value.

During discharging, the energy stored in the capacitor is completely dissipated as heat in the resistor. The power dissipated in the resistor at any instant is $P_R=I^{2}R$, and this power decays exponentially as the current squared decays.

Common Pitfalls

1. Confusing Charge and Current Graphs: A common mistake is to think the charge-time graph for charging is a falling exponential. Remember, charge builds up (rising curve), while current decays (falling curve) during charging. Sketching the graphs side-by-side can solidify this difference.

1. Misapplying the 63% Rule: The rule that one time constant yields ~63% of the final change applies specifically to the quantity approaching its asymptote. For a charging capacitor, after $t=\tau$, $q$ is at 63% of $Q_{max}$. For a discharging capacitor, after $t=\tau$, $q$ is at 37% of $Q_0$ (it has lost 63%). Mixing up "is at" and "has changed by" leads to errors.

1. Forgetting the Pre-factor in Energy Calculations: The energy formula is $U=\frac{1}{2}C{V}^2$, not $C{V}^2$. Omitting the 1/2 is a frequent algebraic error, especially when relating battery energy to stored energy.

1. Using the Wrong $\tau$ for Combined Circuits: In circuits with multiple resistors and/or capacitors, finding the equivalent $R$ and $C$ for calculating $\tau$ must be done carefully. For a single-loop RC circuit, the $R$ is the total resistance through which the capacitor charges/discharges, and $C$ is the equivalent capacitance. These are not always the simple series/parallel equivalents used for static analysis.

Summary

• The time constant $\tau =RC$ governs the rate of exponential change in an RC circuit. After about $5\tau$, charging or discharging is effectively complete.
• During charging, capacitor charge and voltage rise according to $q=Q_{max}(1-e^{-t/\tau })$, while current decays from a maximum as $I=I_0{e}^{-t/\tau }$.
• During discharging, all quantities (charge, voltage, current) decay exponentially from their initial values: $q=Q_0{e}^{-t/\tau }$.
• The energy stored in a fully charged capacitor is $U=\frac{1}{2}C{\mathcal{E}}^2$. During the charging process, an equal amount of energy is dissipated as heat in the resistor.
• Mastering the exponential equations allows you to calculate the state of the circuit (charge, voltage, current) at any point in time during its dynamic evolution.