Capacitor Charging Equations

Why This Matters

When you connect a capacitor to a voltage source through a resistor, something elegant happens: the capacitor doesn't charge instantly. Instead, it follows a predictable exponential curve governed by the circuit's resistance and capacitance. You're being tested on your ability to analyze these transient responses—the behavior of circuits during the transition from one steady state to another. This means understanding why voltage rises while current falls, how the time constant shapes circuit behavior, and what energy and power look like during the charging process.

These equations aren't isolated formulas to memorize—they're interconnected descriptions of the same physical phenomenon. The voltage equation, current equation, and power equation all share the same exponential term because they're all driven by the same underlying RC dynamics. When you see an exam problem, don't just plug numbers into formulas. Ask yourself: What's happening to the charge carriers? Where is energy being stored versus dissipated? Master the relationships between these equations, and you'll handle any RC transient problem thrown at you.

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The Exponential Response Equations

These equations describe how voltage and current evolve over time during charging. The exponential term $e^{-t/RC}$ appears everywhere because it mathematically captures the "diminishing returns" nature of capacitor charging—the closer you get to full charge, the slower the process becomes.

Capacitor Voltage During Charging

• $v_C(t) = V(1 - e^{-t/RC})$—the voltage starts at zero and asymptotically approaches the source voltage $V$
• Rising exponential behavior means the capacitor charges quickly at first, then progressively slower as it approaches full charge
• Never truly reaches $V$ in finite time, which is why we use the time constant to define "practically charged"

Capacitor Current During Charging

• $i(t) = \frac{V}{R}e^{-t/RC}$—current starts at maximum and decays exponentially toward zero
• Initial current $i(0) = \frac{V}{R}$ is determined entirely by Ohm's law since the uncharged capacitor initially acts like a short circuit
• Decaying exponential reflects that as voltage builds across the capacitor, less voltage remains across the resistor to drive current

Resistor Voltage During Charging

• $v_R(t) = Ve^{-t/RC}$—the voltage drop across the resistor mirrors the current's exponential decay
• Starts at $V$ when $t = 0$ because the uncharged capacitor has zero voltage across it
• Kirchhoff's Voltage Law requires $v_R(t) + v_C(t) = V$ at all times—use this to check your work

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The Time Constant

The time constant is your single most important parameter for characterizing RC circuit behavior. It combines resistance and capacitance into one number that tells you everything about the circuit's speed.

Time Constant Definition

• $\tau = RC$—measured in seconds when $R$ is in ohms and $C$ is in farads
• Physical meaning: the time required for voltage or current to complete approximately 63.2% of its total change
• Design parameter that lets engineers control charging speed—increase $R$ or $C$ to slow down the response

The 63.2% Benchmark

• At $t = \tau$, the capacitor voltage reaches $V(1 - e^{-1}) \approx 0.632V$, or 63.2% of final value
• At $t = 5\tau$, the capacitor is considered "fully charged" at approximately 99.3% of $V$
• Exam shortcut: if asked when a capacitor is "essentially charged," answer $5RC$ unless told otherwise

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Steady-State and Initial Conditions

These values represent the boundary conditions that anchor your transient analysis. Every exponential response starts somewhere and ends somewhere—these equations tell you where.

Initial Current

• $i(0) = \frac{V}{R}$—maximum current occurs at $t = 0$ when the capacitor voltage is zero
• Uncharged capacitor acts like a wire initially, so all source voltage appears across the resistor
• Critical for sketching current vs. time graphs—this is your starting point on the vertical axis

Final Capacitor Voltage

• $v_C(\infty) = V$—after sufficient time, the capacitor voltage equals the source voltage
• Fully charged capacitor acts like an open circuit because no current flows when $v_C = V$
• Steady-state analysis means replacing capacitors with open circuits—this is why

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Energy and Power Relationships

These equations describe where energy goes during charging. Spoiler: exactly half the energy from the source gets stored in the capacitor, and the other half is dissipated in the resistor—regardless of component values.

Charge Stored

• $Q = CV$—the total charge accumulated on the capacitor plates at full charge
• Linear relationship means doubling either capacitance or voltage doubles the stored charge
• Instantaneous charge during charging is $q(t) = Cv_C(t) = CV(1 - e^{-t/RC})$

Energy Stored

• $E = \frac{1}{2}CV^2$—energy stored in the electric field between capacitor plates
• Quadratic voltage dependence means doubling voltage quadruples stored energy—voltage matters more than capacitance for energy storage
• This is recoverable energy that can be released back into the circuit during discharge

Instantaneous Power

• $P(t) = \frac{V^2}{R}e^{-t/RC}$—power delivered to the capacitor decreases exponentially
• Maximum power at $t = 0$ when current is highest, then decays as charging slows
• Power to resistor follows the same form since $P_R = i^2R$—the resistor dissipates energy as heat throughout charging

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Quick Reference Table

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Self-Check Questions

1. Why do $v_C(t)$ and $v_R(t)$ both contain the term $e^{-t/RC}$, yet one increases while the other decreases? What circuit law guarantees their sum equals $V$?

2. If you double the resistance in an RC charging circuit, what happens to (a) the time constant, (b) the initial current, and (c) the final capacitor voltage?

3. Compare the capacitor's behavior at $t = 0$ versus $t = \infty$. What circuit element does it "act like" in each case, and why?

4. A capacitor charges through a $10\text{ k}\Omega$ resistor with $\tau = 5\text{ ms}$. What is the capacitance? How long until the capacitor reaches approximately 99% of its final voltage?

5. Explain why exactly half the energy supplied by the voltage source is dissipated in the resistor during charging, regardless of the resistance value. (Hint: think about what happens if you make $R$ very small.)