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RC Circuit Charging and Discharging Worked Examples cheat sheet - grade 11-12

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Physics Grade 11-12

RC Circuit Charging and Discharging Worked Examples Cheat Sheet

A printable reference covering RC time constants, capacitor charging, discharging equations, voltage, current, charge, and worked examples for grades 11-12.

TopicsCircuitsElectricityExponentsFunctionsGraphsPhysicsStudy Guide
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RC circuits show how a resistor and capacitor together control the timing of voltage and current changes. This cheat sheet helps students solve charging and discharging problems using the correct exponential models. It is especially useful for interpreting graphs, finding time constants, and checking units in worked examples.

These skills connect circuit theory to real devices such as camera flashes, timers, sensors, and filters.

The most important idea is the time constant, τ=RC\tau = RC, which sets the speed of the change. During charging, capacitor voltage rises toward the battery voltage while current decreases toward zero. During discharging, capacitor voltage, charge, and current magnitude all decrease exponentially.

After about 5τ5\tau, a capacitor is usually treated as almost fully charged or discharged.

Key Facts

  • The time constant of an RC circuit is τ=RC\tau = RC, where RR is resistance in ohms and CC is capacitance in farads.
  • For charging from an uncharged capacitor, the capacitor voltage is VC(t)=V0(1−e−t/RC)V_C(t) = V_0\left(1 - e^{-t/RC}\right).
  • For charging from an uncharged capacitor, the current is I(t)=V0Re−t/RCI(t) = \frac{V_0}{R}e^{-t/RC}.
  • For discharging from initial voltage ViV_i, the capacitor voltage is VC(t)=Vie−t/RCV_C(t) = V_i e^{-t/RC}.
  • The charge on a capacitor is Q=CVCQ = CV_C, so during charging Q(t)=CV0(1−e−t/RC)Q(t) = CV_0\left(1 - e^{-t/RC}\right).
  • After one time constant during charging, VC=0.632V0V_C = 0.632V_0 and the current has fallen to 0.368I00.368I_0.
  • After one time constant during discharging, VC=0.368ViV_C = 0.368V_i and Q=0.368QiQ = 0.368Q_i.
  • To solve for time in a discharging problem, use t=−RCln⁡(VCVi)t = -RC\ln\left(\frac{V_C}{V_i}\right) when the voltage changes from ViV_i to VCV_C.

Vocabulary

Capacitor
A circuit component that stores electric charge and electrical energy in an electric field.
Resistance
A measure of how strongly a circuit element opposes current, measured in ohms Ω\Omega.
Capacitance
A measure of how much charge a capacitor stores per volt, given by C=QVC = \frac{Q}{V}.
Time Constant
The characteristic time τ=RC\tau = RC that determines how quickly an RC circuit charges or discharges.
Exponential Decay
A decrease in a quantity by the same fraction each time interval, modeled by factors such as e−t/RCe^{-t/RC}.
Steady State
The long-time condition of a circuit after changes have nearly stopped, usually reached after about 5τ5\tau.

Common Mistakes to Avoid

  • Using the charging equation for a discharging problem is wrong because charging approaches a final voltage while discharging approaches zero.
  • Forgetting to convert units is wrong because RR must be in Ω\Omega, CC in F\text{F}, and tt in s\text{s} for τ=RC\tau = RC to work correctly.
  • Assuming the capacitor is fully charged after one time constant is wrong because after 1τ1\tau it has reached only about 63.2%63.2\% of its final voltage.
  • Treating current as constant is wrong because RC circuit current changes exponentially, such as I(t)=V0Re−t/RCI(t) = \frac{V_0}{R}e^{-t/RC} during charging.
  • Dropping the negative sign when solving logarithms is wrong because equations such as t=−RCln⁡(VCVi)t = -RC\ln\left(\frac{V_C}{V_i}\right) must give a positive time.

Practice Questions

  1. 1 A 2200 Ω2200\ \Omega resistor is connected in series with a 470 μF470\ \mu\text{F} capacitor. Find the time constant τ\tau.
  2. 2 An uncharged capacitor in an RC circuit with R=10.0 kΩR = 10.0\ \text{k}\Omega, C=100 μFC = 100\ \mu\text{F}, and V0=12.0 VV_0 = 12.0\ \text{V} is charging. Find VCV_C after t=1.00 st = 1.00\ \text{s}.
  3. 3 A capacitor initially at 9.0 V9.0\ \text{V} discharges through a 2.0 kΩ2.0\ \text{k}\Omega resistor with C=500 μFC = 500\ \mu\text{F}. How long does it take for the voltage to drop to 3.0 V3.0\ \text{V}?
  4. 4 Explain why the current is largest at the instant charging begins but becomes nearly zero after a long time.

Understanding RC Circuit Charging and Discharging Worked Examples

A capacitor does not resist current in the same way as a resistor. It stores separated electric charge on two conducting plates. At the instant a switch closes, an uncharged capacitor has no voltage across its plates.

It therefore behaves much like a wire for that brief instant, so the resistor sets the largest possible current. As charge builds up, the capacitor voltage pushes against the battery.

This leaves less voltage across the resistor, which reduces the current. The changing current is the reason the voltage curve bends instead of rising at a constant rate.

Worked examples become easier when you first describe the circuit state in words. Decide whether the capacitor begins uncharged, partly charged, or fully charged. Identify the voltage being asked for.

It may be the capacitor voltage, the resistor voltage, or the supply voltage. In a series charging circuit, the supply voltage is shared between the resistor and capacitor at every instant. The resistor voltage starts large and falls.

The capacitor voltage starts small and rises. Their values must add to the battery voltage. This simple check catches many mistakes.

Unit conversion matters because the time constant must come out in seconds. Resistance is often given in kilo ohms or mega ohms. Capacitance is often given in microfarads, nanofarads, or millifarads.

Convert both quantities before multiplying. For example, a resistance of two hundred kilo ohms means two hundred thousand ohms. A capacitance of ten microfarads means ten millionths of a farad.

Their product is two seconds. A large resistance limits charge flow.

A large capacitance needs more charge to produce a given voltage. Either change makes the circuit respond more slowly.

Graphs provide a useful way to judge answers before calculating. An exponential curve changes fastest at the beginning, then levels out. It never reaches its final value exactly in the ideal model.

On a charging graph, the slope of capacitor voltage is steepest at the start. On a discharging graph, the voltage falls by the same fraction during each equal time interval, not by the same number of volts.

Real circuits can differ from textbook models because batteries have internal resistance, capacitors leak charge, and meters affect the circuit slightly. Still, the ideal model is accurate enough for most school experiments and timing calculations.