Resistor-Capacitor (RC) Circuits
RC circuits combine charge storage with resistance, so circuit variables evolve in time instead of jumping directly to steady state. Topic 11.8 also includes equivalent capacitance. For a first-order resistor-capacitor model, Kirchhoff’s loop rule with I=dq/dt and ΔV_C=q/C produces a differential equation whose exponential solution is controlled by the time constant τ=RC.
Topic sequence · 8 of 8
Learning objectives
Objective 1
Calculate equivalent capacitance for series and parallel capacitor combinations and apply the corresponding charge/voltage constraints.
Objective 2
Derive and interpret the differential equations and exponential solutions for charging and discharging RC circuits, including τ=RC.
Objective 3
Analyze initial, transient, and long-time behavior using graphs of charge, capacitor voltage, current, resistor voltage, and stored energy.
- Topic 10.3 capacitance, Q=CΔV, and capacitor energy U_C=Q²/(2C)=½C(ΔV)².
- Topic 11.1 current I=dQ/dt.
- Topic 11.3 resistance and Ohm’s law.
- Topic 11.6 Kirchhoff’s loop rule; Topic 11.7 current conservation for branched RC networks.
- Calculus: separation of variables, exponential functions, derivatives, integrals and semi-log graph interpretation.
Definitions first
Representations you must read and create
Representation 1
Series capacitor chain: same charge magnitude on each ideal series capacitor, while the total terminal voltage divides according to ΔV_i=Q/C_i.
Representation 2
Parallel capacitor network: every capacitor shares the same terminal voltage, while charges add to Q_total=C_eqΔV.
Representation 3
Charging graphs aligned on one time axis: q and V_C rise concave down to asymptotes; I and |V_R| decay exponentially from maxima.
Representation 4
Discharging graphs: q and V_C decay exponentially; signed current may be negative depending on the chosen current/charge convention.
Representation 5
Initial-versus-steady circuit snapshots: at t=0+ an initially uncharged capacitor has V_C=0; after t≫τ a charging capacitor branch carries zero DC current while finite V_C can remain.
Representation 6
Semi-log plot for discharge: ln|V_C| versus t is linear with slope −1/(RC), providing an experimental route to τ and an unknown R or C.
Core relationships and calculus
Variables: C F
Use: capacitors connected across the same two nodes and therefore sharing the same ΔV
Variables: C F
Use: series capacitor chain; C_eq is less than the smallest positive component capacitance
Variables: Q C
Use: charge-conservation result for the ideal series-capacitor configuration
Variables: τ s; R Ω; C F
Use: time constant of a simple first-order RC path
Variables: ℰ V; R Ω; q C; C F
Use: series charging circuit with q(0)=0
Variables: q C; t s
Use: initially uncharged capacitor charging through R from an ideal source
Variables: I A
Use: charging current magnitude in the same simple series circuit
Variables: V
Use: capacitor voltage during simple charging
Variables: q C
Use: capacitor discharging through R from initial charge Q₀
Variables: I A
Use: signed discharge current when positive current is defined in the direction of increasing q
Variables: V
Use: capacitor voltage decay with V₀=Q₀/C
Variables: X matching the modeled quantity
Use: compact way to check any single-exponential approach to equilibrium
For charging, apply the loop rule ℰ−IR−q/C=0 and substitute I=dq/dt: R dq/dt=ℰ−q/C. Rearranging gives dq/(Cℰ−q)=dt/(RC). Integrate from q=0 at t=0 to q(t): −ln[(Cℰ−q)/(Cℰ)]=t/(RC). Exponentiating gives q=Cℰ(1−e^(−t/RC)); differentiating gives I=(ℰ/R)e^(−t/RC). For discharge, the source is absent and the loop equation gives R dq/dt+q/C=0, so dq/q=−dt/(RC) and q=Q₀e^(−t/RC). The derivative immediately supplies the signed current.
Equivalent capacitance comes from charge and voltage constraints. Parallel capacitors share ΔV, so Q_total=ΣQ_i=Σ(C_iΔV)=(ΣC_i)ΔV; hence C_eq=ΣC_i. Series capacitors carry equal charge magnitude Q, while ΔV_total=ΣΔV_i=Σ(Q/C_i)=QΣ(1/C_i). Since ΔV_total=Q/C_eq, 1/C_eq=Σ1/C_i. For the transient, insert I=dq/dt and V_C=q/C into Kirchhoff’s loop rule and solve the resulting first-order differential equation. The exponential is therefore derived from the circuit model rather than memorized independently.
Do not interpret “uncharged capacitor acts like a wire at t=0+” as a universal topological replacement in every possible network. The precise statement is that an initially uncharged ideal capacitor has V_C(0+)=0; use the actual circuit topology to determine currents. Likewise, “capacitor acts like an open circuit at long-time DC steady state” means current through that capacitor branch is zero, not that its voltage is zero. Use τ=R_eqC_eq only after correctly identifying the effective first-order resistance and capacitance for the configuration being analyzed. LR and LC dynamics belong to Unit 13.
Physical examples
Example 1
Two capacitors in parallel always have C_eq larger than either individual positive capacitance because their effective plate area/storage capacity is increased.
Example 2
Two positive capacitors in series always have C_eq smaller than the smaller component because the same series charge requires the total voltage to be shared across the elements.
Example 3
After one time constant, an initially uncharged capacitor in a simple charging circuit reaches 1−e⁻¹≈63.2% of its final charge and voltage.
Example 4
After one time constant of simple discharge, charge and voltage remain at e⁻¹≈36.8% of their initial values.
Example 5
Immediately after connecting an initially uncharged capacitor through a finite resistor to a battery, V_C=0 and the resistor initially carries the largest current ℰ/R.
Example 6
Long after a capacitor has charged in a DC circuit, current through its branch is zero in the ideal steady-state model; the capacitor can still retain a nonzero potential difference and stored energy.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Separate topology from dynamics: reduce any valid resistor/capacitor subnetwork first, then identify the relevant R and C for the transient.
Reasoning pattern 2
Check endpoints before algebra: establish t=0+ and t→∞ currents and capacitor voltages. The exponential solution must connect those limits.
Reasoning pattern 3
At t=τ remember the exact exponential factors: charging reaches 1−e⁻¹≈0.632 of its final change; discharge remains at e⁻¹≈0.368 of the initial value.
Reasoning pattern 4
Keep current sign conventions explicit. During discharge, dq/dt is negative if q is defined as the positive plate charge magnitude that is decreasing.
Reasoning pattern 5
For series capacitors, use equal charge magnitude; for parallel capacitors, use equal potential difference. These rules are the opposite pattern from resistor combinations.
Reasoning pattern 6
Graph questions should show correct initial value, asymptote, monotonic direction and concavity—not merely “an exponential-looking curve.”
Reasoning pattern 7
Use energy carefully: U_C=½CV_C² changes as V_C changes, so capacitor energy does not share the same exponential time dependence as V_C.
Deeper calculus and limiting-case check
For simple discharge, V_C(t)=V₀e^(−t/RC). Taking a natural logarithm gives ln|V_C|=ln|V₀|−t/(RC), so a graph of ln|V_C| versus t is linear with slope −1/τ. This is powerful experimentally because it converts an exponential decay into a straight-line fit. Energy provides another nontrivial check: U_C=½CV_C², so during simple discharge U_C(t)=U₀e^(−2t/RC), which decays with an exponent twice that of voltage or charge. In the classic charging circuit with an ideal constant-voltage source, half of the battery energy delivered during charging is stored in the final capacitor and half is dissipated in the resistor, independent of R; R controls the rate, not that final partition.
Model-selection workflow: decide what the problem is before choosing an equation
For Resistor-Capacitor (RC) Circuits, the fastest reliable route is not equation hunting. Start by identifying the nodes, branches, and circuit state, encode it with the circuit schematic/graph, choose the Kirchhoff/constitutive or transient equation, and then test the ideal-element and topology assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat RC circuit and equivalent capacitance C_eq as physical quantities first and symbols second.
Choose a representation
Build Series capacitor chain: same charge magnitude on each ideal series capacitor, while the total terminal voltage divides according to ΔV_i=Q/C_i.. If the prompt supplies Parallel capacitor network: every capacitor shares the same terminal voltage, while charges add to Q_total=C_eqΔV., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is parallel capacitors: C_eq = ΣC_i. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Do not interpret “uncharged capacitor acts like a wire at t=0+” as a universal topological replacement in every possible network. The precise statement is that an initially uncharged ideal capacitor has V_C(0+)=0; use the actual circuit topology to determine currents. Likewise, “capacitor acts like an open circuit at long-time DC steady state” means current through that capacitor branch is zero, not that its voltage is zero. Use τ=R_eqC_eq only after correctly identifying the effective first-order resistance and capacitance for the configuration being analyzed. LR and LC dynamics belong to Unit 13. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Series capacitor chain: same charge magnitude on each ideal series capacitor, while the total terminal voltage divides according to ΔV_i=Q/C_i. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | parallel capacitors: C_eq = ΣC_i | capacitors connected across the same two nodes and therefore sharing the same ΔV. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | series capacitors: 1/C_eq = Σ(1/C_i) | series capacitor chain; C_eq is less than the smallest positive component capacitance. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | For charging, apply the loop rule ℰ−IR−q/C=0 and substitute I=dq/dt: R dq/dt=ℰ−q/C. Rearranging gives dq/(Cℰ−q)=dt/(RC). Integrate from q=0 at t=0 to q(t): −ln[(Cℰ−q)/(Cℰ)]=t/(RC). Exponentiating gives q=Cℰ(1−e^(−t/RC)); differentiating gives I=(ℰ/R)e^(−t/RC). For discharge, the source is absent and the loop equation gives R dq/dt+q/C=0, so dq/q=−dt/(RC) and q=Q₀e^(−t/RC). The derivative immediately supplies the signed current. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A 12.0 V ideal battery charges C₁=6.0 μF and C₂=3.0 μF connected in parallel through R=200 kΩ. First reduce the capacitor network: C_eq=9.0 μF. The time constant is τ=RC=(2.00×10⁵)(9.0×10⁻⁶)=1.80 s. At t=τ, V_C=12(1−e⁻¹)=7.59 V and I=(12/200000)e⁻¹=22.1 μA. Because the capacitors are parallel, both have the same 7.59 V at that instant; Q₁=C₁V=45.5 μC and Q₂=C₂V=22.8 μC, totaling 68.3 μC, equal to C_eqV. As t→∞, V_C→12.0 V, total charge approaches 108 μC, and the current tends to zero. These endpoint and charge-sum checks should agree with the exponential calculation. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 11.8
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship parallel capacitors: C_eq = ΣC_i should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Series capacitor chain: same charge magnitude on each ideal series capacitor, while the total terminal voltage divides according to ΔV_i=Q/C_i. and Parallel capacitor network: every capacitor shares the same terminal voltage, while charges add to Q_total=C_eqΔV.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a known capacitor and a resistor—or the high internal resistance of a voltmeter as suggested in the course framework—to record capacitor voltage during discharge. Measure V_C at closely spaced times, plot V_C versus t, and also plot ln|V_C| versus t. From the fitted slope m=−1/(RC), determine an unknown R or C and compare with the nominal value. Repeat with a different resistance and test whether τ scales linearly with R. Include capacitor tolerance/leakage, meter input resistance, timing uncertainty, residual initial voltage, and the effect of not starting exactly at t=0.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using resistor combination rules for capacitors. Capacitors add directly in parallel and by reciprocal sum in series. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying an initially uncharged capacitor has zero current because it has zero charge. In a charging circuit, zero initial V_C can coincide with maximum initial current. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying a fully charged capacitor has zero voltage because it acts like an open circuit. At steady DC, branch current is zero while capacitor voltage can be finite. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Memorizing τ=RC without identifying which effective resistance and capacitance control the actual first-order transient. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Giving charging and discharging exponentials without checking t=0 and t→∞ limits. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting the sign of discharge current when I=dq/dt and the chosen q decreases. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming capacitor energy decays with e^(−t/RC); because U∝V², simple discharge energy follows e^(−2t/RC). | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 10.3 supplies capacitance and stored-energy models; Topic 11.8 adds network combinations and time evolution. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 11.1 provides I=dQ/dt, which converts circuit current into the derivative that creates the RC differential equation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 11.6 supplies the loop equation used to derive both charging and discharging behavior. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write parallel capacitors: C_eq = ΣC_i and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 11.8 mastery ladder
Interactive RC transient and capacitor network
Combine two capacitors, calculate τ=RC_eq, and inspect either charging from zero or discharge from the entered initial voltage. Check parallel to place C₁ and C₂ in parallel; otherwise they are in series.
Experimental connection
Use a known capacitor and a resistor—or the high internal resistance of a voltmeter as suggested in the course framework—to record capacitor voltage during discharge. Measure V_C at closely spaced times, plot V_C versus t, and also plot ln|V_C| versus t. From the fitted slope m=−1/(RC), determine an unknown R or C and compare with the nominal value. Repeat with a different resistance and test whether τ scales linearly with R. Include capacitor tolerance/leakage, meter input resistance, timing uncertainty, residual initial voltage, and the effect of not starting exactly at t=0.
Connections across Physics C
Connection 1
Topic 10.3 supplies capacitance and stored-energy models; Topic 11.8 adds network combinations and time evolution.
Connection 2
Topic 11.1 provides I=dQ/dt, which converts circuit current into the derivative that creates the RC differential equation.
Connection 3
Topic 11.6 supplies the loop equation used to derive both charging and discharging behavior.
Connection 4
Topic 11.7 remains valid in branched RC networks, where capacitor current changes with time.
Connection 5
Topics 13.5 and 13.6 later introduce LR and LC dynamics; comparing which quantity cannot change instantaneously helps distinguish capacitor and inductor behavior.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using resistor combination rules for capacitors. Capacitors add directly in parallel and by reciprocal sum in series.
Avoid this
Saying an initially uncharged capacitor has zero current because it has zero charge. In a charging circuit, zero initial V_C can coincide with maximum initial current.
Avoid this
Saying a fully charged capacitor has zero voltage because it acts like an open circuit. At steady DC, branch current is zero while capacitor voltage can be finite.
Avoid this
Memorizing τ=RC without identifying which effective resistance and capacitance control the actual first-order transient.
Avoid this
Giving charging and discharging exponentials without checking t=0 and t→∞ limits.
Avoid this
Forgetting the sign of discharge current when I=dq/dt and the chosen q decreases.
Avoid this
Assuming capacitor energy decays with e^(−t/RC); because U∝V², simple discharge energy follows e^(−2t/RC).
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is the RC time constant?
For a simple first-order RC circuit, τ=RC. It sets the characteristic time over which charge, voltage and current approach their new steady-state values.
What happens after one time constant while charging?
The capacitor has completed about 63.2% of the total change from its initial value to its final asymptotic value.
What happens after one time constant while discharging?
Charge and capacitor voltage are about 36.8% of their initial values in the simple exponential model.
Why does an uncharged capacitor act like a wire initially?
More precisely, V_C cannot jump from its initial zero value in a finite-resistance circuit, so V_C(0+)=0. The rest of the circuit determines the initial current.
Why does a charged capacitor act like an open circuit at long-time DC steady state?
Once its voltage and charge stop changing, I=dQ/dt=0 through that capacitor branch even though the capacitor may retain voltage and energy.
How do capacitors combine in series and parallel?
Parallel capacitances add directly. Series capacitances add by reciprocals, and the ideal series capacitors carry the same charge magnitude.
Do I need calculus for RC circuits in Physics C?
Yes. The course expects the fundamental differential equation derived from Kirchhoff’s loop rule and the exponential charging/discharging behavior that follows from it.