A charging capacitor gradually reduces the driving current
You will be able to: Derive the first-order charging equation and interpret the time constant.
Why does charging slow down even though the battery voltage stays fixed?
Close a switch connecting an uncharged capacitor to a battery through a resistor. At first the resistor takes the full battery voltage. As capacitor voltage rises, less voltage remains to drive current through the resistor.
A useful starting point: Parallel capacitors add charge at a common voltage →
Words and symbols before equations
- Time constant τ
- RC, measured in seconds.
- Capacitor charge q(t)
- Charge on its positive plate at time t.
- Exponential approach
- A process whose remaining gap shrinks by the same fraction in equal time intervals.
What this picture assumes
Initially uncharged ideal capacitor charges through positive R from constant emf. τ = RC; kΩ·mF = s. I is positive toward the positive capacitor plate. The graph time axis is seconds; the selected state is not an animation.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- t = 2 s; τ = 2 s; V_C = 6.321 V; q = 6.321 mC; signed dq/dt = 1.839 mA; U = 19.98 mJ. Positive current is toward the capacitor + plate.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Loop balance gives ε−IR−q/C = 0. With I = dq/dt, the equation is dq/dt = (ε−q/C)/R and initial condition q(0) = 0.
Separating variables and integrating gives q = Cε(1−e^(−t/RC)). Therefore V_C = ε(1−e^(−t/τ)) and I = (ε/R)e^(−t/τ). Current goes onto one plate and off the other through the external circuit, not through the ideal dielectric.
At t = τ the capacitor reaches about 63.2% of its final charge, not full charge. The limiting values q→Cε and I→0 are approached asymptotically. The time slider uses actual seconds, so changing R or C changes the progress at the same elapsed time.
A worked example, step by step
A 10 V battery charges C = 1 mF through R = 2 kΩ. Find τ, V_C and I at t = 2 s.
- τ = (2000)(0.001) = 2 s.
- At one time constant e^(−1) ≈ 0.3679.
- V_C = 10(1−0.3679) = 6.321 V and q = 6.321 mC.
- I = (10/2 kΩ)(0.3679) = 1.839 mA.
One time constant is not completion, and an uncharged capacitor resembles a wire only at the initial instant in this ideal RC model.
Does doubling C change the initial current?
Compare with an explanation
No, not for the same R and ideal battery with initially zero capacitor voltage; it changes the timescale and final charge.
Predict. Change one thing. Explain.
Hold elapsed time fixed while increasing R. Compare current, capacitor voltage and τ. Then set t = 0 and t = τ for checks.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t = 2 s; τ = 2 s; V_C = 6.321 V; q = 6.321 mC; signed dq/dt = 1.839 mA; U = 19.98 mJ. Positive current is toward the capacitor + plate.
Initially uncharged ideal capacitor charges through positive R from constant emf. τ = RC; kΩ·mF = s. I is positive toward the positive capacitor plate. The graph time axis is seconds; the selected state is not an animation.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant current, circuit topology, charge or energy conservation, or RC relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor ε = 8 V, R = 4 kΩ and C = 0.5 mF, give τ, initial current, long-time charge and V_C at τ.
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Compare with the answer and four-point rubric
- 1 point: τ = 2 s.
- 1 point: I(0) = 8/4 = 2 mA.
- 1 point: q(∞) = Cε = 4 mC.
- 1 point: V_C(τ) = 8(1−e⁻¹) ≈ 5.057 V.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why does I decrease?
Rising capacitor voltage leaves less voltage across R.
RECALL 2What is the initial condition here?
An uncharged capacitor: V_C(0) = 0.
RECALL 3When is the capacitor exactly fully charged?
Only in the ideal infinite-time limit.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A charging capacitor gradually reduces the driving current
- R dq/dt + q/C = ε.
- q = Cε(1−e^(−t/RC)).
- I = (ε/R)e^(−t/RC).
Remember: One time constant is not completion, and an uncharged capacitor resembles a wire only at the initial instant in this ideal RC model.
Conditions: Initially uncharged ideal capacitor charges through positive R from constant emf. τ = RC; kΩ·mF = s. I is positive toward the positive capacitor plate. The graph time axis is seconds; the selected state is not an animation.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.8.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.8, objectives 11.8.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 4: Electric Circuits, numbered Unit 11 in the official combined Physics C sequence. Topics 11.1–11.8 retain their official identifiers. Models include signed charge flow, prescribed current-density and resistivity integrals, DC resistor networks, nonideal batteries and meters, capacitor combinations and finite-resistance RC transients. Circuit schematics use conventional-current references and explicit node connectivity. Initial capacitor voltage and positive time constants are stated. Unequal ideal sources are never directly wired in parallel. The optional spatial wire view supplements a complete 2D current-density explanation. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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