Discharge reverses current while voltage decays
You will be able to: Derive exponential discharge and distinguish signed plate current from discharge-current magnitude.
What happens after the battery is removed and a resistor closes the loop?
A charged capacitor can briefly drive a resistor after the battery is removed. Charge leaves its positive plate, so its stored charge decreases even though a conventional current flows around the discharge path.
A useful starting point: A charging capacitor gradually reduces the driving current →
Words and symbols before equations
- Initial voltage V₀
- Capacitor voltage immediately before discharge begins.
- Plate current dq/dt
- Signed positive when charge enters the labeled positive plate.
- Discharge magnitude
- Positive rate at which charge leaves that plate through the resistor.
What this picture assumes
Battery is removed; capacitor starts at V₀ and discharges through R. The schematic arrow shows the positive discharge direction. Readout dq/dt is instead positive into the labeled positive plate, so it is negative during discharge.
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 = 3.679 V; q = 3.679 mC; signed dq/dt = -1.839 mA; U = 6.767 mJ. Discharge-current magnitude = 1.839 mA.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Without the battery, loop balance gives R dq/dt + q/C = 0 when current is defined into the positive plate. Separate dq/q = −dt/(RC) and integrate using q(0)=CV₀.
The result is q = CV₀e^(−t/τ), V_C = V₀e^(−t/τ), and dq/dt = −(V₀/R)e^(−t/τ). The positive discharge-current magnitude is V_C/R.
The capacitor’s voltage remains continuous at switching when current is finite. Charge and voltage decay with τ, while energy U = ½CV_C² decays as e^(−2t/τ).
| Property | Charging from zero | Discharging from V₀ |
|---|---|---|
| Voltage | ε(1−e^(−t/RC)) | V₀e^(−t/RC) |
| Plate-current sign | Positive into + plate | Negative into + plate |
| At t = RC | 63.2% of final voltage | 36.8% of initial voltage |
| Long-time branch current | Approaches zero | Approaches zero |
A worked example, step by step
A 10 V capacitor with C = 1 mF discharges through 2 kΩ. Find V_C and signed plate current after 2 s.
- τ = RC = 2 s.
- V_C = 10e⁻¹ ≈ 3.679 V.
- q = CV_C = 3.679 mC.
- dq/dt = −V_C/R ≈ −1.839 mA; the discharge-current magnitude is 1.839 mA.
A negative plate-current sign describes decreasing stored charge, not negative current magnitude or negative capacitor energy.
At t = τ, what fraction of initial energy remains?
Compare with an explanation
e⁻² ≈ 13.5%, because energy depends on voltage squared.
Predict. Change one thing. Explain.
Compare the voltage and energy fractions at the same elapsed time. Increase R and predict whether the same elapsed time removes more or less charge.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t = 2 s; τ = 2 s; V_C = 3.679 V; q = 3.679 mC; signed dq/dt = -1.839 mA; U = 6.767 mJ. Discharge-current magnitude = 1.839 mA.
Battery is removed; capacitor starts at V₀ and discharges through R. The schematic arrow shows the positive discharge direction. Readout dq/dt is instead positive into the labeled positive plate, so it is negative during discharge.
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 V₀ = 6 V, R = 3 kΩ and C = 1 mF, calculate τ and voltage, charge and discharge magnitude at t = 3 s.
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Compare with the answer and four-point rubric
- 1 point: τ = 3 s.
- 1 point: V_C = 6/e ≈ 2.207 V.
- 1 point: q = 2.207 mC.
- 1 point: Discharge magnitude is 2.207/3 ≈ 0.736 mA, with dq/dt negative.
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 1Does current pass through the ideal dielectric?
No; external currents change charge on the plates.
RECALL 2What remains continuous at switching?
Capacitor voltage for finite current.
RECALL 3Does energy decay with the same exponential as voltage?
No; energy has twice the exponent.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Discharge reverses current while voltage decays
- q = CV₀e^(−t/RC).
- dq/dt = −(V₀/R)e^(−t/RC).
- U = U₀e^(−2t/RC).
Remember: A negative plate-current sign describes decreasing stored charge, not negative current magnitude or negative capacitor energy.
Conditions: Battery is removed; capacitor starts at V₀ and discharges through R. The schematic arrow shows the positive discharge direction. Readout dq/dt is instead positive into the labeled positive plate, so it is negative during discharge.
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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