Preserve capacitor voltage across a switch change
You will be able to: Analyze a precharged capacitor’s initial current and final state.
Is every capacitor a short circuit immediately after switching?
A capacitor initially at 8 V is connected through a resistor to a 5 V supply with the same polarity. Its voltage cannot instantly become zero: it initially sends charge back toward the supply.
A useful starting point: Discharge reverses current while voltage decays →
Words and symbols before equations
- V_i
- Capacitor voltage immediately before and after switching, assuming finite current.
- V_f
- Final voltage set by the connected ideal source here.
- Signed resistor current
- Positive from source toward the capacitor’s labeled positive plate.
What this picture assumes
At t = 0 a precharged capacitor is connected through positive R to source V_f (the source-voltage setting). Voltage is continuous. Current reference is source-to-capacitor; negative means reverse flow. No instantaneous short of unequal voltage sources is modeled.
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 = 9.264 V; q = 9.264 mC; signed dq/dt = 0.3679 mA; U = 42.91 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
Finite current means dq/dt is finite, so capacitor charge and voltage cannot jump instantaneously. The wire-like initial approximation applies only when the capacitor starts at zero voltage.
For a source V_f connected through R, solve the same RC equation with V_C(0)=V_i: V_C(t)=V_f+(V_i−V_f)e^(−t/RC). The current is I=(V_f−V_C)/R.
If V_i>V_f, current is negative in the chosen charging direction. At long times V_C approaches V_f and capacitor-branch current approaches zero. Other parallel resistor branches, if present, can still carry steady current; this investigation has one series RC branch.
A worked example, step by step
Let V_i = 8 V, V_f = 5 V, R = 1 kΩ and C = 1 mF. Find initial current and V_C at 1 s.
- Capacitor voltage initially remains 8 V.
- I(0⁺) = (5−8)/1 kΩ = −3 mA.
- τ = 1 s; V_C(1) = 5+3/e ≈ 6.104 V.
- I(1) ≈ −1.104 mA, approaching zero from the negative side.
An initially charged capacitor is not generally a short. Preserve its voltage before applying circuit rules.
If V_i = V_f, is there a transient here?
Compare with an explanation
No. Initial current is zero and the capacitor already matches its final voltage.
Predict. Change one thing. Explain.
Move initial voltage below, equal to and above the final supply voltage. Predict current direction and whether the capacitor gains or loses charge.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t = 2 s; τ = 2 s; V_C = 9.264 V; q = 9.264 mC; signed dq/dt = 0.3679 mA; U = 42.91 mJ. Positive current is toward the capacitor + plate.
At t = 0 a precharged capacitor is connected through positive R to source V_f (the source-voltage setting). Voltage is continuous. Current reference is source-to-capacitor; negative means reverse flow. No instantaneous short of unequal voltage sources is modeled.
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 questionCompare initial and final states for V_i = 2 V, V_f = 10 V, R = 4 kΩ and C = 0.5 mF.
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Compare with the answer and four-point rubric
- 1 point: Initial V_C = 2 V.
- 1 point: Initial I = (10−2)/4 = 2 mA.
- 1 point: τ = 2 s and the voltage rises exponentially toward 10 V.
- 1 point: Final branch current tends to zero and final charge tends to 5 mC.
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 can I reverse?
The capacitor can begin at a higher voltage than the source.
RECALL 2When does the initial short approximation apply?
For an initially uncharged capacitor in a finite-resistance circuit.
RECALL 3What does the long-time open approximation describe?
Zero conduction current in the capacitor branch, not necessarily every circuit branch.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Preserve capacitor voltage across a switch change
- V_C(0⁺) = V_C(0⁻) for finite current.
- V_C = V_f+(V_i−V_f)e^(−t/RC).
- I = (V_f−V_C)/R.
Remember: An initially charged capacitor is not generally a short. Preserve its voltage before applying circuit rules.
Conditions: At t = 0 a precharged capacitor is connected through positive R to source V_f (the source-voltage setting). Voltage is continuous. Current reference is source-to-capacitor; negative means reverse flow. No instantaneous short of unequal voltage sources is modeled.
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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