Compare just after switching with long afterward
You will be able to: Use capacitor voltage continuity and long-time limits in a circuit with two branches.
Can part of a circuit keep conducting after a capacitor charges?
A 12 V source feeds two branches: one contains a 6 kΩ resistor; the other contains a 3 kΩ resistor and an initially uncharged 100 μF capacitor in series. The capacitor branch changes with time, while the direct resistive branch keeps conducting.
A useful starting point: Stored charge can drive a current →
Words and symbols before equations
- t=0⁺
- The instant just after a switch changes position.
- Voltage continuity
- With finite current, an ideal capacitor’s voltage cannot change instantaneously.
- Long-time DC limit
- After transients decay, ideal capacitor current is zero.
- Branch-specific reasoning
- A blocked capacitor branch does not mean every branch in the circuit is blocked.
What this picture assumes
Ideal 12 V source across a 6 kΩ resistor-only branch and a separate R–100 μF series branch. Capacitor initially uncharged. The ideal source fixes both branch voltages; τ=R×100 μF for this topology.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- τ=0.3 s. I_source=2+4=6 mA; V_C=0 V. Long-time limit is 2 mA (dashed line), not zero.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
AP scope: explain RC trends and graphs qualitatively and calculate initial and final states. The simulation supplies intermediate values for exploration; deriving or memorizing time-dependent exponential equations is not required in AP Physics 2.
Initially uncharged means V_C(0⁺)=0, so the capacitor acts like a zero-voltage connection for the instant-of-switching calculation. This wire-like shortcut is valid only for an initially uncharged ideal capacitor, not for a precharged one.
At long time under a constant source, the capacitor branch behaves as an open circuit for current. In the stated topology, the other resistive branch remains directly across the source and keeps its original current.
The charging branch’s time constant is its own series R₂C here, because the ideal source fixes branch voltage. Its current decays from ε/R₂ while the source current decays toward ε/R₁, not zero. A more complicated topology requires finding the resistance seen by the capacitor rather than multiplying by an arbitrary resistor.
A worked example, step by step
For the two branches described, calculate source current at t=0⁺ and after a long time.
- Direct branch: I₁=12/6000=2 mA at all times.
- Initially capacitor V_C=0, so I₂(0⁺)=12/3000=4 mA.
- Initial source current=2+4=6 mA.
- Long-time I₂→0, source I→2 mA, and capacitor V_C→12 V.
“A charged capacitor is an open circuit” is a long-time DC branch statement; it does not remove every other path.
Would a capacitor initially at 5 V act like an ideal wire just after switching?
Compare with an explanation
No. It initially keeps its 5 V; use that voltage in the loop equation.
Predict. Change one thing. Explain.
Compare time zero, one τ and several τ. Track each branch separately and verify the source current equals their sum. Explain why it settles above zero.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
τ=0.3 s. I_source=2+4=6 mA; V_C=0 V. Long-time limit is 2 mA (dashed line), not zero.
Ideal 12 V source across a 6 kΩ resistor-only branch and a separate R–100 μF series branch. Capacitor initially uncharged. The ideal source fixes both branch voltages; τ=R×100 μF for this topology.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant current, voltage or energy 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 questionA 6 V ideal source feeds a 3 kΩ resistor branch and a (2 kΩ + initially uncharged capacitor) branch. (a) Find initial direct-branch current. (b) Find initial capacitor-branch current. (c) Find initial source current. (d) Find long-time source current and capacitor voltage.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: I₁=2 mA.
- 1 point: I₂=3 mA.
- 1 point: Source current=5 mA.
- 1 point: Source current approaches 2 mA; capacitor voltage approaches 6 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 1When is the initial wire-like capacitor shortcut valid?
When its initial voltage is zero and the ideal finite-current model applies.
RECALL 2What happens to capacitor current in long-time DC?
It approaches zero.
RECALL 3Why can source current remain nonzero?
Other conducting branches may remain across the source.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Compare just after switching with long afterward
- Finite current implies capacitor voltage continuity.
- Initially uncharged: V_C(0⁺)=0.
- Long-time DC: I_C→0.
- Optional model formula, not required AP time-dependent mathematics: I_source=ε/R₁+(ε/R₂)exp(−t/(R₂C)).
Remember: “A charged capacitor is an open circuit” is a long-time DC branch statement; it does not remove every other path.
Conditions: Ideal 12 V source across a 6 kΩ resistor-only branch and a separate R–100 μF series branch. Capacitor initially uncharged. The ideal source fixes both branch voltages; τ=R×100 μF for this topology.
Refresh Kid · AP Physics 2 Unit 3 (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. Refresh Kid calls this the third AP Physics 2 unit; College Board numbers it Unit 11; the first unit in this course is official Unit 9. 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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