State what stays fixed before moving the plates
You will be able to: Compare fixed-charge and fixed-voltage plate separation while including battery work.
Why does pulling plates apart have different energy outcomes?
Pull two attracting capacitor plates apart slowly. If the battery is disconnected, charge cannot leave. If it stays connected, charge returns to the battery as capacitance falls. Both cases require a pulling force, but their stored-energy changes differ.
A useful starting point: Charging work is the area under the voltage curve →
Words and symbols before equations
- Isolated capacitor
- Disconnected plates retain their equal opposite free charges, neglecting leakage.
- Connected capacitor
- An ideal battery maintains the voltage while allowing charge exchange.
- Battery work W_b
- Energy supplied by the battery to the capacitor system; negative means energy returned.
- Quasistatic motion
- Slow displacement with negligible change in mechanical kinetic energy.
What this picture assumes
Vacuum plates of area 0.010 m² start 1 mm apart, charged to V₀. Slow change, negligible fringing and leakage. Connected mode holds V₀ fixed; isolated mode holds initial free charge fixed. Work signs are into the capacitor system; mechanical kinetic-energy change is negligible.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Initial C = 88.5 pF, U = 4.425 nJ. Final U = 8.85 nJ. ΔU 4.425 = battery 0 + external 4.425 nJ.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
At fixed Q, C = ε₀A/d decreases as d grows. U = Q²d/(2ε₀A) increases linearly. The required external force magnitude is dU/dd = Q²/(2ε₀A); the electric force is attractive.
At fixed voltage V, U = ε₀AV²/(2d) decreases with separation. That does not imply repulsion: the battery exchanges charge and energy. The attractive force magnitude is ½ε₀AV²/d².
For a change C_i to C_f at fixed V, W_b = VΔQ = V²(C_f−C_i), ΔU = ½V²(C_f−C_i), and W_ext = ΔU−W_b. When separation increases, W_b < 0 and W_ext > 0. The model starts with a 1 mm gap charged to the selected initial voltage.
| Quantity | Isolated capacitor | Ideal battery connected |
|---|---|---|
| Free charge | Fixed | Decreases |
| Voltage | Increases | Fixed |
| Stored energy | Increases | Decreases |
| Mechanical work | Positive for slow separation | Positive; energy also returns to battery |
A worked example, step by step
Start with C_i = 100 pF at 10 V and double the gap. Compare stored energy if disconnected versus still connected.
- Initially Q_i = 1 nC and U_i = 5 nJ; doubling gap gives C_f = 50 pF.
- Disconnected: Q stays 1 nC, V_f = 20 V, U_f = 10 nJ, so W_ext = +5 nJ.
- Connected: V stays 10 V, Q_f = 0.5 nC and U_f = 2.5 nJ.
- Connected battery work is 10(−0.5 nC) = −5 nJ; W_ext = −2.5−(−5) = +2.5 nJ.
Differentiating capacitor energy while ignoring a connected battery gives the wrong force conclusion.
Do plates attract in both cases?
Compare with an explanation
Yes. The force remains attractive; battery energy transfer changes the accounting.
Predict. Change one thing. Explain.
Change the gap factor in isolated and connected modes from the same initial charged state. Compare final charge, voltage, stored energy and signed work. Return factor to 1 as a zero-work check.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Initial C = 88.5 pF, U = 4.425 nJ. Final U = 8.85 nJ. ΔU 4.425 = battery 0 + external 4.425 nJ.
Vacuum plates of area 0.010 m² start 1 mm apart, charged to V₀. Slow change, negligible fringing and leakage. Connected mode holds V₀ fixed; isolated mode holds initial free charge fixed. Work signs are into the capacitor system; mechanical kinetic-energy change is negligible.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant conductor equilibrium, charge conservation, capacitance 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 questionAt fixed 12 V, capacitance decreases from 200 pF to 100 pF during slow separation. Find ΔQ, battery work, stored-energy change and external work.
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Compare with the answer and four-point rubric
- 1 point: ΔQ = VΔC = −1200 pC = −1.2 nC.
- 1 point: W_b = VΔQ = −14.4 nJ.
- 1 point: ΔU = ½V²ΔC = −7.2 nJ.
- 1 point: W_ext = ΔU−W_b = +7.2 nJ; the external agent pulls against attraction.
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 1Which variable stays fixed after disconnection?
Free charge on each isolated plate, neglecting leakage.
RECALL 2Which stays fixed with an ideal battery?
The potential difference.
RECALL 3Why include battery work?
Charge transfer can change stored energy independently of mechanical work.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
State what stays fixed before moving the plates
- Fixed Q: U = Q²d/(2ε₀A).
- Fixed V: U = ε₀AV²/(2d).
- ΔU = W_ext + W_b for slow ideal motion.
Remember: Differentiating capacitor energy while ignoring a connected battery gives the wrong force conclusion.
Conditions: Vacuum plates of area 0.010 m² start 1 mm apart, charged to V₀. Slow change, negligible fringing and leakage. Connected mode holds V₀ fixed; isolated mode holds initial free charge fixed. Work signs are into the capacitor system; mechanical kinetic-energy change is negligible.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 3 (official Unit 10) · Objectives 10.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.3, objectives 10.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 3: Conductors and Capacitors, numbered Unit 10 in the official combined Physics C sequence. Topics 10.1–10.4 retain their official identifiers. Models state the electrostatic conditions, geometry approximations and whether charge or voltage stays fixed. Capacitor geometries include parallel plates, concentric spheres and long coaxial cylinders. Dielectric comparisons assume a fully filling ideal linear material. The optional 3D plate view uses explicitly different gap and lateral scales to show the small separation. 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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