SCIENCE · INVESTIGATE AND EXPLAIN

Derive network capacitance from a floating-node inventory

Capacitors in parallel share voltage, and their plate charges add: Ceq = C1 + C2.

Learning outcome: Predict series voltage division for unequal capacitances, compare the equivalent against the smaller component, and sum energies. Then solve the supplied precharged-node counterexample to identify exactly why the equal-charge shortcut needs an initial inventory assumption.

Starting recap

Integrate current or resistance, preserve node topology and derive transient state with a signed energy account.

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Try

For C1 = 2 and C2 = 3 microfarad, Cs = 1.2 microfarad and Ct = 3.2 microfarad. The series charge is 14.4 microcoulomb, with voltage drops 7.2 and 4.8 V. Total final stored energy is 230.4 microjoule.

Open the experiment

Predict

What condition supports equal charge magnitudes in this ideal series pair?

Your prediction and reason

Experiment

Predict series voltage division for unequal capacitances, compare the equivalent against the smaller component, and sum energies. Then solve the supplied precharged-node counterexample to identify exactly why the equal-charge shortcut needs an initial inventory assumption.

Track floating-node charge in a capacitor network

Worked teaching model · synthetic values · 12 V ideal source across a selected two-capacitor series branch in parallel with a fixed 2 microfarad capacitor. Initial floating middle-node net charge is zero. Final static endpoint only; ideal-source switching transient is not modeled.

First series capacitance: 2 microfarad Second series capacitance: 3 microfarad

Reset keeps your prediction, captures and practice history.

Read the node connections before calculating

M is an initially neutral floating node. The fixed capacitor connects directly between T and B, not through M. Source fixes the two labeled boundary potentials. Component orientation and drawing lengths are schematic; source polarities are specified in the caption and node labels.

First node	Second node	Element
T: 12 V	M	2 μF
M	B: 0 V	3 μF
T: 12 V	B: 0 V	2 μF parallel

Common schematic symbols and conventions

The symbol key is not a wiring recommendation. An open switch removes a conducting path; an ideal closed switch supplies a zero-drop connection. A current arrow is a chosen conventional-charge reference, not electron drift. A junction dot signifies connection; a crossing without a dot need not connect.

Potential through the series capacitor branch

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Potential through the series capacitor branch: numerical coordinates

ordered node index	potential (V)
0	12
1	4.8
2	0

Calculated values under the declared model
Quantity	Value	Unit
Series equivalent capacitance	1.2	microfarad
Total equivalent capacitance	3.2	microfarad
Charge magnitude on each series capacitor	14.4	microcoulomb
First capacitor voltage	7.2	V
Second capacitor voltage	4.8	V
Parallel capacitor charge	24	microcoulomb
Total source-delivered charge	38.4	microcoulomb
Sum of stored energies	230.4	microjoule
Equivalent final-state energy	230.4	microjoule

Equal series charge follows from the initially neutral isolated middle node; parallel branches instead share voltage.

With zero net charge initially on the floating middle node, the adjacent plate charges sum to zero. Thus Q/C1 + Q/C2 = 12 V and reciprocal capacitances add. If that node initially carried a nonzero charge, the simple equal-magnitude statement would require modification. The final energy is not a claim about dissipation during ideal-source switching.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

Equal series charge follows from the initially neutral isolated middle node; parallel branches instead share voltage.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Use the correct common quantity

Capacitors in parallel share voltage, and their plate charges add: Ceq = C1 + C2. For a series pair with an initially neutral isolated middle node, the two adjacent middle plates retain zero net charge, so the capacitor charge magnitudes agree. Adding their voltage drops gives 1/Cseries = 1/C1 + 1/C2, less than either individual capacitance.

Reduce this mixed topology

The selected C1 and C2 form one series branch across 12 V, while a fixed 2 microfarad capacitor forms another parallel branch. First derive Cs = C1C2/(C1 + C2); then Ct = Cs + 2 microfarad. The series charge is Cs times 12 V, and each voltage follows from Q/C. The parallel capacitor instead carries its own 24 microcoulomb.

Expose the neutral-node assumption

A precharged floating middle node changes the simple equal-charge conclusion. For example, two 2 microfarad capacitors between top 12 V, middle Vm and bottom 0 V have middle net charge 2(Vm − 12) + 2Vm in microcoulombs. If that net charge is +4 microcoulomb, Vm = 7 V. The top capacitor then has charge magnitude 10 microcoulomb and the bottom 14 microcoulomb: unequal, with the difference accounted for by the middle-node inventory.

Check final energy without inventing a transient

Sum CiVi²/2 for all capacitors and compare Ct times source voltage squared divided by two under the neutral-node initial condition. This is a final static energy identity. A zero-resistance ideal-source connection does not by itself specify how transient energy is radiated or dissipated, so the model makes no such claim.

Worked original case

For C1 = 2 and C2 = 3 microfarad, Cs = 1.2 microfarad and Ct = 3.2 microfarad. The series charge is 14.4 microcoulomb, with voltage drops 7.2 and 4.8 V. Total final stored energy is 230.4 microjoule.

Investigation

Changed-case practice

Try a new practice case

For C1 = C2 = 4 microfarad, what is total capacitance including the fixed 2 microfarad parallel branch, in microfarads?

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax capacitor networks. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which source, topology, initial condition or instrument assumption would invalidate this inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Include the meter in the circuit being inferred Next: Solve the RC differential equation with a signed initial condition

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