Preserve the energy missing from a final-state-only calculation
Final electrostatic states do not automatically explain the energy transferred while reaching them. Here two capacitors have a common bottom reference conductor and isolated top-node charges, initially at V1=6 V and signed V2. Connecting the top nodes through a positive finite resistor conserves their combined charge but allows resistive energy conversion.
Learning outcome: Compare V2=−4,0,+4 V with C1=2 µF. Retain signed total charge, final common voltage and all energy terms. Explain why equal final voltage does not imply equal energy history and why resistance cannot be inferred from the final state alone.
Starting recap
Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.
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For C1=2 µF, C2=3 µF and V2=0, combined top charge is 12 µC and final voltage 2.4 V. Initial energy is 36 µJ, final energy 14.4 µJ and resistor conversion 21.6 µJ. If V2=−4 V instead, total top charge is zero and final field energy is zero, but the initial 60 µJ is still converted rather than erased.
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Which quantity fixes the final common voltage in this isolated model?
Experiment
Compare V2=−4,0,+4 V with C1=2 µF. Retain signed total charge, final common voltage and all energy terms. Explain why equal final voltage does not imply equal energy history and why resistance cannot be inferred from the final state alone.
Account for energy during isolated charge redistribution
Worked teaching model · synthetic values · Two linear capacitors C1 selected and C2=3 µF, initially V1=6 V and selected V2 relative to their common negative reference conductor. Their top nodes are connected through a positive finite resistor in an isolated system, then allowed to equilibrate. No battery, leakage or inductor; resistor dissipates the energy difference. Virtual only.
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Initial second voltage changes the final common voltage
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: charge-conserving final state (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Initial second voltage changes the final common voltage
charge-conserving final state
| second initial voltage (V) | final common voltage (V) |
|---|---|
| -4 | 0 |
| -3.9 | 0.06 |
| -3.8 | 0.12 |
| -3.7 | 0.18 |
| -3.6 | 0.24 |
| -3.5 | 0.3 |
| -3.4 | 0.36 |
| -3.3 | 0.42 |
| -3.2 | 0.48 |
| -3.1 | 0.54 |
| -3 | 0.6 |
| -2.9 | 0.66 |
| -2.8 | 0.72 |
| -2.7 | 0.78 |
| -2.6 | 0.84 |
| -2.5 | 0.9 |
| -2.4 | 0.96 |
| -2.3 | 1.02 |
| -2.2 | 1.08 |
| -2.1 | 1.14 |
| -2 | 1.2 |
| -1.9 | 1.26 |
| -1.8 | 1.32 |
| -1.7 | 1.38 |
| -1.6 | 1.44 |
| -1.5 | 1.5 |
| -1.4 | 1.56 |
| -1.3 | 1.62 |
| -1.2 | 1.68 |
| -1.1 | 1.74 |
| -1 | 1.8 |
| -0.9 | 1.86 |
| -0.8 | 1.92 |
| -0.7 | 1.98 |
| -0.6 | 2.04 |
| -0.5 | 2.1 |
| -0.4 | 2.16 |
| -0.3 | 2.22 |
| -0.2 | 2.28 |
| -0.1 | 2.34 |
| 0 | 2.4 |
| 0.1 | 2.46 |
| 0.2 | 2.52 |
| 0.3 | 2.58 |
| 0.4 | 2.64 |
| 0.5 | 2.7 |
| 0.6 | 2.76 |
| 0.7 | 2.82 |
| 0.8 | 2.88 |
| 0.9 | 2.94 |
| 1 | 3 |
| 1.1 | 3.06 |
| 1.2 | 3.12 |
| 1.3 | 3.18 |
| 1.4 | 3.24 |
| 1.5 | 3.3 |
| 1.6 | 3.36 |
| 1.7 | 3.42 |
| 1.8 | 3.48 |
| 1.9 | 3.54 |
| 2 | 3.6 |
| 2.1 | 3.66 |
| 2.2 | 3.72 |
| 2.3 | 3.78 |
| 2.4 | 3.84 |
| 2.5 | 3.9 |
| 2.6 | 3.96 |
| 2.7 | 4.02 |
| 2.8 | 4.08 |
| 2.9 | 4.14 |
| 3 | 4.2 |
| 3.1 | 4.26 |
| 3.2 | 4.32 |
| 3.3 | 4.38 |
| 3.4 | 4.44 |
| 3.5 | 4.5 |
| 3.6 | 4.56 |
| 3.7 | 4.62 |
| 3.8 | 4.68 |
| 3.9 | 4.74 |
| 4 | 4.8 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Conserved combined top-node charge | 1.20000e-5 | C |
| Final common voltage | 2.4 | V |
| Initial stored energy | 3.60000e-5 | J |
| Final stored energy | 1.44000e-5 | J |
| Resistive energy conversion | 2.16000e-5 | J |
| Energy residual | -6.77626e-21 | J |
| Charge transferred into second top plate | 7.20000e-6 | C |
Charge conservation determines the final voltage; an independent energy ledger exposes finite resistive conversion.
Vfinal=(C1V1+C2V2)/(C1+C2). Expanding initial minus final stored energy yields C1C2(V1−V2)²/[2(C1+C2)], always nonnegative. A positive finite resistor dissipates this difference; changing its resistance changes the relaxation timescale, not the ideal final state or total heat. An instantaneous ideal-wire connection cannot explain the energy deficit by deleting it. Opposite initial polarities are signed charge, not negative capacitance.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Charge conservation determines the final voltage; an independent energy ledger exposes finite resistive conversion.
Vfinal=(C1V1+C2V2)/(C1+C2). Expanding initial minus final stored energy yields C1C2(V1−V2)²/[2(C1+C2)], always nonnegative. A positive finite resistor dissipates this difference; changing its resistance changes the relaxation timescale, not the ideal final state or total heat. An instantaneous ideal-wire connection cannot explain the energy deficit by deleting it. Opposite initial polarities are signed charge, not negative capacitance.
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