Solve net-charge constraints on coupled floating conductors
A floating conductor need not be at zero potential or have zero charge on every attached capacitor. Its conserved net charge is the sum of its plate charges. The original network has 2 nF from node 1 to ground, 3 nF from node 2 to ground and 1 nF between the nodes. Ground fixes the reference; the two node charges are specified.
Learning outcome: At fixed q2=0, compare q1=−3,0,+3 nC. Retain each node plate charge and reconstruct its specified net charge. Verify energy using both branch terms and half charge dot potential; then change q2 and explain why potential changes at both nodes.
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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At q1=3 nC and q2=0, V1=12/11 V and V2=3/11 V. Node 2 has +9/11 nC on its ground capacitor and −9/11 nC on its coupling plate, so its net charge is zero without either capacitor being uncharged. Stored energy is 18/11 nJ by either account.
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For q2=0, must floating node 2 have zero potential?
Experiment
At fixed q2=0, compare q1=−3,0,+3 nC. Retain each node plate charge and reconstruct its specified net charge. Verify energy using both branch terms and half charge dot potential; then change q2 and explain why potential changes at both nodes.
Solve coupled floating-conductor charge balances
Worked teaching model · synthetic values · Node1 has2 nF to ground, node2 has3 nF to ground, and1 nF couples the nodes. Net node charges are prescribed and conserved. Ground is a fixed reference. Matrix charge balances, not simple series equal-charge assumptions, determine potentials. No switching history or physical wiring.
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Induced floating potential at fixed second-node charge
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: matrix solution with coupling (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Induced floating potential at fixed second-node charge
matrix solution with coupling
| first-node charge (nC) | second-node potential (V) |
|---|---|
| -4 | -0.3636364 |
| -3.9 | -0.3545455 |
| -3.8 | -0.3454545 |
| -3.7 | -0.3363636 |
| -3.6 | -0.3272727 |
| -3.5 | -0.3181818 |
| -3.4 | -0.3090909 |
| -3.3 | -0.3 |
| -3.2 | -0.2909091 |
| -3.1 | -0.2818182 |
| -3 | -0.2727273 |
| -2.9 | -0.2636364 |
| -2.8 | -0.2545455 |
| -2.7 | -0.2454545 |
| -2.6 | -0.2363636 |
| -2.5 | -0.2272727 |
| -2.4 | -0.2181818 |
| -2.3 | -0.2090909 |
| -2.2 | -0.2 |
| -2.1 | -0.1909091 |
| -2 | -0.1818182 |
| -1.9 | -0.1727273 |
| -1.8 | -0.1636364 |
| -1.7 | -0.1545455 |
| -1.6 | -0.1454545 |
| -1.5 | -0.1363636 |
| -1.4 | -0.1272727 |
| -1.3 | -0.1181818 |
| -1.2 | -0.1090909 |
| -1.1 | -0.1 |
| -1 | -0.09090909 |
| -0.9 | -0.08181818 |
| -0.8 | -0.07272727 |
| -0.7 | -0.06363636 |
| -0.6 | -0.05454545 |
| -0.5 | -0.04545455 |
| -0.4 | -0.03636364 |
| -0.3 | -0.02727273 |
| -0.2 | -0.01818182 |
| -0.1 | -0.009090909 |
| 0 | 0 |
| 0.1 | 0.009090909 |
| 0.2 | 0.01818182 |
| 0.3 | 0.02727273 |
| 0.4 | 0.03636364 |
| 0.5 | 0.04545455 |
| 0.6 | 0.05454545 |
| 0.7 | 0.06363636 |
| 0.8 | 0.07272727 |
| 0.9 | 0.08181818 |
| 1 | 0.09090909 |
| 1.1 | 0.1 |
| 1.2 | 0.1090909 |
| 1.3 | 0.1181818 |
| 1.4 | 0.1272727 |
| 1.5 | 0.1363636 |
| 1.6 | 0.1454545 |
| 1.7 | 0.1545455 |
| 1.8 | 0.1636364 |
| 1.9 | 0.1727273 |
| 2 | 0.1818182 |
| 2.1 | 0.1909091 |
| 2.2 | 0.2 |
| 2.3 | 0.2090909 |
| 2.4 | 0.2181818 |
| 2.5 | 0.2272727 |
| 2.6 | 0.2363636 |
| 2.7 | 0.2454545 |
| 2.8 | 0.2545455 |
| 2.9 | 0.2636364 |
| 3 | 0.2727273 |
| 3.1 | 0.2818182 |
| 3.2 | 0.2909091 |
| 3.3 | 0.3 |
| 3.4 | 0.3090909 |
| 3.5 | 0.3181818 |
| 3.6 | 0.3272727 |
| 3.7 | 0.3363636 |
| 3.8 | 0.3454545 |
| 3.9 | 0.3545455 |
| 4 | 0.3636364 |
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 |
|---|---|---|
| Floating node1 potential | 1.090909 | V |
| Floating node2 potential | 0.2727273 | V |
| Node1-to-ground plate charge | 2.181818 | nC |
| Node2-to-ground plate charge | 0.8181818 | nC |
| Coupling plate charge on node1 side | 0.8181818 | nC |
| Reconstructed node1 net charge | 3 | nC |
| Reconstructed node2 net charge | 0 | nC |
| Energy from half charge dot potential | 1.63636e-9 | J |
| Energy from three branches | 1.63636e-9 | J |
| Energy-account residual | 2.06795e-25 | J |
A floating node can have zero net charge and a nonzero potential because its capacitor plate charges need not vanish individually.
In nC/nF units the balances are3V1−V2=q1 and−V1+4V2=q2. The determinant11 is positive, giving V1=(4q1+q2)/11 and V2=(q1+3q2)/11. Summing branch energies equals(1/2)(q1V1+q2V2), with nC converted to C. A zero net node charge is a sum constraint, not an assertion that every connected capacitor is uncharged.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A floating node can have zero net charge and a nonzero potential because its capacitor plate charges need not vanish individually.
In nC/nF units the balances are3V1−V2=q1 and−V1+4V2=q2. The determinant11 is positive, giving V1=(4q1+q2)/11 and V2=(q1+3q2)/11. Summing branch energies equals(1/2)(q1V1+q2V2), with nC converted to C. A zero net node charge is a sum constraint, not an assertion that every connected capacitor is uncharged.
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