Derive a curvature correction to capacitance
Capacitance is a relation between conductor charge and a potential difference for a specified geometry and material model. The spherical capacitor has concentric surfaces at a and b=a+d, with equal and opposite charges. The vacuum field between them is radial, E=KQ/r², and the voltage must be integrated across the gap rather than approximated as one field value times d.
Learning outcome: Compare d=0.1,0.2,0.5 m at fixed a=1 m. Retain exact capacitance, the chosen planar approximation and its signed relative error. Derive a gap-to-radius criterion for a maximum 5% relative error instead of choosing a visually thin gap.
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 a=1 m and d=0.2 m, exact C=2/3 nF while the inner-area planar approximation gives 5/9 nF. Its relative error is −1/6. At maintained 9 V, plate charge magnitude is 6 nC, stored energy 27 nJ and inner gap-side field 54 N/C.
Open the experimentPredict
The inner-area planar approximation has relative error at a=1 m,d=0.2 m equal to:
Experiment
Compare d=0.1,0.2,0.5 m at fixed a=1 m. Retain exact capacitance, the chosen planar approximation and its signed relative error. Derive a gap-to-radius criterion for a maximum 5% relative error instead of choosing a visually thin gap.
Integrate radial voltage and compare a curvature approximation
Worked teaching model · synthetic values · Concentric spherical capacitor with equal/opposite charges and maintained potential difference9 V. Inner radius a and outer radius a+d. Vacuum, ideal conductors. Compare planar approximation using inner-sphere area, not an unspecified effective area. No charged apparatus.
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Curvature correction at fixed inner radius
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: exact spherical relation (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Curvature correction at fixed inner radius
exact spherical relation
| radial gap (m) | capacitance (nF) |
|---|---|
| 0.1 | 1.222222 |
| 0.105 | 1.169312 |
| 0.11 | 1.121212 |
| 0.115 | 1.077295 |
| 0.12 | 1.037037 |
| 0.125 | 1 |
| 0.13 | 0.965812 |
| 0.135 | 0.9341564 |
| 0.14 | 0.9047619 |
| 0.145 | 0.8773946 |
| 0.15 | 0.8518519 |
| 0.155 | 0.827957 |
| 0.16 | 0.8055556 |
| 0.165 | 0.7845118 |
| 0.17 | 0.7647059 |
| 0.175 | 0.7460317 |
| 0.18 | 0.7283951 |
| 0.185 | 0.7117117 |
| 0.19 | 0.6959064 |
| 0.195 | 0.6809117 |
| 0.2 | 0.6666667 |
| 0.205 | 0.6531165 |
| 0.21 | 0.6402116 |
| 0.215 | 0.627907 |
| 0.22 | 0.6161616 |
| 0.225 | 0.6049383 |
| 0.23 | 0.5942029 |
| 0.235 | 0.5839243 |
| 0.24 | 0.5740741 |
| 0.245 | 0.5646259 |
| 0.25 | 0.5555556 |
| 0.255 | 0.546841 |
| 0.26 | 0.5384615 |
| 0.265 | 0.5303983 |
| 0.27 | 0.5226337 |
| 0.275 | 0.5151515 |
| 0.28 | 0.5079365 |
| 0.285 | 0.5009747 |
| 0.29 | 0.4942529 |
| 0.295 | 0.4877589 |
| 0.3 | 0.4814815 |
| 0.305 | 0.4754098 |
| 0.31 | 0.4695341 |
| 0.315 | 0.4638448 |
| 0.32 | 0.4583333 |
| 0.325 | 0.4529915 |
| 0.33 | 0.4478114 |
| 0.335 | 0.4427861 |
| 0.34 | 0.4379085 |
| 0.345 | 0.4331723 |
| 0.35 | 0.4285714 |
| 0.355 | 0.4241002 |
| 0.36 | 0.4197531 |
| 0.365 | 0.4155251 |
| 0.37 | 0.4114114 |
| 0.375 | 0.4074074 |
| 0.38 | 0.4035088 |
| 0.385 | 0.3997114 |
| 0.39 | 0.3960114 |
| 0.395 | 0.3924051 |
| 0.4 | 0.3888889 |
| 0.405 | 0.3854595 |
| 0.41 | 0.3821138 |
| 0.415 | 0.3788487 |
| 0.42 | 0.3756614 |
| 0.425 | 0.372549 |
| 0.43 | 0.369509 |
| 0.435 | 0.366539 |
| 0.44 | 0.3636364 |
| 0.445 | 0.360799 |
| 0.45 | 0.3580247 |
| 0.455 | 0.3553114 |
| 0.46 | 0.352657 |
| 0.465 | 0.3500597 |
| 0.47 | 0.3475177 |
| 0.475 | 0.3450292 |
| 0.48 | 0.3425926 |
| 0.485 | 0.3402062 |
| 0.49 | 0.3378685 |
| 0.495 | 0.335578 |
| 0.5 | 0.3333333 |
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 |
|---|---|---|
| Outer radius | 1.2 | m |
| Exact spherical capacitance | 6.66667e-10 | F |
| Inner-area planar approximation | 5.55556e-10 | F |
| Planar relative signed error | -0.1666667 | ratio |
| Plate charge magnitude | 6.00000e-9 | C |
| Stored energy at 9 V | 2.70000e-8 | J |
| Inner surface gap-side field | 54 | N/C |
The voltage is a radial field integral, and finite curvature changes capacitance relative to the chosen planar area.
Between surfaces E=KQ/r², hence DeltaV=KQ(1/a−1/bouter). Thus C=abouter/[K(bouter−a)]. Using inner area4πa² in epsilon0 A/d gives Cplane=a²/(Kd), and Cplane/Cexact=a/(a+d). The relative error is−d/(a+d). The limit d/a→0 is a controlled approximation, not an exact equality at every gap.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The voltage is a radial field integral, and finite curvature changes capacitance relative to the chosen planar area.
Between surfaces E=KQ/r², hence DeltaV=KQ(1/a−1/bouter). Thus C=abouter/[K(bouter−a)]. Using inner area4πa² in epsilon0 A/d gives Cplane=a²/(Kd), and Cplane/Cexact=a/(a+d). The relative error is−d/(a+d). The limit d/a→0 is a controlled approximation, not an exact equality at every gap.
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