Separate a plate capacitance from a stray contribution
Capacitance is the proportionality C=Q/ΔV for the chosen linear device. An ideal capacitor has equal opposite charges on its plates, so the usual stored charge Q is a plate-charge magnitude, not the net charge of the entire two-plate object. For broad vacuum plates with negligible edge fields, C=ε0A/d. Convert area into square metres and separation into metres before using that equation.
Learning outcome: Compare gaps 1,2,3 mm while keeping the stray contribution fixed. Plot charge against voltage and retain slope and intercept separately. Explain why doubling the gap does not halve the total measured slope when a nonzero fixed stray capacitance remains.
Starting recap
Use signed charge, common coordinate axes, SI units, vector components and algebraic ratios. Separate a source field from probe force and a fitted parameter from independently tested evidence.
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At gap 2 mm, C_plate=ε0(.02)/.002=88.541878128 pF. Adding 20 pF stray gives slope 108.541878128 pF. At 4 V the indicated charge is 439.167512512 pC, including the 5 pC readout offset; plate energy alone is about 7.08335×10⁻¹⁰ J.
Open the experimentPredict
A constant charge-readout offset changes which feature of a linear Q–V fit?
Experiment
Compare gaps 1,2,3 mm while keeping the stray contribution fixed. Plot charge against voltage and retain slope and intercept separately. Explain why doubling the gap does not halve the total measured slope when a nonzero fixed stray capacitance remains.
Separate plate geometry from stray parallel capacitance
Worked teaching model · synthetic values · Ideal vacuum parallel plates area .02 m², epsilon0=8.8541878128e−12 F/m, negligible edge fields. A fixed independent stray capacitance is in parallel. Read charge at 2,4,6 V; charge sensor also adds a known 5 pC offset. No practical capacitor handling.
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Charge versus maintained potential difference
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: plate plus stray plus offset (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Charge versus maintained potential difference
plate plus stray plus offset
| voltage (V) | indicated charge (pC) |
|---|---|
| 0 | 5 |
| 0.15 | 21.2813 |
| 0.3 | 37.5626 |
| 0.45 | 53.8438 |
| 0.6 | 70.1251 |
| 0.75 | 86.4064 |
| 0.9 | 102.688 |
| 1.05 | 118.969 |
| 1.2 | 135.25 |
| 1.35 | 151.532 |
| 1.5 | 167.813 |
| 1.65 | 184.094 |
| 1.8 | 200.375 |
| 1.95 | 216.657 |
| 2.1 | 232.938 |
| 2.25 | 249.219 |
| 2.4 | 265.501 |
| 2.55 | 281.782 |
| 2.7 | 298.063 |
| 2.85 | 314.344 |
| 3 | 330.626 |
| 3.15 | 346.907 |
| 3.3 | 363.188 |
| 3.45 | 379.469 |
| 3.6 | 395.751 |
| 3.75 | 412.032 |
| 3.9 | 428.313 |
| 4.05 | 444.595 |
| 4.2 | 460.876 |
| 4.35 | 477.157 |
| 4.5 | 493.438 |
| 4.65 | 509.72 |
| 4.8 | 526.001 |
| 4.95 | 542.282 |
| 5.1 | 558.564 |
| 5.25 | 574.845 |
| 5.4 | 591.126 |
| 5.55 | 607.407 |
| 5.7 | 623.689 |
| 5.85 | 639.97 |
| 6 | 656.251 |
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 |
|---|---|---|
| Geometric plate capacitance | 88.5419 | pF |
| Measured charge-voltage slope | 108.542 | pF |
| Known charge-reading intercept | 5 | pC |
| Total indicated charge at 4 V | 439.168 | pC |
| Plate field at 4 V | 2000 | V/m |
| Plate energy at 4 V | 7.08335e-10 | J |
| Stray energy at 4 V | 1.6e-10 | J |
The fitted slope includes both capacitances; a charge offset changes the intercept instead.
C=epsilon0 A/d and Q=CV use SI area and separation. Opposite plates carry +Q and −Q, so the capacitor remains neutral overall. A dielectric changes polarization and effective permittivity; here vacuum is stipulated. A slope alone cannot distinguish larger plate area, smaller gap, dielectric or stray parallel capacitance without independent geometry/material evidence. Energy is one half CV², not QV for gradual charging.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The fitted slope includes both capacitances; a charge offset changes the intercept instead.
C=epsilon0 A/d and Q=CV use SI area and separation. Opposite plates carry +Q and −Q, so the capacitor remains neutral overall. A dielectric changes polarization and effective permittivity; here vacuum is stipulated. A slope alone cannot distinguish larger plate area, smaller gap, dielectric or stray parallel capacitance without independent geometry/material evidence. Energy is one half CV², not QV for gradual charging.
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