A researcher measures the capacitance of a parallel-plate capacitor for several plate…

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

A researcher measures the capacitance of a parallel-plate capacitor for several plate separations while keeping the geometric area

A

of each plate and the dielectric material between the plates unchanged. The inverse separation

1/d

and the corresponding measured capacitance

C

are recorded in the table below.

$1/d$ ( $\times 10^3\,\text{m}^{-1}$ )	$C$ ( $\times 10^{-9}\,\text{F}$ )
1.0	0.89
1.5	1.34
2.0	1.78
2.5	2.24
3.0	2.66

Assume all measurements are subject to identical fractional uncertainties. (a) Sketch a graph of

C

versus

1/d

using the data. Describe verbally whether the relationship appears linear and how you infer this from the plotted points. (b) Using an appropriate straight-line fit to the data, determine the numerical value of the slope

S

of the best-fit line. Show your work, including any averaging method you use. (c) Explain how the slope

S

can be used to estimate the dielectric constant

\kappa

of the material filling the space between the plates. Clearly outline each logical step in your reasoning. (d) Calculate a numerical estimate for

\kappa

using your value of

S

and the stated plate area

A = 2.0 \times 10^{-2}\,\text{m}^2

. Express your answer with an appropriate number of significant figures. (e) Discuss two possible experimental sources of systematic error that could cause the estimated

\kappa

to differ from the material's accepted value, and describe qualitatively the direction (higher or lower) in which each source would shift the estimate.

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