A laboratory instrument requires a parallel-plate capacitor with an effective…

PHYSICS · Multi-part · CBSE · Physics · Electrostatic Potential and Capacitance · Electrostatic Potential and Capacitance

Problem

A laboratory instrument requires a parallel-plate capacitor with an effective capacitance of

50\,\text{pF}

. The space between the plates is to be completely filled with a polymer dielectric whose dielectric constant is

\kappa=3.0

. Sheets of the dielectric material can withstand a maximum electric field (breakdown strength) of

3.0\times10^{7}\,\text{V}\!/\!\text{m}

before electrical breakdown occurs. For mechanical reasons, each conducting plate must be a square with a maximum side length of

5.0\,\text{cm}

. Neglect edge effects throughout this problem. (a) Derive an algebraic expression for the plate separation

d

that simultaneously satisfies the required capacitance and remains below the dielectric’s breakdown strength when a potential difference

\Delta V

is applied. Your expression may include

\kappa

, the vacuum permittivity

\varepsilon_0

, the plate area

A

, the desired capacitance

C

, and the breakdown field

E_{\text{bd}}

. (b) Using

C=50\,\text{pF}

and the maximum allowed plate area, calculate the plate separation that meets the capacitance requirement. Then determine the maximum potential difference that can be applied without exceeding the breakdown limit, and state whether the breakdown strength or the capacitance requirement is the stricter design constraint. (c) Suppose the plate area could be doubled while all other specifications remain unchanged. Quantitatively determine how the required plate separation and the maximum safe potential difference would change. Explain the physical basis for each change. (d) Identify one practical manufacturing challenge that could arise when assembling this capacitor and propose a method to mitigate it.

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