Small distances can be measured using a capacitor whose plate separation 𝓍 is variable. Consider an air-filled parallel-plate capacitor with fixed plate area A = 25 mm² separated by a variable distance 𝓍. Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance C in the range 1.0 pF to 1000.0 pF with an accuracy of ∆C = 0.1 pF. Define ∆𝓍 to be the accuracy (magnitude) to which 𝓍 can be determined, and determine a formula for ∆𝓍.

Verified step by step guidance

1. Start by recalling the formula for the capacitance of a parallel-plate capacitor: $C=\frac{\epsilon }{x}$, where $c$ is the capacitance, $e$ is permittivity and $x$ distance

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store electrical charge per unit voltage. It is defined by the formula C = ε(A/x), where C is the capacitance, ε is the permittivity of the dielectric material (air in this case), A is the area of the plates, and x is the separation between them. Understanding this relationship is crucial for determining how changes in plate separation affect the measured capacitance.

Measurement Accuracy

Measurement accuracy refers to the degree to which a measured value reflects the true value. In this context, the accuracy of the capacitance measuring instrument is given as ∆C = 0.1 pF, which indicates the smallest change in capacitance that can be reliably detected. This concept is essential for determining how precisely the plate separation x can be inferred from the measured capacitance.

Error Propagation

Error propagation is the process of determining the uncertainty in a calculated result based on the uncertainties in the measurements used to obtain it. In this case, the uncertainty in the capacitance measurement (∆C) will affect the calculated uncertainty in the plate separation (∆x). Understanding how to propagate these errors is key to deriving a formula for ∆x based on the given ∆C.