A 10.0μF parallel-plate capacitor with circular plates is...

A 10.0$\mu \mathrm{F}$ parallel-plate capacitor with circular plates is connected to a 12.0 $\mathrm{V}$ battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the 12.0 $\mathrm{V}$ battery after the radius of each plate was doubled without changing their separation?

Key Concepts

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. It is defined by the relationship Q = CV, where Q is the stored charge, C is the capacitance, and V is the voltage across the capacitor. This concept is fundamental to understanding how capacitors operate in electrical circuits.

Parallel?Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a small distance. The capacitance of such a capacitor is determined by the geometry of the plates and their separation, usually given by the formula C = (??A)/d, where ?? is the vacuum permittivity, A is the area of one plate, and d is the distance between the plates. This configuration is commonly used to illustrate the basic principles of capacitance in introductory physics and engineering courses.

Battery Connection and Constant Voltage

When a capacitor is connected to a battery, the voltage across the capacitor is maintained at a constant value equal to the battery’s voltage. In this scenario, any change in the capacitor's physical characteristics (such as plate separation or area) will alter the capacitance, and consequently, the charge stored on the plates (since Q = CV). This concept distinguishes situations where the voltage remains constant from those where the capacitor might be isolated, leading to a constant charge.

Effects of Geometric Modifications on Capacitance

The capacitance of a parallel-plate capacitor is directly proportional to the area of the plates and inversely proportional to the distance between them. Changes in these geometric factors will affect the capacitance and therefore the amount of charge stored for a given voltage. For example, increasing the separation reduces the capacitance, while increasing the area of the plates (such as by doubling the radius, which increases the area by a factor of four) increases the capacitance, showing the sensitivity of capacitor behavior to its physical dimensions.

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