In one type of computer keyboard, each key holds a small...

In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, airfilled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is 42.0 mm$^2$, and the separation between the plates is 0.700 mm before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circuitry can detect a change in capacitance of 0.250 pF, how far must the key be depressed before the circuitry detects its depression?

Key Concepts

Capacitive Sensing

Capacitive sensing involves detecting changes in capacitance that result from alterations in the physical configuration of a capacitor. When a mechanical action, such as pressing a key, changes the distance between the plates, the capacitance varies accordingly. This technique is widely used in touch-sensitive applications and electronic input devices to convert mechanical actions into electrical signals.

Electrostatic Permittivity

Electrostatic permittivity, notably the permittivity of free space (??), is a constant that characterizes how an electric field interacts with the medium between the capacitor plates. This constant appears in the formula for the capacitance of a parallel-plate capacitor and is key to quantifying the capacitor’s ability to store charge for given geometric conditions.

Parallel-Plate Capacitor

A parallel-plate capacitor is a basic electrical component that consists of two conductive plates separated by an insulating medium (dielectric). The capacitance is determined by the area of the plates, the separation between them, and the permittivity of the dielectric. This fundamental concept is essential in understanding how capacitors store energy in the electric field created between the plates.

Capacitance Dependence on Geometry

The capacitance of a parallel-plate capacitor is directly proportional to the area of its plates and inversely proportional to the distance between them. This relationship means that increasing the area or decreasing the separation will raise the capacitance, a concept that is critical when designing sensors or components where physical dimensions change the electrical properties.

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