The circuit in Figure P 26.25 consists of two identical...

The circuit in Figure $\mathrm{P} 26.25$ consists of two identical parallel metallic plates connected by identical metallic springs to a $100-V$ battery. With the switch open, the plates are uncharged, are separated by a distance $d=8.00 \mathrm{~mm}$, and have a capacitance $C=2.00 \mu \mathrm{F}$. When the switch is closed, the distance between the plates decreases by a factor of $0.500 .$ (a) How much charge collects on each plate and (b) what is the spring constant for each spring? (Hint: Use the result of Problem 35.)

Key Concepts

Parallel?Plate Capacitor

A parallel?plate capacitor is composed of two conductive plates separated by a distance, with its capacitance directly related to the area of the plates and inversely related to their separation. The basic relation C = ??A/d (for a vacuum) quantifies how the geometry of the plates determines the ability to store electrical charge for a given voltage, making this concept foundational in understanding capacitor behavior.

Voltage–Charge Relationship

The relationship between voltage and charge in a capacitor is given by Q = CV. This linear relationship shows that for a constant applied voltage, any change in the capacitance (such as by reducing the distance between the plates) will directly affect the stored charge. This concept is essential when analyzing circuits where geometrical changes or other factors modify a capacitor’s capacitance.

Electrostatic Force in Capacitors

Electrostatic forces in capacitors arise due to the attraction between oppositely charged plates. This force can be understood either by differentiating the energy stored in the capacitor with respect to distance or directly from field and charge distributions. It plays a crucial role in systems where the mechanical forces due to charging interact with elastic or other restorative forces in the device.

Hooke’s Law and Elastic Force

Hooke’s Law describes the linear relationship between the force exerted by a spring and its displacement from equilibrium, given by F = kx. This principle is used to determine the spring constant when the spring is part of an electromechanical system, such as in a capacitor with flexible plates. It is a key concept in understanding how mechanical restoring forces balance electrical forces.

Force Equilibrium in Electromechanical Systems

Force equilibrium in an electromechanical system is achieved when the mechanical restoring force (from springs or other elastic elements) exactly balances the electrical attraction between charged components. Analyzing this balance enables determination of unknown system parameters like the spring constant by setting the spring force equal to the electrostatic force acting due to changes in the physical configuration of the capacitor.

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