The circuit in Figure P 26.38 consists of two identical...

The circuit in Figure $\mathrm{P} 26.38$ consists of two identical parallel metal plates connected by identical metal springs to a $100-\mathrm{V}$ 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? (Suggestion: Use the result of Problem $37 . )$

Key Concepts

Electrostatic Forces in Capacitors

This concept involves understanding that a charged capacitor experiences an attractive electrostatic force between its plates, which can be derived from the change in stored electrical energy as the distance between the plates varies. The resulting force is crucial for analyzing the mechanical effects induced by the electric field in electromechanical systems.

Capacitance and Geometric Dependence

This concept explains that the capacitance of a parallel-plate capacitor depends on the physical parameters of the system, particularly the area of the plates and the separation between them. A change in the distance between the plates directly affects the capacitance, which in turn influences how much charge the capacitor can store for a given voltage.

Charge-Voltage Relationship

The fundamental relation Q = CV shows that for a constant voltage source, the charge stored on a capacitor is directly proportional to its capacitance. This principle is essential in determining the amount of charge that accumulates on the plates when the capacitance changes due to modifications in the plate separation.

Hooke's Law and Spring Mechanics

Hooke's law states that the force exerted by a spring is directly proportional to its displacement, characterized by the spring constant k. In electromechanical systems, like the one described, the mechanical response of springs to the electrostatic force (shortening or elongation) is dictated by Hooke's law.

Force Equilibrium in Electromechanical Systems

In systems where mechanical and electrical forces interact, equilibrium is achieved when the mechanical restoring force (from the springs) equals the electrostatic attractive force (between the capacitor plates). This balance is used to derive relationships and solve for quantities such as the spring constant in configurations where a change in capacitor geometry occurs.

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