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CP The parallel-plate air capacitor in Fig. $P 24.76$ consists of two horizontal conducting plates of equal area $A$ . The bottom plate rests on a fixed support, and the top plate is suspended by four springs with spring constant $k$ , positioned at each of the four corners of the top plate as shown in the figure. When uncharged, the plates are separated by a distance $z_{0} .$ A battery is connected to the plates and produces a potential difference $V$ between them. This causes the plate separation to decrease to $z$ . Neglect any fringing effects. (a) Show that the electrostatic force between the charged plates has a magnitude $\epsilon_{0} A V^{2} / 2 z^{2} .$ Hint: See Exercise 24.27 ) (b) Obtain an expression that relates the plate separation $z$ to the potential difference $V$ . The resulting equation will be cubic in $z$ (c) Given the values $A=0.300 \mathrm{m}^{2}, z_{0}=1.20 \mathrm{mm}$ $k=25.0 \mathrm{N} / \mathrm{m},$ and $V=120 \mathrm{V}$ , find the two values of $z$ for which the top plate will be in equilibrium. (Hint: You can solve the cubic equation by plugging a trial value of $z$ into the equation and then adjusting your guess until the equation is satisfied to three significant figures. Locating the roots of the cubic equation graphically can help you pick starting values of $z$ for this trial-and-error procedure. One root of the cubic equation has a nonphysical negative value.) $($ d) For each of the two values of $z$ found in part $(c),$ is the equilibrium stable or unstable? For stable equilibrium a small displacement of the object will give rise to a net force tending to return the object to the equilibrium position. For unstable equilibrium a small displacement gives rise to a net force that takes the object farther away from equilibrium.

Name: Problem 76, Chapter 24: University Physics with Modern Physics
Uploaded: 2020-07-30T21:33:46-08:00
Duration: 5 min 34 s
Channel: Sarah Mccrumb

   CP The parallel-plate air capacitor in Fig. $P 24.76$ consists of two horizontal conducting plates of equal area $A$ . The bottom plate rests on a fixed support, and the top plate is suspended by four springs with spring constant $k$ , positioned at each of the four corners of the top plate as shown in the figure. When uncharged, the plates are separated by a distance $z_{0} .$ A battery is connected to the plates and produces a potential difference $V$ between them. This causes the plate separation to decrease to $z$ . Neglect any fringing effects. (a) Show that the electrostatic force
between the charged plates has a magnitude $\epsilon_{0} A V^{2} / 2 z^{2} .$ Hint: See Exercise 24.27 ) (b) Obtain an expression that relates the plate separation $z$ to the potential difference $V$ . The resulting equation will be cubic in $z$ (c) Given the values $A=0.300 \mathrm{m}^{2}, z_{0}=1.20 \mathrm{mm}$ $k=25.0 \mathrm{N} / \mathrm{m},$ and $V=120 \mathrm{V}$ , find the two values of $z$ for which the top plate will be in equilibrium. (Hint: You can solve the cubic equation by plugging a trial value of $z$ into the equation and then adjusting your guess until the equation is satisfied to three significant figures. Locating the roots of the cubic equation graphically
can help you pick starting values of $z$ for this trial-and-error procedure. One root of the cubic equation has a nonphysical negative value.) $($ d) For each of the two values of $z$ found in part $(c),$ is the equilibrium stable or unstable? For stable equilibrium a small displacement of the object will give rise to a net force tending to return the object to the equilibrium position. For unstable equilibrium a small displacement gives rise to a net force that takes the object farther away from equilibrium.

University Physics with Modern Physics

Hugh D. Young 13th Edition

Chapter 24, Problem 76

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Solved by Expert Sarah Mccrumb

07/30/2020

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CP The parallel-plate air capacitor in Fig. $P 24.76$ consists of two horizontal conducting plates of equal area $A$ . The bottom plate rests on a fixed support, and the top plate is suspended by four springs with spring constant $k$ , positioned at each of the four corners of the top plate as shown in the figure. When uncharged, the plates are separated by a distance $z_{0} .$ A battery is connected to the plates and produces a potential difference $V$ between them. This causes the plate separation to decrease to $z$ . Neglect any fringing effects. (a) Show that the electrostatic force between the charged plates has a magnitude $\epsilon_{0} A V^{2} / 2 z^{2} .$ Hint: See Exercise 24.27 ) (b) Obtain an expression that relates the plate separation $z$ to the potential difference $V$ . The resulting equation will be cubic in $z$ (c) Given the values $A=0.300 \mathrm{m}^{2}, z_{0}=1.20 \mathrm{mm}$ $k=25.0 \mathrm{N} / \mathrm{m},$ and $V=120 \mathrm{V}$ , find the two values of $z$ for which the top plate will be in equilibrium. (Hint: You can solve the cubic equation by plugging a trial value of $z$ into the equation and then adjusting your guess until the equation is satisfied to three significant figures. Locating the roots of the cubic equation graphically can help you pick starting values of $z$ for this trial-and-error procedure. One root of the cubic equation has a nonphysical negative value.) $($ d) For each of the two values of $z$ found in part $(c),$ is the equilibrium stable or unstable? For stable equilibrium a small displacement of the object will give rise to a net force tending to return the object to the equilibrium position. For unstable equilibrium a small displacement gives rise to a net force that takes the object farther away from equilibrium.

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Key Concepts

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 area of the plates, the separation between them, and the medium between them. It is a fundamental system used to explore how geometry and material properties affect charge storage and electric fields in electrostatics.

Electrostatic Force Between Plates

When a voltage is applied across a capacitor, an electric field is established between its plates, leading to a distribution of charges that create an attractive force between the plates. This force can be derived from energy considerations or directly from the electric field and is crucial for understanding interactions in capacitor systems. The expression of the force typically depends on parameters such as the permittivity, area, voltage, and the separation squared.

Mechanical Equilibrium in Coupled Systems

When additional mechanical elements like springs are attached to a capacitor system, the equilibrium condition is found by balancing the electrostatic force with the mechanical restoring force. This concept illustrates how electrical and mechanical forces interact and is essential when analyzing systems where electrical actuation causes physical displacement.

Cubic Equations in Equilibrium Analysis

In coupled electro-mechanical systems, the equilibrium condition often leads to a nonlinear equation—in this case, a cubic equation—in the variable of interest. Solving such equations is necessary to determine the possible equilibrium states of the system. It highlights the complexity arising from the interplay of different physical forces, indicating that multiple equilibrium positions (including nonphysical ones) may exist.

Stability Analysis of Equilibrium

Once the equilibrium states are determined, assessing their stability is crucial. Stability analysis determines whether a small perturbation from equilibrium will result in a restoring force that drives the system back (stable equilibrium) or an amplifying force that drives it further away (unstable equilibrium). This concept is important in predicting the dynamic behavior of systems under the influence of combined forces.

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