Question The parallel-plate air capacitor in Figure con-sists of two horizontal conducting plates of equal area A. The bot-tom 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 in the figure. When uncharged, the plates are separated by a distance Z0. 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 €0A V^2/ 2z^2. (Hint: See Exercise 24.27.) (b) Obtain an expression that relates the plate sep- aration z to the potential difference V. The resulting equation will be cubic in z. (c) Given the values A = 0.300 m^2, Z0 = 1.20 mm, k = 25.0 N/m, and V = 120 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 signifi- cant figures. Locating the roots of the cubic equation graphically can help you pick starting values of z for this trial-and-error proce- dure. One root of the cubic equation has a nonphysical negative value.)
Question The parallel-plate air capacitor in Figure con-sists of two
horizontal conducting plates of equal area A. The bot-tom 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 in the figure. When
uncharged, the plates are separated by a distance Z0. 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 €0A V^2/ 2z^2. (Hint: See
Exercise 24.27.) (b) Obtain an expression that relates the plate sep-
aration z to the potential difference V. The resulting equation will
be cubic in z. (c) Given the values A = 0.300 m^2, Z0 = 1.20 mm,
k = 25.0 N/m, and V = 120 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 signifi-
cant figures. Locating the roots of the cubic equation graphically
can help you pick starting values of z for this trial-and-error proce-
dure. 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 dis-
placement of the object will give rise to a net force tending to
return the object to the equilibrium position. For unstable equilib-
rium a small displacement gives rise to a net force that takes the
object farther away from equilibrium.
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