A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed

Question

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed $v$ . If $x$ is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to $x^{α}$ , where $α$ is ______. [2026]

Smart Achievers · Accepted Answer

(1)

$Capacitance C = \frac{ε_{0} A}{x}$

$Energy stored U = \frac{1}{2} C V^{2}$

$U = \frac{1}{2} (\frac{ε_{0} A}{x}) V^{2}$

$\frac{d U}{d t} = \frac{1}{2} ε_{0} A V^{2} (- \frac{1}{x^{2}}) \frac{d x}{d t}$

$\frac{d U}{d t} \propto \frac{1}{x^{2}}$

$value of α = - 2$

Solution

Q.
A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed $v$ . If $x$ is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to $x^{α}$ , where $α$ is ______. [2026]

1 −2

2 1

3 −1

4 2

Ans.
(1)

$Capacitance C = \frac{ε_{0} A}{x}$

$Energy stored U = \frac{1}{2} C V^{2}$

$U = \frac{1}{2} (\frac{ε_{0} A}{x}) V^{2}$

$\frac{d U}{d t} = \frac{1}{2} ε_{0} A V^{2} (- \frac{1}{x^{2}}) \frac{d x}{d t}$

$\frac{d U}{d t} \propto \frac{1}{x^{2}}$

$value of α = - 2$

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Solution

Q. A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed v. If x is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to xα, where α is ______. [2026]

1 −2

2 1

3 −1

4 2

Ans. (1) Capacitance C=ε0Ax Energy stored U=12CV2 U=12(ε0Ax)V2 dUdt=12ε0AV2(-1x2)dxdt dUdt∝1x2 value of α=-2

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Ans.
(1)

$Capacitance C = \frac{ε_{0} A}{x}$

$Energy stored U = \frac{1}{2} C V^{2}$

$U = \frac{1}{2} (\frac{ε_{0} A}{x}) V^{2}$

$\frac{d U}{d t} = \frac{1}{2} ε_{0} A V^{2} (- \frac{1}{x^{2}}) \frac{d x}{d t}$

$\frac{d U}{d t} \propto \frac{1}{x^{2}}$

$value of α = - 2$