Physics Q40 – JEE-MAIN 2026

Home Physics Electrostatics & CapacitanceQuestion #124

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^{a}

, where

a

is _______.

Clue 1 of 3: Identify the type of capacitor and the constant quantity

The capacitor is a parallel plate air capacitor, and it is connected to a battery, implying the voltage $V$ across it is constant.

🧠 Full Solution Path

Step 1: Express energy in terms of separation✦ Active

The capacitance of a parallel plate capacitor is given by $C = \frac{ϵ_{0} A}{x}$ , where $ϵ_{0}$ is the permittivity of free space, $A$ is the area of the plates, and $x$ is the separation between them. Since the capacitor is connected to a battery, the voltage $V$ across it remains constant. The electrostatic energy stored in the capacitor is $U = \frac{1}{2} C V^{2}$ . Substituting the expression for $C$ into the energy formula:

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

Step 2: Calculate the time rate of change of energy○ Expand

To find the time rate of change of electrostatic energy, we differentiate $U$ with respect to time $t$ . Let $K = \frac{ϵ_{0} A V^{2}}{2}$ be a constant. Then $U = K x^{- 1}$ . Using the chain rule $\frac{d U}{d t} = \frac{d U}{d x} \frac{d x}{d t}$ :

\frac{d U}{d t} = \frac{d}{d x} (K x^{- 1}) \frac{d x}{d t} = K (- 1) x^{- 2} \frac{d x}{d t}

Step 3: Substitute the rate of change of separation and determine proportionality○ Expand

The problem states that the plates are pulled apart at a uniform speed $v$ , which means $\frac{d x}{d t} = v$ . Substituting this into the expression for $\frac{d U}{d t}$ :

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

From this, we can see that the time rate of change of electrostatic energy is proportional to $x^{- 2}$ . Therefore, comparing with $x^{a}$ , we find $a = - 2$ .

💡 Teacher's Secret Hint

Ensure to correctly apply the chain rule and identify all constant terms before differentiation.

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Physics Question 40 – JEE-MAIN 2026

Clue 1 of 3: Identify the type of capacitor and the constant quantity

🧠 Full Solution Path