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A $500 \Omega$ resistor connected to an external battery is placed inside a thermally insulated cylinder fitted with a frictionless piston. The cylinder contains an ideal gas. A current $i$ of $200 \mathrm{~mA}$ flows through the resistor as shown in the figure. The mass of the piston is $10 \mathrm{~kg}$. Assuming $g=10 \mathrm{~m} / \mathrm{s}^2$, the speed at which the piston will move upward, due to heat dissipated by the resistor, so that the temperature of the gas remains unchanged is
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The correct answer is:
$20 \mathrm{~cm} / \mathrm{s}$
As, temperature of gas remains constant, heat supplied by resistance $=$ piston work
$$
\begin{aligned}
\Rightarrow \quad i^2 R t & =p \times \Delta V \Rightarrow i^2 R t=\frac{F}{A} \times A \times \Delta x \\
\Rightarrow \quad \frac{i^2 R}{F} & =\frac{\Delta x}{t}=\text { velocity of piston } \\
\Rightarrow \quad v_{\text {piston }} & =\frac{\left(200 \times 10^{-3}\right)^2 \times 500}{10 \times 10} \\
& =\frac{4 \times 10^{-2} \times 5 \times 10^2}{10^2} \\
& =20 \times 10^{-2} \mathrm{~ms}^{-1}=20 \mathrm{cms}^{-1}
\end{aligned}
$$
$$
\begin{aligned}
\Rightarrow \quad i^2 R t & =p \times \Delta V \Rightarrow i^2 R t=\frac{F}{A} \times A \times \Delta x \\
\Rightarrow \quad \frac{i^2 R}{F} & =\frac{\Delta x}{t}=\text { velocity of piston } \\
\Rightarrow \quad v_{\text {piston }} & =\frac{\left(200 \times 10^{-3}\right)^2 \times 500}{10 \times 10} \\
& =\frac{4 \times 10^{-2} \times 5 \times 10^2}{10^2} \\
& =20 \times 10^{-2} \mathrm{~ms}^{-1}=20 \mathrm{cms}^{-1}
\end{aligned}
$$
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