Question

A current $i=2\text{A}$ flows through the resistance $R=10\Omega$ with what constant speed $v$ (in m/s), the piston (mass $=1\text{kg}$) must move in upward direction so that temperature of ideal gas present inside cylinder may remain unchanged? Take ($g=10{\text{m/s}}^2$)

Answer 1

Due to current $i=2\text{A}$ flowing through the resistance $R$, the rate of energy dissipated in resistance is the source of energy for workdone by the gas.

Rate of energy supplied by resistance.
$\frac{dQ}{dt}=i^{2}R=(2{)}^2\times 10=40\text{J/s}$

Using first law of thermodynamics:
$dQ=dU+dW$
$\Rightarrow \frac{dQ}{dt}=\frac{dU}{dt}+\frac{dW}{dt}.....\left(i\right)$
(for adiabatic walls of cylinder no heat is last, which is supplied by resistance)

Also to maintain the temperature of gas inside cylinder $\Delta T=0$ i..e dU = 0 or $\frac{dU}{dt}=0$

We can write the work done by gas, $dW=PdV$
$\Rightarrow \frac{dW}{dt}=\frac{PdV}{dt}$
Let area of cross -section of pistion is $A$, thus the change in volume of gas due to piston displacement will be;
$dV=Adx$
or, $\frac{dW}{dt}=\frac{P\left(Adx\right)}{dt}=PA\left(\frac{dx}{dt}\right).....\left(ii\right)$

The force on piston due to gas is,
$F_{gas}=P\times A=PA$
Applying equilibrium condition on piston (v = constant, a = 0)
$\Rightarrow PA=mg$
$\therefore P=\frac{mg}{A}$
Substituting in equation (ii),
$\frac{dW}{dt}=\frac{mg}{A}\times A\times \left(\frac{dx}{dt}\right)$
or $\frac{dW}{dt}=mg\left(\frac{dx}{dt}\right)=mgv....\left(iii\right)$
Here $\frac{dx}{dt}=v=\text{constant}$

From Eq. (i) & (iii):
$\frac{dQ}{dt}=0+mgv$
$\Rightarrow 40=(1\times 10)v$
$\therefore v=4\text{m/s}$

$\begin{array}{c}\text{Why this question}?\\ \text{Tip: In this problem the heat is supplied to the}\\ \text{gas by resistance and that heat is completely}\\ \text{available for work done by gas (adiabatic walls}\Delta Q=0).\\ \text{Thus the power dissipated in resistance must be able to}\\ \text{overcome the power of gravity only then piston can}\\ \text{move with constant velocity.}\end{array}$