Question

A rod of negligible heat capacity has length $40cm$, area of cross-section $1.0c{m}^2$ and thermal conductivity $100W/\left(m{-}^{o}C\right).$ The temperature of one end is maintained at $0{}^{o}C$ and that of the other end is slowly and linearly varied from $0{}^{o}C$ to $60{}^{o}C$ in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted (in $J$) through the rod in these 10 minutes.

• A. 112.5
• B. 225
• C. 450
• D. 45

Answer 1

The correct option is C 450

Temperature as a function of time
$T=\frac{t}{10}{}^{o}C$, where $t$ is in seconds.

Using law of thermal conductivity:
$\frac{dQ}{dt}=kA\frac{dT}{dx}$
Here, $\frac{dT}{dx}$ can be approximated to $\frac{T}{l}$ as changes are linear in nature.


${\int }_0^{Q}dQ={\int }_0^{(10\times 60)}\left(\frac{kA}{l}\right)\times \left(0.1t\right)dtQ={\begin{array}{c}\left[\frac{100\times 1\times {10}^{-4}\times 0.1t^2}{0.4\times 2}\right]\end{array}|}_0^{600}=450J$