Question

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

Answer 1

Given, Length $=40cm$

Area of cross section $=1.0c{m}^2$

Thermal conductivity $=100W/m^{\circ }C$

Time $=10min$

Temperature as a function of time is given as

$T=\frac{60}{600}{t}^{\circ }C=\frac{1}{10}{t}^{\circ }C$

$\frac{dQ}{dt}=kA\frac{dT}{dx}$

Integrating both sides, we get

${\int }_0^{Q}dQ={\int }_0^{(10\times 60)}\left(\frac{kA}{0.4}\right)0.1tdt$

$Q=\frac{100\times 1\times {10}^{-4}\times 0.1t^2}{0.8}{|}_0^{600}=450J$