Question

A rod of negligible heat capacity has length 20 cm, area of cross section 1.0 cm2 and thermal conductivity 200 W m−1°C−1. The temperature of one end is maintained at 0°C and that of the other end is slowly and linearly varied from 0°C to 60°C in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these 10 minutes.

Answer 1

The total heat transmitted through the rod in given 10 minutes is 1800J (approx)

Explanation:

To find the total heat transmitted through the rod:

Step 1:

Rate of increase in temp./sec $=\frac{\Delta T}{\Delta t}=\frac{60-0}{10} \times 60=0.1^{\circ} \mathrm{C} / \mathrm{s}$

                 Rate of heat flow $=\frac{\Delta Q}{\Delta t}=\frac{K A \Delta T}{l}$

As we know that,                $Q=\sum \frac{K A \Delta T}{l} \Delta t$

$Q=\frac{K A}{l} \times 0.1+\frac{K A}{l} \times 0.2+\frac{K A}{l} \times 0.3+\cdots+\frac{K A}{l} \times 60$

$Q=\frac{K A}{l}[0.1+0.2+0.3+\cdots+60]$

Step 2:

Sum of n terms of an A.P

$Q=\frac{K A}{l}\left[\frac{600}{2}(0.1+60)\right]$

$Q \approx 1800 J$

Therefore the total heat transmitted through the rod  is 1800J