#### Question

A rod of negligible heat capacity has length 20 cm, area of cross section 1.0 cm^{2} 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.

#### Solution

Given:

Length of the rod,* l* = 20 cm = 0.2 m

Area of cross section of the rod, *A* = 1.0 cm^{2} = 1.0 × 10 ^{-4}m^{2}

Thermal conductivity of the material of the rod, k = 200 W m^{-1}° C^{-1 }

The temperature of one end of the rod is increased uniformly by 60° C within 10 minutes.

This mean that the rate of increase of the temperature of one end is 0.1° C per second

`rArr 60/(10xx60)""^circ C//s`

So, total heat flow can be found by adding heat flow every second.

Rate of flow of heat = `(dQ)/dt`

Q"net" = ∑ `(KA)/d ( T_2 - T_1)×Deltat`

For each interval,

`Deltat = 1`

`Q"net" = KA/d ( 0.1 + (KA)/d xx0.2 + (KA)/d xx0.3 + ...................+ KA/dxx 60.0`

`Q"net" = (KA)/d (0.1 + 0.2 +.........+ 60.0)`

sum of n terms of an AP is given by

`s_n = n/2 ( axxa_n)`

`Q_net = (KA)/d xx 600/2 (0.1+60)`

⇒` Q'"net" = (200xx - 10^-4)/(20xx10^-2) xx 600/2xx60.1`

⇒ Q_{net} = 1800J (approximately)