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.
Given-
Length of the rod, l = 20 cm = 0.2 m
Area of cross section of the rod, A = 1.0 cm2 = 1.0×10-4 m2
Thermal conductivity of the rod, k = 200 W m-1°C-1
Also, the temperature of one end is maintained at 0°C and that of the other end is varied from 0°C to 60°C in 10 minutes.
Hence rate of increase of the temperature at one end is 0.1°C per second.
⇒ 
= 
= 
0.1°C/s
Now, rate of flow of heat –
= 
= 
+
...........+
(
= 
)
We know Arithmetic Progressions –
For a series given by
a+2a+...................na
Sum of n terms is given by
Sum= 
, where a is the first term, d is the difference between first and second term and n is the number of terms.
Substituting the values, for 10 minutes –
Here a= 0,1. d = 0.1 and n = 60 for 10mins,
= 
= 1800J
Hence , the total heat transmitted through the rod in these 10 minutes is 1800J