Q. 34

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 :

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


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