What is the length of bar B for steady state conditions?
Two rods, one made of metal A that has a conductivity 400 J/s*m and a length 10 cm and the other made of metal B has a conductivity of 50 J/s*m. They are soldered together at one end. They have the same area of cross-section. The free end of the first rod is in boiling water at 100 C and the free end of the other rod is at 0 C. The temperature at the joined end is 25 C
To get the answer here we need to ensure that the rates of heat flow through both the rods are equal else steady state conditions will not be reached. Now, we see that the conductivity of the two rods is different. Let us denote the length of the rod of metal B by Lb.
Now we use the equation for heat flow through the rods, and equate them.
ka*A*(100 – T)/ 10 = kb*A*(0 – T)/ Lb
=> Lb (ka*A*(100 – T)) = 10 (kb*A*(T– 0))
=> Lb (400*A*(100 – 25)) = 10 (50*A*(25– 0))
=> Lb (8 * 75) = 10*25
=> Lb = 10*25 / 8*75
=> Lb = 5/12 cm
Notice that rod B is much shorter than rod A. This is explained by the fact that the conductivity of A is a lot higher than that of B.
Therefore the length of the second rod should be 5/12 cm.