A , B and C are the three identical conductors but made from different materials . They are kept in contact as shown . Their thermal conductivities are K , 2K and `(K)/(2)` . The free end of A is at `100^(@) C` and the free end of C is at `0^(@)C` . During steady state , the temperature of the junction of A and B is nearly .... `""^(@) C` .

Options

A. 71
B. 29
C. 63
D. 37

Answer

A

Explanation

Here , `K_(A) = K , K_(B) = 2 K , K_(C) = (K)/(2)`

Let L be the length and A be area of cross-section of each conductor respectively . Let `T_(1)` and `T_(2)` be temperature of A - B and B - C junctions respectively .
At the steady state , `H_(A) = H_B = H_C`
`(K_(A) A (100 - T_(1)))/(L) = (K_(B) A(T_1- T_2))/(L) = (K_(C) A(T_2 - 0))/(L)`
`(KA (100 - T_(1)))/(L) = (2KA (T_1 - T_2))/(L) = ((K)/(2) A (T_2 - 0))/(L)`
`therefore (100 - T_1) = 2 (T_1 - T_2)`
or `100 - T_1 = 2T_1 - 2 T_2`
`2 T_2 = 3 T_1 - 100`
`T_2 = 3/2 T_(1) - 50 " " ... (i)`
and `2 (T_(1) - T_(2)) = (1)/(2) T_(2)`
`2 (T_(1) ( (3)/(2) T_(1) - 50)) = (1)/(2) ((3)/(2) T_(1) - 50) ` (Using (i))
`2 ((2 T_(1) - 3T_(1) + 100)/(2)) = (3T_(1) - 100)/(4) - T_(1) + 100 = (3T_(1) - 100)/(4) - 4 T_(1) + 400 = 3 T_(1) - 100 7 T_(1) = 500`
`T_(1) = (500^(@) C)/(7) = 71^(@) C` .