question_answer

Three rods of same dimensions have thermal conductivities 3K, 2K and K respectively.

They are arranged as shown below

What will be the temperature T of the junction?

A) \[\frac{200}{3}\,{{\,}^{o}}C\] 

B) \[\frac{100}{3}\,{{\,}^{o}}C\]

C) \[75\,{{\,}^{o}}C\]      

D) \[\frac{50}{3}{{\,}^{o}}C\]

Correct Answer: A

Solution :

[a]: Let L and A be length and area of cross- section of each rod. At steady state,\[{{H}_{1}}={{H}_{2}}+{{H}_{3}}\] \[\frac{(100-T)(3K)A}{L}=\frac{(T-50)2KA}{L}+\frac{(T-0)KA}{L}\] \[3(100-T)=2(T-50)+T\] \[300-3T=2r-100+T\] \[6T=400\]or\[T=\frac{400}{6}=\frac{200}{3}{{\,}^{o}}C\]