NEET · Physics · STD 11 - 10.2 transmission of heat
Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity \(2 K\) while that in the middle has thermal conductivity \(K\). The left end of the combination is maintained at temperature \(3 T\) and the right end at \(T\). The rods are thermally insulated from outside. In steady state, temperature at the left junction is \(T_1\) and that at the right junction is \(T_2\). The ratio \(T_1 / T_2\) is
- A \(\frac{3}{2}\)
- B \(\frac{4}{3}\)
- C \(\frac{5}{3}\)
- D \(\frac{5}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{5}{3}\)
Step-by-step Solution
Detailed explanation
\(\text {In steady state } \Rightarrow\left(\frac{dQ }{dt }\right)_1=\left(\frac{dQ }{dt }\right)_2=\left(\frac{dQ }{dt }\right)_3 \)
\( \frac{2 KA \left(3 T- T _1\right)}{L }=\frac{KA \left( T _1- T _2\right)}{L }=\frac{2 KA \left( T _2- T \right)}{L } \)
\( 6 T-2 T_1= T _1- T _2=2 T_2-2 T \)
\( 6 T-2 T_1= T _1- T _2 \)
\( 6 T+ T _2=3 T_1 \ldots\text{(i)} \)
\( T _1- T _2=2 T_2-2 T \)
\( 2 T-3 T_2=- T _1 \ldots\text{(ii)}\)
Multiply equation (2) By '3'
\(6 T-9 T_2=-3 T_{1} \ldots \text { (iii)}\)
Equation (1) - (3)
\(10 T_2=6 T_1\)
\(\frac{T_1}{T_2}=\frac{5}{3}\)
\( \frac{2 KA \left(3 T- T _1\right)}{L }=\frac{KA \left( T _1- T _2\right)}{L }=\frac{2 KA \left( T _2- T \right)}{L } \)
\( 6 T-2 T_1= T _1- T _2=2 T_2-2 T \)
\( 6 T-2 T_1= T _1- T _2 \)
\( 6 T+ T _2=3 T_1 \ldots\text{(i)} \)
\( T _1- T _2=2 T_2-2 T \)
\( 2 T-3 T_2=- T _1 \ldots\text{(ii)}\)
Multiply equation (2) By '3'
\(6 T-9 T_2=-3 T_{1} \ldots \text { (iii)}\)
Equation (1) - (3)
\(10 T_2=6 T_1\)
\(\frac{T_1}{T_2}=\frac{5}{3}\)
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