Three identical heat conducting rods are connected in series as shown in the figure. The rods on the side have thermal conductivity \(2K\) while that in the middle has thermal conductivity \(K\). The left end of the combination is maintained at temperature \(3T\) 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 junctions is \(T_2\). The ratio \(T_1/T_2\) is
Show Hint
Heat current is the same in all three rods because there is no side loss. Write k A (temperature difference) / L for each rod and equate them.
Step 1: Understand the figure
Three identical rods are joined end to end. The two outer rods have conductivity \(2K\) and the middle rod has \(K\). The left end is held at \(3T\) and the right end at \(T\). Let each rod have length \(L\) and area \(A\). \(T_1\) is the temperature at the left junction and \(T_2\) at the right junction.
Step 2: Use the steady state condition
The rods are insulated from the sides, so no heat leaks out. In steady state the same heat current flows through every rod. For one rod the heat current is \(\frac{kA\,\Delta T}{L}\).
Step 3: Write the equal heat flow equation
\[ \frac{2KA(3T-T_1)}{L}=\frac{KA(T_1-T_2)}{L}=\frac{2KA(T_2-T)}{L} \] Cancel \(\frac{KA}{L}\) from all three parts: \[ 2(3T-T_1)=T_1-T_2=2(T_2-T) \]
Step 4: Solve for T1 and T2
From the first and third parts, \(3T-T_1=T_2-T\), so \(T_1+T_2=4T\). From the middle and third parts, \(T_1-T_2=2T_2-2T\), so \(T_1=3T_2-2T\). Put this in the first result: \(3T_2-2T+T_2=4T\), so \(T_2=\frac{3T}{2}\). Then \(T_1=4T-\frac{3T}{2}=\frac{5T}{2}\).
Step 5: Find the ratio and check the options
\[ \frac{T_1}{T_2}=\frac{5T/2}{3T/2}=\frac{5}{3} \] Option (A) 3/2, option (B) 4/3 and option (D) 4/5 do not satisfy the two equations. For example 4/5 would make \(T_1\) smaller than \(T_2\), but heat flows from the hot left end, so \(T_1\) must be larger than \(T_2\).
Final Answer:
The ratio \(T_1/T_2\) is 5/3, which is option (C).
\[ \boxed{\dfrac{5}{3}\ \text{(C)}} \]
