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: Treat each rod like a resistor
Heat flow behaves like current. A rod of length $L$, area $A$ and conductivity $k$ has thermal resistance $R_{th}=\frac{L}{kA}$. The temperature difference plays the role of voltage.
Step 2: Add the resistances in series
Let $R=\frac{L}{KA}$. The outer rods have $\frac{R}{2}$ each and the middle rod has $R$. Total resistance is $\frac{R}{2}+R+\frac{R}{2}=2R$.
Step 3: Find the heat current
The total temperature drop is $3T-T=2T$. So the heat current is $H=\frac{2T}{2R}=\frac{T}{R}$. The same $H$ flows through every rod because there is no side loss.
Step 4: Find the junction temperatures
Drop across the left rod: $H\cdot\frac{R}{2}=\frac{T}{2}$. So $T_1=3T-\frac{T}{2}=\frac{5T}{2}$. Drop across the middle rod: $H\cdot R=T$. So $T_2=T_1-T=\frac{3T}{2}$. Check with the right rod: $T_2-\frac{T}{2}=T$, which matches the cold end.
Step 5: Compare with the options
$\frac{T_1}{T_2}=\frac{5}{3}$. The values $\frac{3}{2}$, $\frac{4}{3}$ and $\frac{4}{5}$ do not match these temperatures, so they are rejected.
Final Answer:
Using the thermal resistance method, the ratio is 5/3.
\[ \boxed{\dfrac{5}{3}\ \text{(C)}} \]
