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 and

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Solution

Q.
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 $\frac{T_{1}}{T_{2}}$ is [2025]

1 $\frac{5}{3}$

2 $\frac{5}{4}$

3 $\frac{3}{2}$

4 $\frac{4}{3}$

Ans.
(1)

According to heat transfer, at junction $T_{1}$

$\frac{2 K A (3 T - T_{1})}{L} = \frac{K A (T_{1} - T_{2})}{L}$

Since all identical rods are connected in series,

So, A and L are same.

$∴$    $6 T - 2 T_{1} = T_{1} - T_{2}$

$6 T = 3 T_{1} - T_{2}$ ...(i)

At junction $T_{2},$

$\frac{K A (T_{1} - T_{2})}{L} = \frac{2 K A (T_{2} - T)}{L}$

or    $T_{1} - T_{2} = 2 T_{2} - 2 T$

or    $2 T = 3 T_{2} - T_{1}$ ...(ii)

From eqn. (i) and (ii),

$\frac{T_{1}}{T_{2}} = \frac{5}{3}$

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