Three identical heat conducting rods are connected in series as...

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

Options

C. $\frac{5}{3}$

\begin{aligned} & \text{\text{ In } steady state } \Rightarrow\left(\frac{\mathrm{dQ}}{\mathrm{dt}}\right)_1=\left(\frac{\mathrm{dQ}}{\mathrm{dt}}\right)_2=\left(\frac{\mathrm{dQ}}{\mathrm{dt}}\right)_3 \\ & \frac{2 \mathrm{KA}\left(3 \mathrm{ T}-\mathrm{T}_1\right)}{\mathrm{L}}=\frac{\mathrm{KA}\left(\mathrm{ T}_1-\mathrm{T}_2\right)}{\mathrm{L}}=\frac{2 \mathrm{KA}\left(\mathrm{ T}_2-\mathrm{T}\right)}{\mathrm{L}} \\ & 6 \mathrm{ T}-2 \mathrm{ T}_1=\mathrm{T}_1-\mathrm{T}_2=2 \mathrm{ T}_2-2 \mathrm{ T} \\ & 6 \mathrm{ T}-2 \mathrm{ T}_1=\mathrm{T}_1-\mathrm{T}_2 \\ & 6 \mathrm{ T}+\mathrm{T}_2=3 \mathrm{ T}_1 ....(i)\\ & \mathrm{ T}_1-\mathrm{T}_2=2 \mathrm{ T}_2-2 \mathrm{ T} \\ & 2 \mathrm{ T}-3 \mathrm{ T}_2=-\mathrm{T}_1.....(ii) \end{aligned}

Multiply equation (2) By ' 3 '

6 \mathrm{ T}-9 \mathrm{ T}_2=-3 \mathrm{ T}_1 ....(iii) Equation (1) - (3)

\begin{aligned} & 10 \mathrm{ T}_2=6 \mathrm{ T}_1 \\ & \frac{\mathrm{ T}_1}{\mathrm{ T}_2}=\frac{5}{3} \end{aligned}