Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1 = 300\,\text{K}$ and $T_2 = 100\,\text{K}$, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1$ and $K_2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200\,\text{K}$, then $K_1/K_2 =$ ________.

Step 1: Identify the given parameters.

The cylinders have equal length $L$, and the radius of the larger cylinder is twice that of the smaller cylinder.

$$L_1 = L_2 = L$$ $$r_2 = 2r_1$$ $$A_2 = 4A_1$$

Step 2: Calculate the temperature difference across each cylinder.

The temperatures of the heat baths are $T_1 = 300\,\text{K}$ and $T_2 = 100\,\text{K}$, and the junction temperature is $T_{\text{junction}} = 200\,\text{K}$.

$$\Delta T_1 = 300\,\text{K} - 200\,\text{K} = 100\,\text{K}$$ $$\Delta T_2 = 200\,\text{K} - 100\,\text{K} = 100\,\text{K}$$

Step 3: Apply the steady-state heat flow condition.

For two cylinders connected in series, the rate of heat flow through both must be equal in the steady state.

$$Q_1 = Q_2$$ $$\frac{K_1 A_1 \Delta T_1}{L} = \frac{K_2 A_2 \Delta T_2}{L}$$

Step 4: Solve for the ratio of thermal conductivities.

Substitute the known values and relations into the heat flow equation.

$$K_1 A_1 = K_2 A_2$$ $$K_1 A_1 = K_2 (4A_1)$$ $$\frac{K_1}{K_2} = 4$$

Answer: $4$