Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures `T_(1) = 300K and T_(2) = 100 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 K, then ` K_(1)//K_(2)=`__________.

Explanation

To solve the problem, we need to analyze the heat conduction through the two cylinders connected in series. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have two conducting cylinders of equal length \( L \) but different radii. The smaller cylinder has a radius \( r_1 \), and the larger cylinder has a radius \( r_2 = 2r_1 \). The temperatures at the ends of the cylinders are \( T_1 = 300 \, K \) and \( T_2 = 100 \, K \). The temperature at the junction of the two cylinders is \( T_j = 200 \, K \).

### Step 2: Use the Heat Conduction Equation
The rate of heat flow through a cylindrical conductor can be given by the equation:
\[ Q = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \]
where \( Q \) is the heat flow, \( K \) is the thermal conductivity, \( A \) is the cross-sectional area, \( T_1 \) and \( T_2 \) are the temperatures at the ends, and \( L \) is the length of the cylinder.

### Step 3: Calculate the Cross-Sectional Areas
The cross-sectional area \( A \) of a cylinder is given by:
\[ A = \pi r^2 \]
For the smaller cylinder:
\[ A_1 = \pi r_1^2 \]
For the larger cylinder:
\[ A_2 = \pi (2r_1)^2 = 4\pi r_1^2 \]

### Step 4: Set Up the Heat Flow Equations
Since the cylinders are in series, the heat flow through both cylinders must be equal:
\[ \frac{K_1 \cdot A_1 \cdot (T_1 - T_j)}{L} = \frac{K_2 \cdot A_2 \cdot (T_j - T_2)}{L} \]
Cancelling \( L \) from both sides gives:
\[ K_1 \cdot A_1 \cdot (T_1 - T_j) = K_2 \cdot A_2 \cdot (T_j - T_2) \]

### Step 5: Substitute the Areas and Temperatures
Substituting \( A_1 \) and \( A_2 \):
\[ K_1 \cdot (\pi r_1^2) \cdot (300 - 200) = K_2 \cdot (4\pi r_1^2) \cdot (200 - 100) \]
This simplifies to:
\[ K_1 \cdot \pi r_1^2 \cdot 100 = K_2 \cdot 4\pi r_1^2 \cdot 100 \]
Cancelling \( \pi r_1^2 \cdot 100 \) from both sides gives:
\[ K_1 = 4K_2 \]

### Step 6: Find the Ratio \( \frac{K_1}{K_2} \)
From the equation \( K_1 = 4K_2 \), we can express the ratio:
\[ \frac{K_1}{K_2} = 4 \]

### Final Answer
Thus, the final answer is:
\[ \frac{K_1}{K_2} = 4 \]