The ratio of the thermal conductivities of two different materials is `1 : 2`. The thermal resistance of the rods of these materials having the same thickness are equal. Find the ratio of the length of rods.

Explanation

To solve the problem step by step, we will use the relationship between thermal resistance, thermal conductivity, and the dimensions of the rods.

### Step 1: Understand the Given Information
We are given:
- The ratio of thermal conductivities \( k_1 : k_2 = 1 : 2 \).
- The thermal resistances of the two rods are equal.
- The thickness of both rods is the same.

### Step 2: Define Thermal Resistance
The thermal resistance \( R_{th} \) for a rod can be expressed as:
\[ R_{th} = \frac{L}{k \cdot A} \]
where:
- \( L \) is the length of the rod,
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area.

### Step 3: Write the Thermal Resistance for Both Rods
Let:
- \( L_1 \) be the length of rod 1,
- \( L_2 \) be the length of rod 2,
- \( k_1 \) be the thermal conductivity of rod 1,
- \( k_2 \) be the thermal conductivity of rod 2.

From the given ratio, we can express:
- \( k_1 = k \)
- \( k_2 = 2k \)

Now, we can write the thermal resistances for both rods:
\[ R_1 = \frac{L_1}{k_1 \cdot A} = \frac{L_1}{k \cdot A} \]
\[ R_2 = \frac{L_2}{k_2 \cdot A} = \frac{L_2}{2k \cdot A} \]

### Step 4: Set the Thermal Resistances Equal
Since the thermal resistances are equal, we can set \( R_1 = R_2 \):
\[ \frac{L_1}{k \cdot A} = \frac{L_2}{2k \cdot A} \]

### Step 5: Simplify the Equation
We can cancel \( A \) and \( k \) from both sides (since they are non-zero):
\[ L_1 = \frac{L_2}{2} \]

### Step 6: Find the Ratio of Lengths
Rearranging the equation gives us:
\[ \frac{L_1}{L_2} = \frac{1}{2} \]

Thus, the ratio of the lengths of the rods is:
\[ L_1 : L_2 = 1 : 2 \]

### Final Answer
The ratio of the lengths of the rods is \( 1 : 2 \).

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