The ratio of the diameters of two metallic rods of the same material is `2 : 1` and their lengths are in the ratio `1 : 4`. If the temperature difference between their ends are equal, the rate of flow of heat in them will be in the ratio

Options

A. `2 : 1`
B. `4 : 1`
C. ` 8 : 1`
D. `16 : 1`

Answer

To find the ratio of the rate of flow of heat (H1 : H2) in two metallic rods with given dimensions and properties, we can follow these steps:

### Step 1: Understand the given ratios
- The diameter ratio of the two rods is given as \( D_1 : D_2 = 2 : 1 \).
- The length ratio of the two rods is given as \( L_1 : L_2 = 1 : 4 \).
- The materials of the rods are the same, so their thermal conductivity \( K_1 = K_2 \).
- The temperature difference across both rods is the same, \( \Delta T_1 = \Delta T_2 \).

### Step 2: Write the formula for heat flow
The rate of flow of heat (H) through a rod can be expressed by the formula:
\[ H = \frac{\Delta T}{R} \]
where \( R \) is the thermal resistance. The thermal resistance \( R \) can be defined as:
\[ R = \frac{L}{K \cdot A} \]
where:
- \( L \) is the length of the rod,
- \( K \) is the thermal conductivity,
- \( A \) is the cross-sectional area of the rod.

### Step 3: Express H1 and H2
For the two rods, we can express the rates of heat flow \( H_1 \) and \( H_2 \) as:
\[ H_1 = \frac{\Delta T_1}{R_1} = \frac{\Delta T_1 \cdot K_1 \cdot A_1}{L_1} \]
\[ H_2 = \frac{\Delta T_2}{R_2} = \frac{\Delta T_2 \cdot K_2 \cdot A_2}{L_2} \]

### Step 4: Set up the ratio H1 : H2
Taking the ratio of \( H_1 \) to \( H_2 \):
\[ \frac{H_1}{H_2} = \frac{\Delta T_1 \cdot K_1 \cdot A_1 \cdot L_2}{\Delta T_2 \cdot K_2 \cdot A_2 \cdot L_1} \]

### Step 5: Simplify the ratio
Since \( \Delta T_1 = \Delta T_2 \) and \( K_1 = K_2 \), these terms cancel out:
\[ \frac{H_1}{H_2} = \frac{A_1 \cdot L_2}{A_2 \cdot L_1} \]

### Step 6: Calculate the areas A1 and A2
The cross-sectional area \( A \) of a rod is given by:
\[ A = \frac{\pi D^2}{4} \]
Thus, the areas for the two rods are:
\[ A_1 = \frac{\pi (D_1)^2}{4} = \frac{\pi (2D)^2}{4} = \frac{4\pi D^2}{4} = \pi D^2 \]
\[ A_2 = \frac{\pi (D_2)^2}{4} = \frac{\pi (D)^2}{4} \]
So, the ratio of areas \( \frac{A_1}{A_2} = \frac{\pi D^2}{\frac{\pi D^2}{4}} = 4 \).

### Step 7: Substitute the values into the ratio
Now substituting \( A_1 \) and \( A_2 \) into the ratio:
\[ \frac{H_1}{H_2} = \frac{A_1 \cdot L_2}{A_2 \cdot L_1} = \frac{4 \cdot L_2}{L_1} \]
Given \( L_1 : L_2 = 1 : 4 \), we have \( L_2 = 4L_1 \):
\[ \frac{H_1}{H_2} = \frac{4 \cdot 4L_1}{L_1} = 16 \]

### Final Answer
Thus, the ratio of the rate of flow of heat in the two rods is:
\[ H_1 : H_2 = 16 : 1 \]