Heat is flowing through two cylindrical rods of the same material. The diamters of the rods are in the ratio `1: 2` and the length in the ratio `2 : 1`. If the temperature difference between the ends is same then ratio of the rate of flow of heat through them will be

Options

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

Explanation

To solve the problem, we will use the formula for the rate of heat transfer through a cylindrical rod, which is given by:

\[ \frac{Q}{T} = \frac{K \cdot A \cdot \Delta \theta}{L} \]

Where:
- \( Q \) is the heat transferred,
- \( T \) is the time,
- \( K \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta \theta \) is the temperature difference,
- \( L \) is the length of the rod.

### Step 1: Understand the given ratios
We are given:
- The diameters of the rods are in the ratio \( D_1 : D_2 = 1 : 2 \).
- The lengths of the rods are in the ratio \( L_1 : L_2 = 2 : 1 \).

### Step 2: Express the areas in terms of diameter
The cross-sectional area \( A \) of a cylindrical rod is given by:

\[ A = \frac{\pi D^2}{4} \]

Thus, for the two rods:
- For rod 1: \( A_1 = \frac{\pi D_1^2}{4} \)
- For rod 2: \( A_2 = \frac{\pi D_2^2}{4} \)

### Step 3: Substitute the diameter ratio
From the diameter ratio \( D_1 : D_2 = 1 : 2 \), we can express \( D_1 \) and \( D_2 \) as:
- Let \( D_1 = 1 \) and \( D_2 = 2 \).

Now, substituting these values into the area formulas:
- \( A_1 = \frac{\pi (1)^2}{4} = \frac{\pi}{4} \)
- \( A_2 = \frac{\pi (2)^2}{4} = \frac{\pi \cdot 4}{4} = \pi \)

### Step 4: Substitute the lengths
From the length ratio \( L_1 : L_2 = 2 : 1 \), we can express \( L_1 \) and \( L_2 \) as:
- Let \( L_1 = 2 \) and \( L_2 = 1 \).

### Step 5: Write the heat transfer equations for both rods
Now, substituting the areas and lengths into the heat transfer formula:

For rod 1:
\[ \frac{Q_1}{T} = \frac{K \cdot A_1 \cdot \Delta \theta}{L_1} = \frac{K \cdot \frac{\pi}{4} \cdot \Delta \theta}{2} \]

For rod 2:
\[ \frac{Q_2}{T} = \frac{K \cdot A_2 \cdot \Delta \theta}{L_2} = \frac{K \cdot \pi \cdot \Delta \theta}{1} \]

### Step 6: Calculate the ratio of heat transfer rates
Now, we can find the ratio \( \frac{Q_1/T}{Q_2/T} \):

\[ \frac{Q_1/T}{Q_2/T} = \frac{\frac{K \cdot \frac{\pi}{4} \cdot \Delta \theta}{2}}{\frac{K \cdot \pi \cdot \Delta \theta}{1}} = \frac{\frac{\pi}{4} \cdot \Delta \theta}{2} \cdot \frac{1}{\pi \cdot \Delta \theta} = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

### Final Ratio
Thus, the ratio of the rate of flow of heat through the two rods is:

\[ \frac{Q_1}{Q_2} = \frac{1}{8} \]

### Answer
The ratio of the rate of flow of heat through the two rods is \( 1 : 8 \).

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