The lengths and radii of two rods made of same material are in the ratios 1 : 2 and 2 : 3 respectively. If the temperature difference between the ends for the two rods be the same, then in the steady state, the amount of heat flowing per second through them will be in the ratio

Options

A. `1 : 3`
B. `4 : 3`
C. `8 : 9`
D. `3 : 2`

Answer

To solve the problem, we need to determine the ratio of the amount of heat flowing per second through two rods made of the same material, given their lengths and radii in specific ratios.

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- Lengths of the rods are in the ratio \( L_1 : L_2 = 1 : 2 \).
- Radii of the rods are in the ratio \( R_1 : R_2 = 2 : 3 \).

2. **Understand the Formula for Heat Transfer**:
The amount of heat flowing per second (Q/t) through a rod is given by:
\[ \frac{Q}{t} = \frac{K \cdot A \cdot (\theta_1 - \theta_2)}{L} \]
Where:
- \( K \) = thermal conductivity (same for both rods since they are made of the same material)
- \( A \) = cross-sectional area of the rod
- \( \theta_1 - \theta_2 \) = temperature difference (same for both rods)
- \( L \) = length of the rod

3. **Calculate the Cross-Sectional Area**:
The cross-sectional area \( A \) of a rod with radius \( R \) is given by:
\[ A = \pi R^2 \]
Therefore, for the two rods:
- \( A_1 = \pi R_1^2 \)
- \( A_2 = \pi R_2^2 \)

4. **Express Heat Flow for Each Rod**:
For rod 1:
\[ \frac{Q_1}{t} = \frac{K \cdot \pi R_1^2 \cdot (\theta_1 - \theta_2)}{L_1} \]
For rod 2:
\[ \frac{Q_2}{t} = \frac{K \cdot \pi R_2^2 \cdot (\theta_1 - \theta_2)}{L_2} \]

5. **Set Up the Ratio of Heat Flow**:
Since \( K \), \( \pi \), and \( (\theta_1 - \theta_2) \) are the same for both rods, we can simplify the ratio:
\[ \frac{Q_1/t}{Q_2/t} = \frac{R_1^2 / L_1}{R_2^2 / L_2} \]

6. **Substitute the Ratios**:
- From the given ratios, we have:
- \( L_1 = 1x \) and \( L_2 = 2x \) (where \( x \) is a common length unit)
- \( R_1 = 2y \) and \( R_2 = 3y \) (where \( y \) is a common radius unit)

Now substituting these values:
\[ \frac{Q_1/t}{Q_2/t} = \frac{(2y)^2 / (1x)}{(3y)^2 / (2x)} = \frac{4y^2 / x}{9y^2 / 2x} \]

7. **Simplify the Ratio**:
\[ = \frac{4y^2 \cdot 2x}{9y^2 \cdot x} = \frac{8}{9} \]

8. **Final Result**:
The ratio of the amount of heat flowing per second through the two rods is:
\[ \frac{Q_1}{Q_2} = \frac{8}{9} \]

### Conclusion:
The amount of heat flowing per second through the two rods will be in the ratio \( 8 : 9 \).