Two identical rods with different thermal conductivities `k_(1)" & "k_(2)` and different temperatures are first placed along length and then along area, then the ratio of rates of heat flow in both cases is

Options

A. `(k_(1)+k_(2))/(k_(1)+k_(2))`
B. `(4k_(1)k_(2))/((k_(1)+k_(2))^(2))`
C. `((k_(1)+k_(2))^(2))/(4k_(1))`
D. `(4k_(1)k_(2))/((k_(1)+k_(2))^(2))`

Explanation

To solve the problem of finding the ratio of rates of heat flow through two identical rods with different thermal conductivities \( k_1 \) and \( k_2 \) when placed in series and in parallel, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup:**
- We have two identical rods with thermal conductivities \( k_1 \) and \( k_2 \).
- The rods are placed first in series (along the length) and then in parallel (along the area).

2. **Heat Flow in Series:**
- When the rods are placed in series, the total thermal resistance \( R_{\text{series}} \) is given by:
\[ R_{\text{series}} = R_1 + R_2 = \frac{L}{k_1 A} + \frac{L}{k_2 A} \]
- Simplifying this, we find:
\[ R_{\text{series}} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} \right) = \frac{L}{A} \cdot \frac{k_1 + k_2}{k_1 k_2} \]

3. **Calculating Heat Flow in Series:**
- The rate of heat flow \( Q_{\text{series}} \) through the rods in series can be expressed using Fourier's law:
\[ Q_{\text{series}} = \frac{\Delta T}{R_{\text{series}}} = \frac{\Delta T \cdot A \cdot k_1 k_2}{L (k_1 + k_2)} \]

4. **Heat Flow in Parallel:**
- When the rods are placed in parallel, the total thermal resistance \( R_{\text{parallel}} \) is given by:
\[ \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{k_1 A}{L} + \frac{k_2 A}{L} \]
- Simplifying this, we find:
\[ R_{\text{parallel}} = \frac{L}{A} \cdot \frac{1}{\frac{k_1 + k_2}{L}} = \frac{L}{A} \cdot \frac{1}{\frac{k_1 + k_2}{L}} = \frac{L}{A (k_1 + k_2)} \]

5. **Calculating Heat Flow in Parallel:**
- The rate of heat flow \( Q_{\text{parallel}} \) through the rods in parallel can be expressed as:
\[ Q_{\text{parallel}} = \frac{\Delta T}{R_{\text{parallel}}} = \frac{\Delta T \cdot A (k_1 + k_2)}{L} \]

6. **Finding the Ratio of Heat Flows:**
- Now, we can find the ratio of heat flow in series to heat flow in parallel:
\[ \frac{Q_{\text{series}}}{Q_{\text{parallel}}} = \frac{\frac{\Delta T \cdot A \cdot k_1 k_2}{L (k_1 + k_2)}}{\frac{\Delta T \cdot A (k_1 + k_2)}{L}} = \frac{k_1 k_2}{(k_1 + k_2)^2} \]

### Final Result:
The ratio of rates of heat flow in both cases is:
\[ \frac{Q_{\text{series}}}{Q_{\text{parallel}}} = \frac{k_1 k_2}{(k_1 + k_2)^2} \]