Two different metal rods of equal lengths and equal areas of cross-section have their ends kept at the same temperature `theta_(1)` and `theta_(2)`. if `K_(1)` and `K_(2)` are their thermal conductivities, `rho_(1)` and `rho_(2)` their densities and `S_(1)` and `S_(2)` their specific heats, then the rate of flow of heat in the two rods will be the same if:

Options

A. `(k_(1))/(k_(2))=(rho_(1)s_(1))/(rho_(1)s_(2))`
B. `(k_(1))/(k_(2))=(rho_(1)s_(2))/(rho_(2)s_(1))`
C. `(k_(1))/(k_(2))=(theta_(1))/(theta_(2))`
D. `k_(1)=k_(2)`

Explanation

To solve the problem of determining the condition under which the rate of heat flow through two different metal rods is the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**: We have two metal rods (Rod 1 and Rod 2) of equal lengths (L) and equal cross-sectional areas (A). The ends of both rods are maintained at the same temperatures, θ1 and θ2.

2. **Use the Heat Transfer Formula**: The rate of heat transfer (dq/dt) through a rod can be expressed using Fourier's law of heat conduction:
\[ \frac{dq}{dt} = -k \cdot A \cdot \frac{\Delta T}{L} \]
where:
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference (θ2 - θ1),
- \( L \) is the length of the rod.

3. **Write the Heat Transfer Equations for Both Rods**:
- For Rod 1:
\[ \frac{dq_1}{dt} = -k_1 \cdot A \cdot \frac{\theta_2 - \theta_1}{L} \]
- For Rod 2:
\[ \frac{dq_2}{dt} = -k_2 \cdot A \cdot \frac{\theta_2 - \theta_1}{L} \]

4. **Set the Heat Transfer Rates Equal**: Since we want the rate of heat flow in both rods to be the same, we set the two equations equal to each other:
\[ -k_1 \cdot A \cdot \frac{\theta_2 - \theta_1}{L} = -k_2 \cdot A \cdot \frac{\theta_2 - \theta_1}{L} \]

5. **Cancel Common Factors**: Since the area (A) and the temperature difference (θ2 - θ1) are the same for both rods, we can cancel these terms from both sides of the equation:
\[ k_1 = k_2 \]

6. **Conclusion**: The rate of flow of heat in the two rods will be the same if their thermal conductivities are equal:
\[ k_1 = k_2 \]