Two different metal rods of the same length have their ends kept at the same temperature `theta_(1)`, and `theta_(2)`with `theta_(2) gt theta_(1)`. If `A_(1)` and `A_(2)` are their cross-sectional areas and `K_(1)` and `K_(2)` their thermal conductivities, the rate of flow of heat in the two rods will be the same it:

Options

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

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 Heat Transfer Formula**:
The rate of heat transfer (dQ/dt) through a rod is given by the formula:
\[ \frac{dQ}{dt} = \frac{K \cdot A \cdot (T_2 - T_1)}{L} \]
where:
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area,
- \( T_2 - T_1 \) is the temperature difference across the rod,
- \( L \) is the length of the rod.

2. **Set Up the Equations for Both Rods**:
For the first rod (with thermal conductivity \( K_1 \) and area \( A_1 \)):
\[ \frac{dQ}{dt} = \frac{K_1 \cdot A_1 \cdot (T_2 - T_1)}{L} \]
For the second rod (with thermal conductivity \( K_2 \) and area \( A_2 \)):
\[ \frac{dQ}{dt} = \frac{K_2 \cdot A_2 \cdot (T_2 - T_1)}{L} \]

3. **Equate the Heat Transfer Rates**:
Since the problem states that the rate of heat flow in both rods is the same, we can set the two equations equal to each other:
\[ \frac{K_1 \cdot A_1 \cdot (T_2 - T_1)}{L} = \frac{K_2 \cdot A_2 \cdot (T_2 - T_1)}{L} \]

4. **Cancel Common Terms**:
The temperature difference \( (T_2 - T_1) \) and the length \( L \) are common in both equations, so we can cancel them out:
\[ K_1 \cdot A_1 = K_2 \cdot A_2 \]

5. **Rearrange the Equation**:
Rearranging the equation gives us:
\[ \frac{A_1}{A_2} = \frac{K_2}{K_1} \]

### Final Result:
Thus, the condition for the rate of heat flow in the two rods to be the same is:
\[ \frac{A_1}{A_2} = \frac{K_2}{K_1} \]