Two rods P and Q of equal lengths have thermal conductivities `K_(1) and K_(2)` and cross-sectional areas `A_(1) and A_(2)` respectively. If the temperature difference across the ends of each rod is the same, then the condition for which the rate of flow of heat through each of them will be equal, is

Options

A. `(A_(1))/(A_(2))=(K_(2))/(K_(1))`
B. `(A_(1))/(A_(2))=(K_(1))/(K_(2))`
C. `(A_(1))/(A_(2))=sqrt((K_(1))/(K_(2)))`
D. `(A_(1))/(A_(2))=((K_(2))/(K_(1)))^(2)`

Answer

To solve the problem, we need to find the condition under which the rate of heat flow through two rods P and Q will be equal, given their thermal conductivities and cross-sectional areas.

### Step-by-Step Solution:

1. **Understand the Heat Flow Formula**:
The rate of heat flow (heat current) through a rod can be expressed using the formula:
\[ I = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \]
where:
- \( I \) is the rate of heat flow,
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area,
- \( (T_1 - T_2) \) is the temperature difference across the rod,
- \( L \) is the length of the rod.

2. **Identify Given Information**:
For rods P and Q:
- Rod P: \( K_1 \) (thermal conductivity), \( A_1 \) (cross-sectional area)
- Rod Q: \( K_2 \) (thermal conductivity), \( A_2 \) (cross-sectional area)
- Both rods have the same length \( L \) and the same temperature difference \( (T_1 - T_2) \).

3. **Set Up the Equation for Each Rod**:
For rod P, the rate of heat flow is:
\[ I_P = \frac{K_1 \cdot A_1 \cdot (T_1 - T_2)}{L} \]
For rod Q, the rate of heat flow is:
\[ I_Q = \frac{K_2 \cdot A_2 \cdot (T_1 - T_2)}{L} \]

4. **Equate the Heat Flow Rates**:
Since we want the rates of heat flow to be equal, we set \( I_P = I_Q \):
\[ \frac{K_1 \cdot A_1 \cdot (T_1 - T_2)}{L} = \frac{K_2 \cdot A_2 \cdot (T_1 - T_2)}{L} \]

5. **Simplify the Equation**:
Since \( (T_1 - T_2) \) and \( L \) are the same for both rods, we can cancel them out:
\[ K_1 \cdot A_1 = K_2 \cdot A_2 \]

6. **Rearranging the Equation**:
Rearranging gives us the condition:
\[ \frac{A_1}{A_2} = \frac{K_2}{K_1} \]

### Final Condition:
The condition for the rate of flow of heat through each rod to be equal is:
\[ \frac{A_1}{A_2} = \frac{K_2}{K_1} \]