Two rods `A` and `B` are of equal lengths. Their ends of kept between the same temperature and their area of cross-section are `A_(1)` and `A_(2)` and thermal conductivities `K_(1)` and `K_(2)`. The rate of heat transmission in the two rods will be equal, if

Options

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

Explanation

To solve the problem, we need to determine the condition under which the rate of heat transmission through two rods A and B is equal. We will use the formula for heat conduction, which is derived from Fourier's law of heat conduction.

### Step-by-Step Solution:

1. **Understand the Heat Transfer Formula**:
The rate of heat transfer (I) through a rod can be expressed as:
\[ I = \frac{\Delta T}{R} \]
where \(\Delta T\) is the temperature difference across the rod and \(R\) is the thermal resistance.

2. **Define Thermal Resistance**:
The thermal resistance \(R\) for a rod can be given by:
\[ R = \frac{L}{K \cdot A} \]
where:
- \(L\) is the length of the rod,
- \(K\) is the thermal conductivity of the material,
- \(A\) is the cross-sectional area of the rod.

3. **Set Up the Equations for Both Rods**:
For rod A:
\[ R_1 = \frac{L}{K_1 \cdot A_1} \]
For rod B:
\[ R_2 = \frac{L}{K_2 \cdot A_2} \]

4. **Equate the Rates of Heat Transfer**:
Since the ends of both rods are kept at the same temperature, the rate of heat transfer through both rods is equal:
\[ I_A = I_B \]
This implies:
\[ \frac{\Delta T}{R_1} = \frac{\Delta T}{R_2} \]

5. **Cancel Out the Temperature Difference**:
Since \(\Delta T\) is the same for both rods, we can cancel it out:
\[ R_1 = R_2 \]

6. **Substituting the Resistance Values**:
Substituting the expressions for \(R_1\) and \(R_2\):
\[ \frac{L}{K_1 \cdot A_1} = \frac{L}{K_2 \cdot A_2} \]

7. **Cancel the Lengths**:
Since the lengths \(L\) are equal and non-zero, we can cancel \(L\) from both sides:
\[ \frac{1}{K_1 \cdot A_1} = \frac{1}{K_2 \cdot A_2} \]

8. **Cross Multiply**:
Cross multiplying gives us:
\[ K_1 \cdot A_1 = K_2 \cdot A_2 \]

### Conclusion:
The rate of heat transmission in the two rods will be equal if:
\[ K_1 \cdot A_1 = K_2 \cdot A_2 \]