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Step-by-Step Solution
Step 1: Identify the formula for the rate of heat conduction
The rate of heat conduction (also called heat current) through a rod of length $L$, cross-sectional area $A$, and thermal conductivity $K$ with a temperature difference $\Delta T$ between its ends is given by:
$H = \dfrac{K \, A \, \Delta T}{L}.$
Step 2: Write the expressions for the two rods
Let rod 1 have thermal conductivity $K_1$ and cross-sectional area $A_1$. Let rod 2 have thermal conductivity $K_2$ and cross-sectional area $A_2$. Both rods have the same length $L$ and maintain the same temperature difference $\Delta T$ between their ends.
Thus, the heat current through rod 1 is:
$H_1 = \dfrac{K_1 \, A_1 \,\Delta T}{L},$
and the heat current through rod 2 is:
$H_2 = \dfrac{K_2 \, A_2 \,\Delta T}{L}.$
Step 3: Apply the given condition for the ratio of heat currents
It is given that the rate of heat conduction in rod 1 is four times that in rod 2. Mathematically, this can be written as:
$H_1 = 4 \, H_2.$
Step 4: Substitute the expressions and simplify
Using the expressions from Step 2 in the relation $H_1 = 4 \, H_2$:
$\dfrac{K_1 \, A_1 \,\Delta T}{L} = 4 \,\dfrac{K_2 \, A_2 \,\Delta T}{L}.$
Cancel the common factors $\dfrac{\Delta T}{L}$ on both sides:
$K_1 \, A_1 = 4 \, K_2 \, A_2.$
Step 5: State the conclusion
Thus, the condition for the rate of heat conduction to be four times in rod 1 compared to rod 2 is satisfied when:
$\boxed{K_1 A_1 = 4 K_2 A_2}.$
This corresponds to option (1) in the given choices.
