Two metal rods $1$ and $2$ of same lengths have same temperature difference between their ends. Their thermal conductivities are $K_1$ and $K_2$ and cross sectional areas $A_1$ and $A_2$ , respectively. If the rate of heat conduction in $1$ is four times that in $2$, then
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Let $L$ be length of each rod.
Rate of heat flow in rod $1$ for the temperature difference $\Delta T$ is
${H_1} = \frac{{{K_1}{A_1}\Delta T}}{L}$
Rate of heat flow in rod $2$ for the same difference $\Delta T$ is
${H_2} = \frac{{{K_2}{A_2}\Delta T}}{L}$
As per question, ${H_1} = 4{H_2}$
$\frac{{{K_1}{A_1}\Delta T}}{L} = 4\frac{{{K_2}{A_2}\Delta T}}{L}\,\,;\,\,{K_1}{A_1} = 4{K_2}{A_2}$
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