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The three rods shown in figure have identical dimensions. Heat flows from the hot end at a rate of $40 \,W$ in the arrangement $(a)$. Find the rates of heat flow when the rods are joined as in arrangement $(b)$ is ......... $W$ (Assume $K_al=200 \,W / m ^{\circ} C$ and $\left.K_{c u}=400 \,W / m ^{\circ} C \right)$
$75$
$200$
$400$
$4$
Solution
(c)
$(a)$ $\because$ The rods have identical dimensions.
Let their area of crossection be $=A$ and length be $=L$
So each rod would have heat resistance of
$R=\frac{L}{K A}$
$R_{\text {eff }}=R_1+R_2+R_3$
$=\frac{L}{K_{A l} A}+\frac{L}{K_{C u} A}+\frac{L}{K_{A l} A}$
$R_{\text {eff }}=\frac{L}{A} \times \frac{5}{400}$ $\left[\because K_N=200, K_{C u}=400\right]$
$H_1=\frac{\Delta T}{R_{\text {eff }}}=\frac{100}{\frac{L}{A} \times \frac{5}{400}} \quad \ldots (1)$
$(b)$ When rods are connected in parallel
$\frac{1}{R_{\text {eff }}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$
$R_{\text {eff }}=\frac{L}{A} \times 800$
$H_2=\frac{100}{\frac{L}{A} \times 800} \quad \ldots (2)$
$(2)$ by $(1)$
$\frac{40}{H_2}=\frac{400}{800 \times 5} \Rightarrow H_2=400\,W$
