Three rods A, B and C of thermal conductiviti | vedclass

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Question

Let $R_{A}, R_{B}$ and $R_{C}$ be the thermal resistance of rods

$A,$ $\mathrm{B}$ and $\mathrm{C}$ respectively. Then

$\mathrm{R}_{\mathrm{A}}=

Vedclass · Accepted Answer

$20$

Vedclass · Answer

Let $R_{A}, R_{B}$ and $R_{C}$ be the thermal resistance of rods

$A,$ $\mathrm{B}$ and $\mathrm{C}$ respectively. Then

$\mathrm{R}_{\mathrm{A}}=\frac{2 l}{\mathrm{KA}} ; \mathrm{R}_{\mathrm{B}}=\frac{l}{2 \mathrm{K} \cdot 2 \mathrm{A}}=\frac{l}{4 \mathrm{KA}}$

and $\mathrm{R}_{\mathrm{C}}=\frac{l}{4 \mathrm{K} \cdot 2 \mathrm{A}}=\frac{l}{8 \mathrm{KA}}$

since thermal current

$\mathrm{H}=\frac{\mathrm{Q}}{\mathrm{t}}=\frac{	heta_{1}-	heta_{2}}{l / \mathrm{KA}}=\frac{	heta_{1}-	heta_{2}}{\mathrm{R}}$

or $\quad \mathrm{RH}=	heta_{1}-	heta_{2}$

$	herefore \quad 100-	heta=\mathrm{R}_{\mathrm{A}} \mathrm{H}_{\mathrm{A}}$     $...(i)$

$	heta-50=\mathrm{R}_{\mathrm{B}} \mathrm{H}_{\mathrm{B}}$        $...(ii)$

$	heta-0=\mathrm{R}_{\mathrm{c}} \mathrm{H}_{\mathrm{c}}=\mathrm{R}_{\mathrm{C}}\left(\mathrm{H}_{\mathrm{A}}-\mathrm{H}_{\mathrm{B}}ight)$    $...(iii)$

On substituting values of $\mathrm{R}_{\mathrm{A}}, \mathrm{R}_{\mathrm{B}}$ and $\mathrm{R}_{\mathrm{C}},$ we get

$100-	heta=\frac{2 l}{\mathrm{KA}} \cdot \mathrm{H}_{\mathrm{A}} \Rightarrow \mathrm{H}_{\mathrm{A}}=\frac{\mathrm{K} \mathrm{A}}{2 l}(100-	heta)$

$	heta-50=\frac{l}{4 \mathrm{KA}} \mathrm{H}_{\mathrm{B}} \Rightarrow \mathrm{H}_{\mathrm{B}}=\frac{4 \mathrm{KA}}{l}(	heta-50)$

$	heta=\frac{l}{8 \mathrm{KA}}\left(\mathrm{H}_{\mathrm{A}}-\mathrm{H}_{\mathrm{B}}ight)$

$=\frac{1}{8 \mathrm{KA}}\left[\frac{\mathrm{KA}}{2 l}(100-	heta)-\frac{4 \mathrm{KA}}{l}(	heta-50)ight]$

$	heta=20^{\circ} \mathrm{C}$

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