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(A) For no heat to flow through the central rod (connecting $C$ and $D$),the temperatures at $C$ and $D$ must be equal,i.e.,$	heta_C = 	heta_D$.For

Vedclass · Accepted Answer

$k_1 k_4 = k_2 k_3$

Vedclass · Answer

(A) For no heat to flow through the central rod (connecting $C$ and $D$),the temperatures at $C$ and $D$ must be equal,i.e.,$	heta_C = 	heta_D$.For the upper path $ACB$,the rate of heat flow is given by:$\left( \frac{Q}{t} ight)_{AC} = \left( \frac{Q}{t} ight)_{CB} \implies \frac{k_1 A (	heta_A - 	heta_C)}{\ell} = \frac{k_2 A (	heta_C - 	heta_B)}{\ell}$$\implies \frac{	heta_A - 	heta_C}{	heta_C - 	heta_B} = \frac{k_2}{k_1} \quad \dots(i)$For the lower path $ADB$,the rate of heat flow is given by:$\left( \frac{Q}{t} ight)_{AD} = \left( \frac{Q}{t} ight)_{DB} \implies \frac{k_3 A (	heta_A - 	heta_D)}{\ell} = \frac{k_4 A (	heta_D - 	heta_B)}{\ell}$$\implies \frac{	heta_A - 	heta_D}{	heta_D - 	heta_B} = \frac{k_4}{k_3} \quad \dots(ii)$Since $	heta_C = 	heta_D$,the left-hand sides of equations $(i)$ and $(ii)$ are equal.Therefore,$\frac{k_2}{k_1} = \frac{k_4}{k_3} \implies k_1 k_4 = k_2 k_3$.

Five rods of identical dimensions are arranged as shown in the figure. Their thermal conductivities are $k_1, k_2, k_5, k_4$ and $k_3$. When $A$ and $B$ are maintained at different temperatures,no heat flows through the central rod if:

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