Five rods of same dimensions are arranged as shown in the figure. They have thermal conductivities K₁, K₂, K₃, K₄ and K₅. When points A and B are maintained at different temperatures, no heat flows through the central rod if

For no current flow between C and D

${\left(\frac{\mathrm{Q}}{\mathrm{t}}\right)}_{\mathrm{AC}}={\left(\frac{\mathrm{Q}}{\mathrm{t}}\right)}_{\mathrm{CB}}$

$\Rightarrow \frac{{\mathrm{K}}_1\mathrm{A}\left({\mathrm{\theta }}_{\mathrm{A}}-{\mathrm{\theta }}_{\mathrm{C}}\right)}{\mathrm{l}}=\frac{{\mathrm{K}}_2\mathrm{A}\left({\mathrm{\theta }}_{\mathrm{C}}-{\mathrm{\theta }}_{\mathrm{B}}\right)}{\mathrm{l}}$

$\Rightarrow \frac{{\mathrm{\theta }}_{\mathrm{A}}-{\mathrm{\theta }}_{\mathrm{C}}}{{\mathrm{\theta }}_{\mathrm{C}}-{\mathrm{\theta }}_{\mathrm{B}}}=\frac{{\mathrm{K}}_2}{{\mathrm{K}}_1}....\left(\mathrm{i}\right)$

also ${\left(\frac{\mathrm{Q}}{\mathrm{t}}\right)}_{\mathrm{AD}}={\left(\frac{\mathrm{Q}}{\mathrm{t}}\right)}_{\mathrm{DB}}$

$\Rightarrow \frac{{\mathrm{K}}_3\mathrm{A}\left({\mathrm{\theta }}_{\mathrm{A}}-{\mathrm{\theta }}_{\mathrm{D}}\right)}{\mathrm{l}}=\frac{{\mathrm{K}}_4\mathrm{A}\left({\mathrm{\theta }}_{\mathrm{D}}-{\mathrm{\theta }}_{\mathrm{B}}\right)}{\mathrm{l}}$
$\Rightarrow \frac{{\mathrm{\theta }}_{\mathrm{A}}-{\mathrm{\theta }}_{\mathrm{D}}}{{\mathrm{\theta }}_{\mathrm{D}}-{\mathrm{\theta }}_{\mathrm{B}}}=\frac{{\mathrm{K}}_4}{{\mathrm{K}}_3}....\left(\mathrm{ii}\right)$
It is given that ${\mathrm{\theta }}_{\mathrm{C}}={\mathrm{\theta }}_{\mathrm{D}}$, hence from equation (i) and (ii) we get $\frac{{\mathrm{K}}_2}{{\mathrm{K}}_1}=\frac{{\mathrm{K}}_4}{{\mathrm{K}}_3}\Rightarrow {\mathrm{K}}_1{\mathrm{K}}_4={\mathrm{K}}_2{\mathrm{K}}_3$