Five rods of same dimensions are arranged as shown in the figure. They have thermal conductivities `K _(1) , K_(2) , K _(3) , K_( 4)` and `K_( 5)` . When points A and B are maintained at different temperatures, no heat flows through the central rod if

Options

A. `K_(1) = K_(4) and K_(2) = K_(3)`
B. `K_(1)K_(4) = K_(2)K_(3)`
C. `K_(1)K_(2) = K_(3)K_(4)`
D. None of the above

Answer

B

Explanation

For no current flow between C and D
`(Q/t)_(AC)=(Q/t)_(CB)`
`rArr ((K_(1)A(theta_(A)-theta_(C)))/l=(K_(2)A(theta_(C)-theta_(B)))/l`
`(theta_(A)-theta_(C))/l = K_(2)/K_(1)` ..................(i)
Also `(Q/t)_(AD) = (Q/t)_(DB)`
`rArr (K_(3)A(theta_(A)-theta_(D))/l = K_(4)A(theta_(D)-theta_(B))/l`
`(theta_(A)-theta_(D))/(theta_(D)-theta_(B)) = K_(4)/K_(3)`.................(ii)
It is given that `theta_(C)=theta_(D)`, hence from Eqs (i) and (ii),
We get `(K_(2)/K_(1)) = (K_(4)/K_(3) rArr K_(1)K_(4)=K_(2)K_(3))`