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_1K_4=K_2K_3`
C. `K_1K_2=K_3K_4`
D. `(K_1)/(K_4)=(K_2)/(K_3)`

Answer

B

Explanation

For no current flow between C and D –
`(Q/t)_(AC)=(Q/t)_(CB)`
`implies (K_1 A (T_A-T_C))/(l)=(K_2A(T_C-T_B))/(l)`
`(T_A-T_C)/(T_C-T_B)=(K_2)/(K_1)` …(i)
Also, `(Q/t)_(AD)=(Q/t)_(DB)`
`(K_3A(T_(A)-T_(D))/(l))=(K_4A(T_D - T_B)/(l))`
`implies (T_A-T_D)/(T_D-T_B)=(K_4)/(K_3)` ....(ii)
given, `T_C=T_D`
By (i) and (ii) - `(K_2)/(K_1)=(K_4)/(K_3) implies K_1 K_4 = K_2 K_3`
Note : This is balance Wheatstone bridge, so product of opposite arm is equal i.e. `K_1K_4=K_2K_3`