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)=k_(2)k_(3)`
B. `k_(1)=k_(4) and k_(2)=k_(3)`
C. `(k_(1))/(k_(4))=(k_(2))/(k_(3))`
D. `k_(1)k_(2)=k_(3)k_(4)`

Answer

A

Explanation

Let x and A be the length and cross-sectional area of each rod.
`therefore` Thermal resistance of the rod, AC,
`R_(AC)=(x)/(K_(1)A)`
Similarly,
`R_(BC)=(x)(K_(2)A),R_(AD)=(x)/(K_(3)A),R_(DB)=(x)/(K_(4)A)`
Now, there will be no heat flow through the central rod CD if C and D are at the same temperature. using electrical analogy for heat conduction, we get
`(R_(AC))/(R_(BC))=(R_(AD))/(R_(DB))` (Wheatstone bridge principle)
`therefore(x//K_(1)A)/(x//K_(2)A)=(x//K_(3)A)/(x//K_(4)A)implies(K_(2))/(K_(1))=(K_(4))/(K_(3))`
or, `K_(1)K_(4)=K_(2)K_(3)`.