Three identical rods A,B and C are placed end to end. A temperature difference is maintained between the free ends of A and C. The thermal conductivity of B is `thrice` that of C and `half` of that of A. The effective thermal conductivity of the system is ........... (`K_A` is the thermal conductivity of rod A).

Options

A. `(1)/(3)K_(A)`
B. `3K_(A)`
C. `2K_(A)`
D. `(2)/(3)K_(A)`

Answer

A

Explanation

Let L be length and A be area of cross-section of each rod respectively.
Given: `K_(B)=3K_(C),K_(B)=(1)/(2)K_(A)`
Thermal resistance of rod A,`R_(A)=(L)/(K_(A)A)`
Thermal resistance of rod B,`R_(B)=(L)/(K_(B)A)`
Thermal resistance of rod C,`R_(C)=(L)/(K_(C)A)`
The electrical equivalent circuit is as shown in the figure.

Here,the rods are connected in series
`therefore R_(S)=R_(A)+R_(B)+R_(C)`
Let `K_(eff)` effective thermal conductivity of the system.
`therefore (3L)/(K_(eff)A)=(1)/(K_(A)A)+(L)/(K_(B)A)+(L)/(K_(C)A)`
`(3)/(K_(eff)A)=(1)/(K_(A))+(1)/((K_(A))/(2))+(1)/((K_(B))/(3))=(1)/(K_(A))+(1)/(K_(A)/(2))+(1)/(K_(A)/(6))`
`(3)/(K_(eff)A)=(1)/(K_(A))[1+2+6]`
`(3)/(K_(eff))=(9)/(K_(A))impliesK_(eff)=(3)/(9)K_(A)=(1)/(3)K_(A)`