A cylinder A of a material of thermal conductivity `K_(1)` is surrounded by a cylindrical shell of inner radius equal to the cylinder. The shell is made of a material of thermal conductivity `K_(2)`. The two ends of the combined system are maintained at two different temperature. if the effective thermal conductivity of the system is `(4K_(1) + 5K_(2))/(9)`, find the rate of outer radius of the shell to the radius of the cylinder.

Options

A. `2.0`
B. `5//2`
C. `5//3`
D. `3//2`

Answer

D

Explanation

Radius of cylinder = `R` and outer radius of shell =`nR`
`R_(eq) = (R_(1)R_(2))/(R_(1)+R_(2)) =((1)/(K_(1)piR^(2))xx(1)/(K_(2)(n^(2)-1)piR^(2)))/((1)/(K_(1)piR^(2))+(1)/(K_(2)(n^(2)-1)piR^(2)))`
But `R_(eq) = (1)/(K pi n^(2)R^(2))`
on solving , `K = ([K_(1)+K_(2)(n^(2)-1)])/(n^2)`
`=(4K_(1)+5K_(2))/(9)` (given)
`=(K_(1)+5K_(2)//4)/(9//4)`
`n=3//2`.