A cylinder of radius `R` made of a material of thermal conductivity `K_(1)` is surrounded by cylindrical shell of inner radius `R` and outer radius `2R` made of a material of thermal con-ductivity `K_(2)` The two ends of the combined system are maintained at two differnet tem-peratures There is no loss of heat across the cylindrical surface and system is in steady state What is the effective thermal conductivity of the system
.

Explanation

Two cylinders are in parallel, therefore equivalent thermal resistance R is given by
`1/R=1/(R_(1))+1/(R_(2))`
But `R=l/(KA)`
`:.(kA)/l=(k_(1)A_(1))/(l_(1))+(k_(2)A_(2))/(l_(2))`
Hre `l_(1)=l_(2)=l` and `A_(1)=piR^(2)`
`A_(2)=pi(2R)^(2)-piR^(2)=3piR^(2)` and `a=pi(2R)^(2)=4piR^(2)`
`:. (k4piR^(2))/l=(k_(1)piR^(2))/l+(k_(2)3piR^(2))/l`
i.e. `k=(k_(1)+3k_(2))/4`