Seven rods `A, B, C, D, E, F and G` are joined as shown in figure. All the rods have equal cross-sectional area `A` and length `l`. The thermal conductivities of the rods are `K_(A)=K_(C)=K_(0)`, `K_(B)=K_(D)=2K_(0)`, `K_(E)=3K_(0)`, `K_(F)=4K_(0)` and `K_(G)=5K_(0)` . The rod E is kept at a constant temperature `T_(1)` and the rod G is kept at a constant temperature `T_(2)(T_(2)gtT_(1))` . (a) Show that the rod F has a uniform temperature `T=(T_(1)+2T_(2))//3` . (b) Find the rate of heat flowing from the source which maintains the temperature `T_(2)` .

Answer

`(3K_(0)A(theta_(2)-theta_(1)))/(8l)`

Explanation

`(R_(A))/(R_(C)) = (k_(C))/(k_(A)) = (1)/(2)`
& `(R_(B))/(R_(D)) = (k_(D))/(k_(B)) = (1)/(2`
`:. (R_(A))/(R_(C)) = (R_(B))/(R_(D)) rArr` Balanced `W.S.B`.
`rArr ((theta_(2)-theta))/(R_(B)) = (theta_(2)-theta_(1))/(R_(A)+R_(B)) rArr theta =(3 theta_(1)+theta_(2))/(4)`
`rArr` Rate of heat flow from the source
`=((theta_(2)-theta_(1)))/((((R_(A)+R_(B))(R_(C)+R_(D)))/(R_(A)+R_(B+R_(C)+R_(D)))))`
`= ((theta_(2)-theta_(1)))/(((((1)/(k_(A))+(1)/(K_(B)))((1)/(k_(C))+(1)/(k_(D)))(l)/(A))/((1)/(k_(A))+(1)/(k_(B))+(1)/(k_(C))+(1)/(k_(D)))))`
`=(3k_(0)A(theta_(2)-theta_(1)))/(8l)`