Two ends of a rod of uniform cross sectional area are kept at temperature `3T_(0)` and `T_(0)` as shown. Thermal conductivity of rod varies as `k=alphaT`, (where `alpha` is a constant and T is absolute temperature). In steady state, the temperature of the middle section of the rod is

Options

A. `sqrt(7)T_(0)`
B. `sqrt(5)T_(0)`
C. `2T_(0)`
D. `sqrt(3)T_(0)`

Answer

B

Explanation

Ins steady state
`(DeltaQ)/(Deltat)=-kA(dT)/(dx)`
`(DeltaQ)/(Deltat)=-alphaTA(dT)/(dx)`
`(DeltaQ)/(Deltat)underset(0)overset(l)intdx=-alphaAint_(3T_(0))^(T_(0))TdT`
`(DeltaQ)/(Deltat)l=4alphaAT_(0)^(2)` ...(i)
Similarly,
`(DeltaQ)/(Deltat)underset(0)overset(l//2)intdx=-alphaAunderset(3T_(0))overset(T)intTdT`
`(DeltaQ)/(Deltat)(l)/(2)=alphaA((9T_(0)^(2)-T^(2)))/(2)` ...(ii)
Dividing (ii) by (i)
`9T_(0)^(2)-T^(2)=4T_(0)^(2)`
`T^(2)=5T_(0)^(2)`
`T^(2)=5T_(0)^(2)`
`T=sqrt(5)T_(0)`.