One end of a thermally insulated rod is kept at a temperature `T_(1)` and the other at `T_(2)`. The rod is composed of two sections of lengths `l_(1) and l_(2)` and thermal conductivities `K_(1) and K_(2)` respectively. The temperature at the interface of the two sections is

Options

A. `(K_(2)l_(2)T_(1) + K_(1)l_(1)T_(2))//(K_(1)l_(1) + K_(2)l_(2))`
B. `(K_(2)l_(1)T_(1) + K_(1)l_(2)T_(2))//(K_(2)l_(1) + K_(1)l_(2))`
C. `(K_(1)l_(2)T_(1) + K_(2)l_(1)T_(2))//(K_(1)l_(2) + K_(2)l_(1))`
D. `(K_(1)l_(1)T_(1) + K_(2)l_(2)T_(2))//(K_(1)l_(1) + K_(2)l_(2))`

Answer

C

Explanation

Let the temperature of the interface be `T_(0)`.
Thermal current =`("Temperature difference")/("Thermal resitance")`
Thermal resistance = `(1)/(K)(l)/(A)`
`implies (T_(1) - T_(0))/(l_(1)//K_(1)A) = (T_(0)- T_(2))/(l_(2) //K_(2)A)`
`implies T_(0) = (K_(1)l_(2)T_(1) - K_(2)l_(1)T_(2))/(K_(1)l_(2) + K_(2)l_(1))`