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

Options

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

Answer

D

Explanation

If T is the temperature of the interface.
`(k_(1)A(T_(1)-T))/(l_(1)) = (k_(2)A(T-T_(2)))/(l_(2))`
or `k_(1)l_(2)AT_(1)+k_(2)l_(1)AT_(2) = k_(2)Al_(1)T+k_(1)A l_(2)T`
or `T =(k_(1)l_(2)T_(1) + k_(2)l_(1)T_(2))/(k_(1)l_(2)+k_(2)l_(1))`