Three rods of identical cross-sectional area and made from the same metal from the sides of an isosceles triangle ABC, right-angled at B. The point A and B are maintained at temperatures T and `sqrt(2)` T respectively. In the steady state, the temperature of the point C is `T_c.` Assuming that only heat conduction takes places, `T_c//T` is

Options

A. `(1)/((sqrt2 + 1))`
B. `(2)/((sqrt2 + 1))`
C. `(1)/(2(sqrt2 - 1))`
D. `(1)/(sqrt 3(sqrt2 1))`

Answer

B

Explanation

(2) `R_(AB) = R_(BC) = R`
`R_(AC) = sqrt(2) R`
Since `T_(B) gt T_(A)`, heat will flow from `B` to `A` through two paths `BA` and `BCA`
In path `BCA`, current will be same in `BC` and `CA`
`i= (sqrt 2 T - T_(C ))/(R_(BC)) = (T_(C ) - T)/(R_(AC))`
`(sqrt 2 T - T_(C ))/(R ) = (T_(C ) - T)/(sqrt 2 R)`
`2T - 2 sqrt 2 T_(C ) = T_(C ) - T`
`(sqrt2 + 1) T_(C ) = 3 T`
`(T_(C ))/(T) = (3)/((sqrt 2 + 1))`