A circuit shown in Fig, has resistances `R_(1) = 20 Omega` and `R_(2) = 30 Omega`. At what value of the resitanace `R_(x)` will the thermal power generated in it practically independent of small variations of that resistance? The voltage between the points `A` and `B` is supposed to be constant in this case

Explanation

Let, `varphi_(A) - varphi_(B) = varphi`
Now, thermal power generated in the resistanace `R_(x)`,
`P = t^(2) R_(x) = [(varphi)/(R_(1) + (R_(2) R_(x))/(R_(2) + R_(x))) (R_(2))/(R_(2) + R_(x))]^(2) R_(x)`,
For `P` to be independent of `R_(x)`,
`(dp)/(d R_(x)) = 0`, which yeilds
`R_(x) = (R_(1) R_(2))/(R_(1) + R_(2)) = 12 Omega`