A circuit shown in the figure has resistances `20 Omega` and `30 Omega`. At what value of resistance `R_x` will the thermal power generated in it be practically independent of small variations of that resistance? The voltage between points `A` and `B` is supposed to be constant in this case.

Explanation

Let V be the potential difference applied across across A and B.
Let `I,I_(1),I_(2)` be the currents through `R_(1),R_(2)` and `R_(x)` respectively. Here `R_(2)` and `R_(x)` are connected in parallel, their effective resistance is
`R_(P) = (R_(2) R_(x))/(R_(2) + R_(x))`
Total resistance between A and B is
`R_(1) = (R_(2) R_(x))/(R_(2) + R_(x))`
Current through the circuit,
`I = V/(R_(1) + (R_(2) R_(x))/(R_(2) + R_(x)))`
Potential difference across C and D
`V_(G//D) I R_(P) = I = V/(R_(1) + (R_(2) R_(x))/(R_(2) + R_(x))) xx (R_(2) R_(x))/(R_(2) + R_(x))`
`(VR_(2)R_(x))/(R_(1) (R_(2) + R_(x)) + R_(2)R_(x))`
Current through resistance `R_(x)` will be
`I_(2) = V_(CD)/R_(x) = (VR_(2))/(R_(1) (R_(2) + R_(x)) + R_(2)R_(x))`
`= (VR_(2))/(R_(1) (R_(2) + R_(x)) + R_(2)R_(x))`
Thermal power generated in resistance `R_(x)` will be
`P = I_(2)^(2) R_(x) = (V^(2)R_(2)^(2)R_(x))/[R_(1)R_(2)+(R_(1) + R_(2))+R_(x)]^(2)`
For P to be independent of `R_(x),dP/dR_(x) = 0`
So, `d/dR_(x) [ (V^(2)R_(2)^(2)R_(x))/[R_(1)R_(2)+(R_(1) + R_(2))+R_(x)]^(2) ] = 0`
On simplifying, we get
`R_(1)R_(2) - (R_(1) + R_(2)) R_(x) = 0 or R_(x) = R_(1)R_(2)/(R_(1) + R_(2))`
`:. R_(x) = (20 xx 30)/(20 + 30) = 12 Omega`