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.

Answer

`12 omega`

Explanation

The equivalent resistance between the points is
`R_(0) = R_(1) + (R_(2) R_(x))/(R_(2) + R_(x))`
Power generated by `R_(x) is I_(x)^(2) R_(x)`.
From fig. `S 7.3`, `I = ((V)/( R_(1) + ( R_(2)R_(x))/( R_(2) + R_(x))))`
` I_(x) = ((V)/( R_(1) + ( R_(2)R_(x))/( R_(2) + R_(x))))((R^(2))/( R_(2) + R_(x)))`
`P_(x) = I_(x)^(2) R_(x) = [(VR_(2))/( R_(1) R_(2) + R_(1) R_(x) + R_(2) R_(x))]^(2) R_(x)`
For the maximum value , `( dP_(x))/( dR_(x)) = 0 `
or `R_(1) R_(2) + R_(1) R_(x) + R_(2) R_(x) = 2( R_(1) + R _(2)) R_(x)`
or `R_(x) = ( R_(1) R_(2))/( R_(1) + R_(2)) = ( 20 xx 30)/( 20 + 30) = 12 Omega`