resistance - How do we predict if power loss will increase or decrease in a DC circuit?

I want to know whether it is possible to predict if power loss will increase or decrease in a circuit with a fixed EMF, a fixed resistor in series and a variable resistor in series.

If we change the variable resistor's resistance, how do we know if power loss over the variable resistor will increase or decrease? This is presuming we already know the values for EMF, the resistance of the fixed resistor and the original power loss before changing the resistance.

For example, if we double resistance, what happens to the power loss of the in the variable resistor?

Furthermore, we can assume no internal resistance.

I think its something related to the maximum power loss when load resistance equals internal resistance rule.

Answer

If you look at this problem qualitatively, if the variable load resistance is either infinite or zero there is no power dissipation in the variable resistor load. So if we had an expression for the power dissipation in the variable load as a function of variable load resistance it would have a maxima at the series R0 resistance (we could guess, since that's like source resistance) and would decrease on either side.

^{simulate this circuit – Schematic created using CircuitLab}

We can derive this expression- current is V/(R0 +R), and power dissipation in the variable load R is \$I^2 R\$, so load dissipation is \$V^2 R\over {(R0 + R)^2 }\$

Plot some typical values to get an idea: (V =5V, R0 = 5\$\Omega\$)

If you differentiate that expression wrt R to find the maxima you get \$-V^2 (R - R0)\over {(R + R0)^3}\$ , which is 0 at R = R0, as expected.

The curve is not monotonic- it decreases on either side of the series resistor value.