Question

for the circuit in figure 5 - 2, what was happening to the total power in the circuit as the resistance of $r_2$ was increasing? explain your answer.

Explanation:

Step1: Recall power - resistance relationship

The power in a circuit is given by $P=\frac{V^{2}}{R_{total}}$ (assuming a constant - voltage source $V$). When resistors are in series, $R_{total}=R_1 + R_2$, and when in parallel, $\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}$.

Step2: Analyze series - circuit case

If $R_1$ and $R_2$ are in series, $R_{total}=R_1 + R_2$. As $R_2$ increases, $R_{total}$ increases. Since $P = \frac{V^{2}}{R_{total}}$ and $V$ is constant, the total power $P$ decreases because power is inversely proportional to resistance.

Step3: Analyze parallel - circuit case

If $R_1$ and $R_2$ are in parallel, $\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}=\frac{R_2 + R_1}{R_1R_2}$, so $R_{total}=\frac{R_1R_2}{R_1 + R_2}$. As $R_2$ increases, $R_{total}$ increases. Using $P=\frac{V^{2}}{R_{total}}$, the total power $P$ decreases because power is inversely proportional to resistance.

Answer:

The total power in the circuit decreases as the resistance of $R_2$ increases, regardless of whether $R_1$ and $R_2$ are in series or parallel, because the total resistance of the circuit increases and power is inversely proportional to resistance when the voltage is constant.