The user wants me to solve a physics problem by comparing the power dissipated in three different resistor configurations connected to a constant voltage source.

Step-by-step Solution:

1. Understand the Relationship between Power, Voltage, and Resistance

The power P dissipated by a resistor configuration is related to the voltage V across it and its equivalent resistance $R_eq$. Since the voltage of the battery is constant for all three configurations (V = 3V), the most convenient formula to use is:

$P = \frac{V^2}{R_{eq}}$

This formula shows that for a constant voltage, the power dissipated is inversely proportional to the equivalent resistance ($P \propto 1/R_{eq}$). This means the configuration with the smallest equivalent resistance will dissipate the most power, and the one with the largest equivalent resistance will dissipate the least.

2. Calculate the Equivalent Resistance for each Configuration

Let's calculate the equivalent resistance for R1, R2, and R3.

• Configuration R1: This configuration has a single resistor with resistance $R = 1 \Omega$. So, the equivalent resistance $R_{eq1}$ is: $R_{eq1} = 1 \Omega$

• Configuration R2: This configuration has two $1 \Omega$ resistors connected in parallel. The equivalent resistance $R_{eq2}$ is calculated as: $\frac{1}{R_{eq2}} = \frac{1}{1 \Omega} + \frac{1}{1 \Omega} = \frac{2}{1 \Omega}$ $R_{eq2} = \frac{1}{2} \Omega = 0.5 \Omega$

• Configuration R3: This configuration has two $1 \Omega$ resistors connected in series. The equivalent resistance $R_{eq3}$ is the sum of the individual resistances: $R_{eq3} = 1 \Omega + 1 \Omega = 2 \Omega$

3. Compare the Equivalent Resistances

Now, let's compare the calculated equivalent resistances:

$R_{eq2} < R_{eq1} < R_{eq3}$ $(0.5 \Omega < 1 \Omega < 2 \Omega)$

4. Compare the Power Dissipated

Since power is inversely proportional to resistance ($P \propto 1/R_{eq}$), the order of power dissipation will be the reverse of the order of the equivalent resistances:

$P_2 > P_1 > P_3$

5. (Optional) Calculate the Exact Power Values

We can also calculate the exact power for each configuration using V = 3V:

• For R1: $P_1 = \frac{V^2}{R_{eq1}} = \frac{(3V)^2}{1 \Omega} = 9 W$
• For R2: $P_2 = \frac{V^2}{R_{eq2}} = \frac{(3V)^2}{0.5 \Omega} = 18 W$
• For R3: $P_3 = \frac{V^2}{R_{eq3}} = \frac{(3V)^2}{2 \Omega} = 4.5 W$

Comparing these values confirms our conclusion: 18 W > 9 W > 4.5 W, which means $P_2 > P_1 > P_3$.

6. Conclusion

The correct relationship between the powers is $P_2 > P_1 > P_3$, which corresponds to option C.