To solve this problem, we need to determine the minimum value of $R$ such that the power consumed by the $6\Omega$ resistor is maximized. Here's the step-by-step process:

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1. Understand the Circuit Configuration:

• Parallel combination: $30 \Omega$ and $R$.
• Series connection: The parallel combination is connected to a $6 \Omega$ resistor and powered by a $120V$ source.

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2. Recall Power Maximization Rule:

The power $P$ dissipated in a resistor is given by: $P = \frac{V^2}{R}$ where $V$ is the voltage across the resistor, and $R$ is its resistance. To maximize the power in the $6 \Omega$ resistor, the voltage across it must also be maximized.

For maximum voltage across $6 \Omega$:

1. The total current in the circuit should be minimized (reducing voltage drop across the parallel combination).
2. The equivalent resistance $R_\text{eq}$ of the parallel combination should approach infinity, meaning $R$ must be very high.

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3. Compute Equivalent Resistance of the Parallel Combination:

The equivalent resistance of $30 \Omega$ and $R$ is: $R_\text{parallel} = \frac{30R}{30 + R}$

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4. Total Resistance in the Circuit:

The total resistance in the circuit is: $R_\text{total} = R_\text{parallel} + 6$

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5. Current in the Circuit:

The current $I$ from the source is: $I = \frac{V_\text{source}}{R_\text{total}} = \frac{120}{\frac{30R}{30 + R} + 6}$

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6. Power in the $6 \Omega$ Resistor:

The voltage across the $6 \Omega$ resistor is: $V_{6} = I \times 6$ The power dissipated in the $6 \Omega$ resistor is: $P_{6} = \frac{V_{6}^2}{6} = 6I^2$

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7. Maximize $P_{6}$:

For $P_{6}$ to be maximized, $R_\text{parallel}$ must be minimized. The minimum $R_\text{parallel}$ occurs when $R \to \infty$, making $R_\text{parallel} \approx 30 \Omega$.

With $R \to \infty$:

• $R_\text{total} = R_\text{parallel} + 6 = 30 + 6 = 36 \Omega$.
• Current $I = \frac{120}{36} = 3.33 \, \text{A}$.
• Power in $6 \Omega$: $P_{6} = 6 \times (3.33)^2 = 6 \times 11.1 = 66.6 \, \text{W}.$

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Conclusion:

The minimum value of $R$ for maximum power in the $6 \Omega$ resistor is infinite. Practically, $R$ must be very large to approximate this condition.

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Questions to Explore:

1. What happens to the circuit behavior if $R$ is very small?
2. How does $R_\text{parallel}$ affect the overall circuit current?
3. How would the maximum power change if the supply voltage was reduced?
4. Can you derive the critical value of $R$ that gives a specific fraction of maximum power?
5. What is the efficiency of the circuit under maximum power conditions?

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Tip:

To maximize power in any series-parallel combination, focus on balancing resistances to shift the voltage drop toward the resistor of interest.