To which two points in the circuit of FIGURE P28.45 should a 12 V battery be connected to dissipate the most power?

검증된 단계별 안내

Step 1: Recall the formula for power dissipation in a resistor: \( P = \frac{V^2}{R} \), where \( P \) is power, \( V \) is voltage, and \( R \) is resistance. To maximize power dissipation, the resistance \( R \) across the battery should be minimized.

Step 2: Analyze the circuit. The circuit is a triangle configuration with resistors of 30 \( \Omega \), 40 \( \Omega \), and 20 \( \Omega \) between points 1, 2, and 3. To find the points where the battery should be connected, calculate the equivalent resistance between different pairs of points.

Step 3: Calculate the equivalent resistance between points 1 and 2. The 30 \( \Omega \) resistor is directly between these points, and the 40 \( \Omega \) and 20 \( \Omega \) resistors form a series connection between points 1 and 2. Combine the series resistances and then calculate the parallel equivalent resistance.

Step 4: Repeat the process for points 1 and 3, and points 2 and 3. For each pair, determine the equivalent resistance by considering the series and parallel combinations of resistors.

Step 5: Compare the equivalent resistances for all pairs of points. The pair with the smallest equivalent resistance will dissipate the most power when the 12 V battery is connected. Connect the battery to this pair of points.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = IR. Understanding this law is crucial for analyzing circuits, as it helps determine how voltage, current, and resistance interact.

Power Dissipation in Resistors

The power (P) dissipated by a resistor in an electrical circuit can be calculated using the formula P = I²R, where I is the current through the resistor and R is its resistance. This concept is essential for determining how much power is converted to heat in the resistors when a voltage is applied. Maximizing power dissipation involves understanding how to configure the circuit to optimize current flow through the resistors.

Series and Parallel Resistor Configurations

Resistors can be arranged in series or parallel configurations, affecting the total resistance and current in the circuit. In a series configuration, the total resistance is the sum of individual resistances, while in parallel, the total resistance decreases as more paths for current are added. Recognizing these configurations is vital for analyzing the circuit in the question, as it influences how the battery's voltage will distribute across the resistors.