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Ch 28: Fundamentals of Circuits
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 28: Fundamentals of CircuitsProblema 45a
Capitolo 28, Problema 45a

To which two points in the circuit of FIGURE P28.45 should a 12 V battery be connected to dissipate the most power?
Circuit diagram showing three resistors: 30Ω, 40Ω, and 20Ω, arranged in a triangle with connection points labeled 1, 2, and 3.

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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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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.
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Resistance and Ohm's Law

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.
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Power in Circuits

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.
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Combining Resistors in Series & Parallel
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