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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 49
Capitolo 28, Problema 49

Suppose you have resistors 2.5 Ω, 3.5 Ω, and 4.5 Ω and a 100 V power supply. What is the ratio of the total power delivered to the resistors if they are connected in parallel to the total power delivered if they are connected in series?

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Step 1: Recall the formula for total resistance in a series connection. For resistors connected in series, the total resistance \( R_{\text{series}} \) is the sum of the individual resistances: \( R_{\text{series}} = R_1 + R_2 + R_3 \). Substitute \( R_1 = 2.5 \, \Omega \), \( R_2 = 3.5 \, \Omega \), and \( R_3 = 4.5 \, \Omega \) into the formula.
Step 2: Recall the formula for total resistance in a parallel connection. For resistors connected in parallel, the total resistance \( R_{\text{parallel}} \) is given by \( \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \). Substitute \( R_1 = 2.5 \, \Omega \), \( R_2 = 3.5 \, \Omega \), and \( R_3 = 4.5 \, \Omega \) into the formula.
Step 3: Use the formula for power delivered to a circuit: \( P = \frac{V^2}{R} \), where \( P \) is the power, \( V \) is the voltage, and \( R \) is the total resistance. For the series connection, substitute \( R_{\text{series}} \) into the formula to calculate the total power delivered in the series case.
Step 4: Similarly, for the parallel connection, substitute \( R_{\text{parallel}} \) into the formula \( P = \frac{V^2}{R} \) to calculate the total power delivered in the parallel case.
Step 5: To find the ratio of the total power delivered in parallel to the total power delivered in series, divide the power in the parallel case by the power in the series case: \( \text{Ratio} = \frac{P_{\text{parallel}}}{P_{\text{series}}} \). Simplify the expression using the relationship between \( R_{\text{series}} \) and \( R_{\text{parallel}} \).

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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 as V = IR. Understanding Ohm's Law is essential for calculating current in both series and parallel circuits.
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Resistance and Ohm's Law

Power in Electrical Circuits

The power (P) delivered to a resistor in an electrical circuit is calculated using the formula P = IV, where I is the current through the resistor and V is the voltage across it. In series circuits, the same current flows through all components, while in parallel circuits, the voltage across each component is the same. This distinction is crucial for determining total power in different configurations.
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Power in Circuits

Series and Parallel Circuits

In a series circuit, resistors are connected end-to-end, resulting in a total resistance that is the sum of individual resistances. Conversely, in a parallel circuit, resistors are connected across the same voltage source, leading to a total resistance that is less than the smallest individual resistor. The configuration affects both the total current and power delivered, making it vital to understand these differences for the problem at hand.
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Combining Capacitors in Series & Parallel
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