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

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.

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.