Ch. 26 - DC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 26 - DC Circuits

문제 94

25장, 문제 94

The performance of the starter circuit in a car can be significantly degraded by a small amount of corrosion on a battery terminal. Figure 26–88a depicts a properly functioning circuit with a battery (12.5-V emf, 0.02-Ω internal resistance) attached via corrosion-free cables to a starter motor of resistance R_s = 0.15Ω. Sometime later, corrosion between a battery terminal and a starter cable introduces an extra series resistance of only R_C = 0.10Ω into the circuit as suggested in Fig. 26–88b. Let P₀ be the power delivered to the starter in the circuit free of corrosion, and let P be the power delivered to the starter with corrosion. Determine the ratio P/P₀.

검증된 단계별 안내

Step 1: Start by understanding the circuit. In the first case (corrosion-free), the total resistance in the circuit is the sum of the internal resistance of the battery (Rᵢ = 0.02Ω) and the resistance of the starter motor (Rₛ = 0.15Ω). In the second case (with corrosion), an additional resistance R_C = 0.10Ω is introduced in series, so the total resistance becomes Rᵢ + Rₛ + R_C.

Step 2: Use Ohm's Law to calculate the current in each case. The current in the corrosion-free circuit is I₀ = ε / (Rᵢ + Rₛ), where ε is the emf of the battery (12.5 V). For the circuit with corrosion, the current is I = ε / (Rᵢ + Rₛ + R_C).

Step 3: Recall that the power delivered to the starter motor is given by P = I²Rₛ. For the corrosion-free circuit, the power is P₀ = I₀²Rₛ. For the circuit with corrosion, the power is P = I²Rₛ.

Step 4: Substitute the expressions for I₀ and I into the power formulas. For P₀, substitute I₀ = ε / (Rᵢ + Rₛ) into P₀ = I₀²Rₛ. For P, substitute I = ε / (Rᵢ + Rₛ + R_C) into P = I²Rₛ.

Step 5: Calculate the ratio P/P₀. This is given by (I²Rₛ) / (I₀²Rₛ). Simplify the ratio using the expressions for I and I₀, which results in P/P₀ = [(Rᵢ + Rₛ) / (Rᵢ + Rₛ + R_C)]². This is the final expression for the ratio of the power delivered to the starter motor with and without corrosion.

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주요 개념

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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 how changes in resistance, such as those caused by corrosion, affect the current and power in an electrical circuit.

Power in Electrical Circuits

The power (P) delivered in an electrical circuit can be calculated using the formula P = IV, where I is the current and V is the voltage. Additionally, power can also be expressed in terms of resistance as P = I²R or P = V²/R. This concept is essential for determining how the introduction of additional resistance from corrosion affects the overall power delivered to the starter motor.

Series Resistance

In a series circuit, the total resistance is the sum of the individual resistances. When corrosion introduces an additional resistance, it increases the total resistance in the circuit, which in turn reduces the current flowing through the circuit according to Ohm's Law. This concept is vital for understanding how the performance of the starter motor is impacted by the added resistance from corrosion.

관련 실천

교과서 질문

Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery ε = 15.000 V with a 10.00-MΩ resistor in series with an unknown resistor R. As shown in Fig. 26–92, a particular voltmeter reads V₁ = 366 mV when connected across the 10.00 -MΩ resistor and this meter reads V₂ = 7.317 V when connected across R. Determine the value of R. [Hint: Define R_V as the voltmeter’s internal resistance.]

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교과서 질문

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교과서 질문

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848

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교과서 질문

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교과서 질문

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778

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