Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 32: AC Circuits

문제 64

32장, 문제 64

A generator consists of a 12-cm by 16-cm rectangular loop with 500 turns of wire spinning at 60 Hz in a 25 mT uniform magnetic field. The generator output is connected to a series RC circuit consisting of a 120 Ω resistor and a 35 μF capacitor. What is the average power delivered to the circuit?

검증된 단계별 안내

Calculate the area of the rectangular loop using the formula \( A = l \times w \), where \( l = 12 \ \text{cm} = 0.12 \ \text{m} \) and \( w = 16 \ \text{cm} = 0.16 \ \text{m} \).

Determine the maximum emf (electromotive force) generated by the loop using the formula \( \mathcal{E}_{\text{max}} = N \cdot B \cdot A \cdot \omega \), where \( N = 500 \) (number of turns), \( B = 25 \ \text{mT} = 0.025 \ \text{T} \), \( A \) is the area calculated in step 1, and \( \omega = 2 \pi f \) with \( f = 60 \ \text{Hz} \).

Find the root mean square (RMS) value of the emf using the relationship \( \mathcal{E}_{\text{rms}} = \frac{\mathcal{E}_{\text{max}}}{\sqrt{2}} \).

Calculate the total impedance of the RC circuit using the formula \( Z = \sqrt{R^2 + \left( \frac{1}{\omega C} \right)^2} \), where \( R = 120 \ \Omega \), \( \omega = 2 \pi f \), and \( C = 35 \ \mu\text{F} = 35 \times 10^{-6} \ \text{F} \).

Determine the average power delivered to the circuit using the formula \( P_{\text{avg}} = \frac{\mathcal{E}_{\text{rms}}^2}{Z} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in the wire. This principle, described by Faraday's Law, states that the induced EMF is proportional to the rate of change of magnetic flux through the loop. In this question, the spinning loop in a magnetic field generates an alternating current due to its motion.

AC Circuit Analysis

AC circuit analysis involves understanding how alternating current (AC) behaves in circuits, particularly with components like resistors and capacitors. The impedance in an RC circuit combines resistance and reactance, affecting the current and voltage relationship. The average power delivered to the circuit can be calculated using the root mean square (RMS) values of voltage and current, along with the phase angle between them.

Power in AC Circuits

In AC circuits, the average power delivered is calculated using the formula P = VIcos(φ), where V is the RMS voltage, I is the RMS current, and φ is the phase angle between the voltage and current. The presence of reactive components like capacitors can cause the current to lag or lead the voltage, affecting the power factor. Understanding this concept is crucial for determining how much power is effectively used in the circuit.

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