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?

Verified step by step guidance

1. Calculate the area of the rectangular loop using the formula \( A = l \times w \), where \( l = 12 \ \text{cm} = 0.12 \ \text{m} \) and \( w = 16 \ \text{cm} = 0.16 \ \text{m} \).
2. 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 \( f = 60 \ \text{Hz} \).
3. Find the root mean square (RMS) value of the emf using the relationship \( \mathcal{E}_{\text{rms}} = \frac{\mathcal{E}_{\text{max}}}{\sqrt{2}} \).
4. 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 \( C = 35 \ \mu\text{F} = 35 \times 10^{-6} \ \text{F} \).
5. Determine the average power delivered to the circuit using the formula \( P_{\text{avg}} = \frac{\mathcal{E}_{\text{rms}}^2}{Z} \).

Key Concepts

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.