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Faraday's Law of electromagnetic induction is fundamental to understanding how electric generators work. It states that an electromotive force (emf) is induced in a conductor when the magnetic flux through the conductor changes. The induced emf is directly proportional to the rate of change of magnetic flux. In the context of the provided exercise, Faraday's Law is applied to calculate the peak emf produced by the generator. The formula is given by: \[ \text{emf}_{\text{peak}} = NAB\omega \sin(\omega t) \]where:

• N is the number of turns in the coil (100 turns in this case)
• A is the area of one loop in the coil
• B is the magnetic field strength (0.50 T here)
• \(\omega\) is the angular frequency in radians per second

Faraday's Law shows how a rotating coil in a magnetic field can produce an AC voltage. When the position of the coil changes, the amount of magnetic field lines passing through it changes, inducing an emf.

The root mean square (rms) voltage of an alternating current system, like the one produced by the generator in the exercise, measures the effective value of an alternating voltage. It is useful because it represents the DC equivalent of an AC voltage in terms of its ability to perform work in a resistive circuit. The relationship between the rms voltage and the peak voltage in an AC circuit is given by:\[ \text{emf}_{\text{rms}} = \frac{\text{emf}_{\text{peak}}}{\sqrt{2}} \] In this exercise, the generator has an rms voltage of 120 V. By rearranging the formula, we can calculate the peak voltage:\[ \text{emf}_{\text{peak}} = 120\sqrt{2} \approx 169.7 \text{ V} \]Understanding rms values is crucial in electricity and electronics because they allow us to compare AC and DC circuits directly and work with real power considerations.

Angular frequency, often symbolized by \( \omega \), is a measure of how quickly the phase of a wave progresses. It is particularly relevant in systems involving rotating objects or periodic changes such as AC currents.In this context, angular frequency is found using the equation:\[ \omega = 2\pi f \]where \( f \) is the frequency of the generator. Given that \( f = 60.0 \text{ Hz} \), we calculate:\[ \omega = 2\pi \times 60.0 = 120\pi \text{ rad/s} \]Utilizing angular frequency allows us to explore the timing-related aspects of oscillating systems and address issues involving rotational speed and periodic waveforms, which are essential for a generator's operation.

A magnetic field, denoted typically by \( B \), is a vector field that describes the magnetic influence on charges, moving charges, and magnetic materials. In the scope of electromagnetic induction, the magnetic field plays a critical role in inducing electromotive force in the loop of wire within the generator coil.The strength of the magnetic field in this problem is given as 0.50 Tesla (T). This value reflects how 'strong' or 'weak' the magnetic effect is on the generator's coil. The interaction between the rotating coil and the stable magnetic field generates an electromagnetic force thanks to Faraday's Law.Magnetic fields form the basis of many technologies, from simple magnets to complex electronic systems like transformers, motors, and generators. They are critical in numerous applications such as data storage, navigation, and even medical technologies.