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Faraday's Law is a fundamental principle in electromagnetism that describes how a changing magnetic field can induce voltage, or electromotive force (emf), in a circuit. This is a key concept in understanding how inductors work. An inductor, such as the one in the original exercise, generates an induced emf when there is a change in current flowing through it.

According to the law, the magnitude of the induced emf is proportional to the rate of change of the magnetic flux. In mathematical terms, for an inductor with inductance \(L\), the induced emf \(\varepsilon(t)\) is expressed as:

\[ \varepsilon(t) = -L \frac{di}{dt} \]

The negative sign indicates Lenz's Law, showing that the induced emf acts in a direction to oppose the change in current that created it. This is crucial for various applications like transformers and electrical generators.

In the given exercise, applying Faraday’s Law helps us find the expression for \(\varepsilon(t)\) based on \(\frac{di}{dt}\), the rate of change of current.

A time-varying current is one that changes its magnitude and direction with respect to time. In mathematical expressions, this current is often represented in terms of sinusoidal functions such as sine or cosine.

For example, the exercise discusses a current given by:

\( i(t) = (124 \text{ mA}) \cos[(240\pi / \text{s}) t] \)

Here, the amplitude of the current is 124 mA, and it oscillates at an angular frequency of \(240\pi\) rad/s. This means the current completes \(120\) cycles per second\((60\text{ Hz})\) as the term \(\frac{240\pi}{2\pi}\) gives the frequency in Hertz.

Time-varying currents like these are vital in AC circuit analysis and are commonly encountered in electrical engineering and physics.

Understanding how this type of current influences an inductor, leads to the calculation of a time-dependent induced emf, as the rate of change of this current directly affects the emf generation.

Maxima and minima of functions refer to the highest or lowest points a function reaches, respectively. Finding these points is particularly important in understanding the behavior of oscillating functions, like those describing time-varying currents.

In the exercise, the cosine current function \( i(t) = (124 \text{ mA}) \cos[(240\pi / \text{s}) t] \) reaches its maximum value when \( \cos[(240\pi / \text{s}) t] = \pm 1 \). This is because the maximum of the cosine function is 1, hence the current reaches \( \pm 124 \text{ mA} \).

Similarly, the sine function describing induced emf \( \varepsilon(t) = 7.44\pi \sin[(240\pi / \text{s}) t] \) reaches its maximum when \( \sin[(240\pi / \text{s}) t] = \pm 1 \). This results in a maximum emf of approximately \(23.40 \text{ V} \).

Determining these maxima and minima is crucial in engineering and physics as it helps identify the peak behaviors of functions, which is essential for optimizing circuit performance and other practical applications.