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Faraday's law of induction is a fundamental principle in electromagnetism, describing how a changing magnetic field within a circuit induces an electromotive force (emf). In simpler terms, it explains how electricity can be generated by moving a coil within a magnetic field. This law is expressed mathematically as \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced emf and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux. The negative sign indicates the direction of the induced emf opposes the change in magnetic flux, a concept known as Lenz's Law.
When applied to mutual inductance, Faraday's law takes the form \( \varepsilon = M \frac{dI}{dt} \). Here, \( M \) is the mutual inductance between two coils, and \( \frac{dI}{dt} \) is the rate at which current changes in one coil. Therefore, as the current in the first coil changes, it creates a changing magnetic field that induces an emf in the second coil. This principle is crucial for understanding transformers and many electrical devices.

Magnetic flux, denoted by \( \Phi \), represents the quantity of magnetic field passing through a given area. Imagine it as the number of magnetic field lines penetrating through a surface. The greater the number of lines crossing the surface, the higher the magnetic flux.
The formula for magnetic flux is \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field's strength, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the magnetic field and the perpendicular to the surface.

• Magnetic flux is measured in Webers (Wb).
• When computing flux through a coil, the number of turns affects the calculation, since more turns means more field lines are captured.

In the context of mutual inductance, flux through each turn of the coil can be found using \( \Phi = \frac{M I_1}{N_2} \). Here, \( M \) refers to mutual inductance, \( I_1 \) is the current in the first coil, and \( N_2 \) is the number of turns in the second coil. This calculation helps in determining the magnetic interaction between the coils.

Induced electromotive force, or induced emf, is the potential difference generated in a circuit due to a change in magnetic flux. This induction is central to devices such as generators, where it converts mechanical energy to electrical energy. The formula linking induced emf to mutual inductance is \( \varepsilon = M \frac{dI}{dt} \), with \( M \) as the mutual inductance and \( \frac{dI}{dt} \) the change rate of current.
In practical terms:

• Induced emf allows power to be transferred across components without direct connection.
• The magnitude of induced emf depends on how quickly the magnetic environment changes.
• Devices utilizing induced emf are often highly efficient, making them essential in modern technology.

The reciprocal nature of mutual inductance—where changing the current in one coil induces emf in another, and vice versa—is a practical manifestation of this principle. Using the problem's data, when the current rate of change in the second coil is 0.360 A/s, the first coil experiences an induced emf calculated as \( 2.46 \times 10^{-3} \) V, demonstrating the power of mutual induction.