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Electromotive force, or emf, is a term used in electromagnetism to describe the potential difference that can cause current to flow in a circuit. In the context of Faraday's Law of Electromagnetic Induction, an induced emf is generated when there is a change in magnetic flux through a coil. This is not a force as its name might suggest, but rather a voltage. Faraday's Law states that the induced emf is equal to the negative rate of change of magnetic flux multiplied by the number of loops in the coil. The formula is given by:

• \( \text{emf} = -n \frac{\Delta \Phi}{\Delta t} \)

Here, \( n \) is the number of loops, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the change in time. This induced emf is what causes a current to flow if the circuit is closed.
The negative sign in the formula signifies the direction of the induced emf as explained by Lenz's Law which we will discuss in a later section. Understanding this concept is fundamental to using Faraday's Law in problems involving electromagnetic induction.

Magnetic flux refers to the amount of magnetic field passing through a given surface area. It is an important concept when discussing electromagnetic induction. The magnetic flux \( \Phi \) through a surface is given by the equation:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

where \( B \) is the magnetic field, \( A \) is the area the field lines pass through, and \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
In the exercise, the magnetic flux changes from \(-58\) to \(+38\) Weber (Wb), a unit of magnetic flux. The greater this change in a given time, the higher the induced emf according to Faraday's Law. Flux helps us to quantify the number or density of magnetic field lines intersecting a coil or surface, and hence it's fundamental when calculating induced emf.
Changes in magnetic flux can occur due to variations in the magnetic field strength, the area of the loop, or the angle between them.

Lenz's Law is a fundamental principle that helps determine the direction of the induced current resulting from a change in magnetic flux. This law is part of the broader Faraday's Law, characterized by the negative sign in the induced emf formula \( \text{emf} = -n \frac{\Delta \Phi}{\Delta t} \). Lenz's Law states that the direction of the induced emf and, therefore, the induced current is such that it opposes the change in magnetic flux that caused it.
In simpler terms, if an increase in magnetic flux occurs, the induced current will create its magnetic field opposing the increase. Conversely, if the magnetic flux decreases, the induced current will create its magnetic field to increase it. This law is a manifestation of the conservation of energy, ensuring that the induced effects oppose the change that produces them.
In our given solution, the negative sign in the computed emf of \(-457.14 \text{ V}\) indicates this opposition, pointing to how the induced current would flow to counteract the increasing magnetic flux.