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Faraday's Law of Electromagnetic Induction is a fundamental principle in electromagnetism. It explains how an electromotive force (emf) is induced in a conductor when it experiences a change in magnetic flux. According to Faraday's Law,

• The induced emf is directly proportional to the rate of change of magnetic flux.
• The equation for Faraday's Law is given by: \( \mathcal{E} = -\frac{d\Phi}{dt} \)

Here, \( \mathcal{E} \) represents the induced emf and \( \Phi \) is the magnetic flux.

In the given exercise, when the coil is rapidly removed from the magnetic field, the magnetic flux becomes zero, causing a rapid change that induces an emf in the coil. This is quantified as the average emf using Faraday's Law formula:\[ \mathcal{E}_{\text{av}} = \frac{NAB}{\Delta t} \]where:

• \( N \) is the number of turns in the coil
• \( A \) is the area of each turn
• \( B \) is the magnetic field strength
• \( \Delta t \) is the time over which the change occurs

Magnetic Flux is a measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. It is represented by the symbol \( \Phi \) and defined as the product of the magnetic field \( B \), the area \( A \) through which the field lines pass, and the cosine of the angle \( \theta \) between the magnetic field and the normal to the area.

• The formula for magnetic flux is \( \Phi = NBA\cos\theta \).
• In our exercise, because the magnetic field is perpendicular to the coil, \( \theta = 0 \), thus \( \cos 0 = 1 \), simplifying the flux to: \( \Phi = NAB \).

When the coil is removed from the field, the magnetic flux decreases from its initial value to zero, as the field no longer intersects the coil. This change in flux is what induces the emf, according to Faraday's Law.

To calculate the charge \( Q \), we only need to consider the initial magnetic flux. Since \( \Phi \) initially was \( NAB \) and becomes zero, the total change in flux \( \Delta \Phi \) is \( NAB \). Therefore, the relationship between charge and flux change is encapsulated in:\[ Q = \frac{NAB}{R} \]

Ohm's Law is a fundamental rule in electronics that relates the voltage, current, and resistance in an electrical circuit. It helps us understand how the quantities of an elastic conductor behave when carrying an electrical current. Ohm's Law is expressed by the formula:

• \( I = \frac{V}{R} \)

Where:

• \( I \) is the current in amperes (A)
• \( V \) is the voltage in volts (V)
• \( R \) is the resistance in ohms (\( \Omega \))

In the context of the coil from the exercise, the induced voltage, or emf, leads to an induced current \( I_{\text{av}} \).

According to Ohm's Law, this relationship is characterized as:\[ I_{\text{av}} = \frac{\mathcal{E}_{\text{av}}}{R} \]Given that the emf \( \mathcal{E}_{\text{av}} \) is \( \frac{NAB}{\Delta t} \), we can substitute this into Ohm's formula:\[ I_{\text{av}} = \frac{NAB}{R \Delta t} \]Finally, the step incorporates this current to calculate the total charge \( Q \) through:\[ Q = I_{\text{av}} \Delta t = \frac{NAB}{R} \]Thus, demonstrating how the average current and resulting charge depends on the coil's inherent characteristics and the changing magnetic field, while being independent of the time interval \( \Delta t \).