The current flowing in a solenoid that is \(20.0 \mathrm{~cm}\) long and has a radius of \(2.00 \mathrm{~cm}\) and 500 turns decreases from \(3.00 \mathrm{~A}\) to \(1.00 \mathrm{~A}\) in \(0.1 .00 \mathrm{~s} .\) Determine the magnitude of the induced electric field inside the solenoid \(1.00 \mathrm{~cm}\) from its center.

Faraday's Law of electromagnetic induction is a fundamental principle that explains how electric currents can be produced by changing magnetic fields. This law is the basic operation behind many electrical generators and transformers. According to Faraday's Law, an electromotive force (EMF) is induced in a circuit whenever there is a change in magnetic flux through that circuit. The size of the induced EMF is directly proportional to the rate of change of the magnetic flux.

Mathematically, Faraday's Law is expressed as:
\[EMF = -N \frac{d\Phi}{dt}\]
Here, \(EMF\) represents the induced electromotive force in volts, \(N\) is the number of turns in the solenoid or coil through which the flux is changing, \(d\Phi\) is the change in magnetic flux in webers over time \(dt\) in seconds. The negative sign indicates the direction of the induced EMF (and the resulting current, if the circuit is closed) which is given by Lenz's Law. This law states that the induced current will flow in a direction that opposes the change in magnetic flux that produced it, which is also a consequence of the law of conservation of energy.

In the context of the exercise, Faraday's Law is used to calculate the magnitude of the induced EMF when the current in the solenoid changes over time.

The concept of magnetic flux (\(\Phi\)) is central to understanding Faraday's Law of electromagnetic induction. Magnetic flux is a measure of the total magnetic field (B) passing through a certain area (A). It's comparable to the amount of water flowing through a pipe - the more water, the greater the flow. Similarly, the greater the magnetic field that intersects a surface area, the larger the magnetic flux.

Mathematically, magnetic flux is given by the equation:
\[\Phi = B \cdot A \cdot \cos(\theta)\]
Where \(\Phi\) is the magnetic flux in webers (Wb), \(B\) is the magnetic field in teslas (T), \(A\) is the area in square meters (m^2) through which the field lines pass, and \(\theta\) is the angle between the magnetic field lines and the normal (perpendicular) to the area. For a solenoid with a uniform magnetic field and an area perpendicular to the field lines, the cosine term is equal to one, simplifying the calculation of magnetic flux.

In the given problem, the magnetic flux change is key to determining the EMF, and subsequently the induced electric field inside the solenoid, as the current changes over time.

The electromotive force, or EMF, is not really a force per se. Despite its name, EMF is a measure of energy provided per unit charge, measured in volts (V). It is the work done on a charge when it completes a circuit, a potential difference that can push electric charges through a conductor. In the context of electromagnetic induction, EMF is the voltage generated when the magnetic environment around a conductor or coil is changing.

When discussing induced EMF as a result of a change in magnetic flux, it's essential to understand that this induced voltage is what drives the flow of current in a closed circuit. The magnitude of the induced EMF can be influenced by several factors, such as the rate of change in magnetic flux, the number of turns in the coil, and the cross-sectional area of the coil.

For the problem provided, using Faraday's Law, we find the magnitude of the induced EMF once we have the change in magnetic flux. The calculation disregards the direction of the induced EMF, focusing only on its absolute value to determine the strength of the induced electric field.

A solenoid is a coil of wire that produces a uniform magnetic field in its interior when electric current passes through it. This magnetic field is strongest in the center and aligns along the length of the coil. The strength of the magnetic field inside a solenoid is determined by several factors: the current flowing through the coils (I), the number of turns of the wire (N), the length of the solenoid (L), and the permeability of free space (\(\mu_0\)).

The formula to calculate the magnetic field inside an ideal solenoid is:
\[B = \mu_0 * \frac{N}{L} * I\]
In this expression, \(\mu_0\) is known as the magnetic constant, with a value of about \(4\pi \times 10^{-7} Tm/A\). By using the initial and final currents, we can determine the change in the magnetic field (\(\Delta B\)), which, when multiplied by the cross-sectional area of the solenoid, gives us the change in magnetic flux (\(\Delta \Phi\)). Knowing this, we can apply Faraday's Law to find the EMF and then the induced electric field at a specific location inside the solenoid.

In the given problem, the magnetic field calculation is pivotal in determining the electric field induced due to the change in the current. As the current flowing through the solenoid's coils changes, it alters the magnetic field inside, leading to electromagnetic induction.