A wedding ring (of diameter \(2.0 \mathrm{~cm}\) ) is tossed into the air and given a spin, resulting in an angular velocity of 17 rotations per second. The rotation axis is a diameter of the ring. Taking the magnitude of the Earth's magnetic field to be \(4.0 \cdot 10^{-5} \mathrm{~T}\), what is the maximum induced potential difference in the ring?

If you've ever wondered how electricity is generated through motion, Faraday's Law is the fundamental principle to understand. Named after the renowned scientist Michael Faraday, the law deals with the concept of electromagnetic induction. Simply put, it states that a change in magnetic environment of a coil of wire will cause a voltage, or emf (electromotive force), to be induced in the coil.

Mathematically expressed, Faraday's Law is represented as:
\[EMF = -N\frac{d\text{Φ}_B}{dt}\]
where EMF is the induced potential difference, N represents the number of turns of wire, and dΦB/dt denotes the rate of change of magnetic flux through the wire. The negative sign alludes to Lenz's Law, indicating the direction of the induced EMF opposes the change in magnetic flux causing it.

This phenomenon is exploited in various applications, from the generators producing electricity in power plants to the principle behind transformers used in power distribution.

Imagine a loop of wire in a magnetic field: how much of this magnetic field passes through the loop? The answer lies in the concept of magnetic flux. It's quantified as the product of the magnetic field and the loop's area through which the field lines pass, and it's vital for understanding electromagnetic phenomena.

Mathematically, it's given by:
\[\text{Φ}_B = B \times A \times \text{cos}(\theta)\]
where ΦB is the magnetic flux, B is the magnitude of the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop's surface. A change in any of these factors—field strength, area, or angle—can induce a potential difference as described by Faraday's Law.

For example, in the exercise mentioned, as the ring spins, there is a change in the angle θ affecting the magnetic flux, which, in turn, leads to the generation of an induced potential difference in the ring.

Think of angular velocity as the rotational equivalent of linear velocity. It represents the rate at which an object spins around a fixed axis. It's crucial for understanding motions in everything from car engines to the Earth's rotation.

It is denoted by the symbol ω and is often measured in radians per second (rad/s). Since there are 2π radians in a full circle (360°), you can convert revolutions per second to radians per second by multiplying by 2π.

The formula is:
\[\omega = \frac{d\theta}{dt}\]
where ω is the angular velocity, dθ is the infinitesimal change in angle, and dt is the change in time. In our exercise, the angular velocity of the wedding ring determines how quickly the ring's position changes relative to the magnetic field, affecting the rate of change of magnetic flux and, by extension, the magnitude of the induced EMF.