1. Lenz's Law

Concept Explanation:
Lenz's Law states that the direction of the induced current (or electromotive force, EMF) in a closed circuit is such that it opposes the change in magnetic flux that caused it. This is a consequence of the conservation of energy, ensuring that the induced current creates a magnetic field that opposes the original change in flux.

Mathematical Formulation:
Lenz's Law can be expressed as:
EMF = -dΦ_B/dt
where:
- EMF is the electromotive force,
- Φ_B is the magnetic flux, which is the product of the magnetic field B, the area A, and the cosine of the angle θ between the field and the normal to the surface (i.e., Φ_B = B × A × cos θ).

The negative sign indicates the opposition to the change in flux.

Example Problem:
A magnetic field perpendicular to a loop of wire is increasing in strength. Explain the direction of the induced current in the loop according to Lenz's Law.

Solution:
As the magnetic field increases, the magnetic flux through the loop increases. According to Lenz's Law, the induced current will flow in a direction that creates a magnetic field opposing this increase. If the increasing magnetic field is directed into the page, the induced current will flow in a counterclockwise direction (as viewed from above) to produce a magnetic field out of the page, opposing the increase.