Detailed Summary
This section details fundamental concepts in electromagnetism, especially focusing on the following:
- Magnetic Flux (ΦB): The magnetic flux through a surface is defined as the product of the magnetic field (B) and the area (A) it penetrates, taking into account the angle (θ) between them:
$$
ΦB = B imes A imes ext{cos}(θ)
$$
-
Faraday’s Law of Induction: It states that the electromotive force (emf) generated in a coil is related to the rate of change of magnetic flux through it:
$$
ε = -N rac{dΦB}{dt}
$$
Here, the negative sign indicates the direction of the induced emf as per Lenz's law.
-
Lenz’s Law: This principle asserts that the induced emf will generate a current that opposes the change in magnetic flux that produced it, thus conserving energy.
-
Motional EMF: When a conductor moves through a magnetic field, an emf is induced, calculated as:
$$
e = Blv$$ where v is the velocity of the conductor.
-
Inductance: It's defined as the ratio of induced magnetic flux linkage to the current.
$$
L = rac{NΦ}{I}
$$
-
Mutual Inductance: This is the ability of one coil to induce emf in another nearby coil due to change in current, expressed through:
$$
ε1 = -M_{12} rac{dI_{2}}{dt}
$$
Mutual inductance values are reciprocal for the two coils.
-
Self-Inductance: A changing current in a coil can induce an emf in itself, represented as:
$$
ε = -L rac{dI}{dt}
$$
where L is the self-inductance of the coil.
-
Inductance of Long Solenoid: The self-inductance of a solenoid filled with a magnetic material is given by:
$$
L = rac{μ_{r} μ_{0} n^{2} A}{l}
$$
-
AC Generators: Mechanical energy is converted into electrical energy in generators through electromagnetic induction. The formula for induced emf in an AC generator is:
$$
e = NBA(2 ext{πn}) ext{sin}(2 ext{πnt})
$$
This section is foundational in understanding the principles governing electromagnetic induction, which underlie many electrical engineering and physics applications.