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= Faraday's Law of Induction: Definition and Mathematical Representation =

== Introduction ==
'''Faraday’s Law of Electromagnetic Induction''' is a fundamental principle of electromagnetism discovered by Michael Faraday. It describes how a changing magnetic field within a closed loop induces an electromotive force (EMF) in the conductor. This principle is the working mechanism behind electric generators, transformers, and inductors.

== Statement of Faraday’s Law ==
The induced electromotive force (EMF) in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.

== Mathematical Formulation ==

=== Integral Form ===
<math>
\mathcal{E} = -\frac{d\Phi_B}{dt}
</math>

Where:
* <math>\mathcal{E}</math> is the induced EMF (in volts),
* <math>\Phi_B</math> is the magnetic flux through the circuit,
* <math>\frac{d\Phi_B}{dt}</math> is the time derivative of the magnetic flux.

The magnetic flux is defined as:

<math>
\Phi_B = \int_S \vec{B} \cdot d\vec{A}
</math>

Where:
* <math>\vec{B}</math> is the magnetic field (in tesla),
* <math>d\vec{A}</math> is an infinitesimal area vector perpendicular to the surface <math>S</math>.

=== Differential Form ===
Using Maxwell's equations, the differential form of Faraday's Law is:

<math>
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
</math>

This expresses that a time-varying magnetic field produces a circulating electric field.

== Physical Interpretation ==
* A changing magnetic field induces an electric field.
* The induced EMF drives a current if the circuit is closed.
* The negative sign indicates the direction of the induced EMF opposes the change in flux (as per Lenz's Law).

== Lenz’s Law ==
Lenz’s Law gives the direction of the induced EMF:

<math>
\mathcal{E} = -\frac{d\Phi_B}{dt}
</math>

The negative sign shows that the induced current will create a magnetic field opposing the change in the original magnetic flux.

== Applications ==
* Electric generators – convert mechanical energy to electrical energy using induction.
* Transformers – transfer electric power between circuits via changing magnetic flux.
* Inductive sensors – detect position or motion using electromagnetic principles.
* Electromagnetic brakes – generate resistance via induction in metallic conductors.
* Induction cooking – use changing magnetic fields to generate heat directly in cookware.

== Examples ==

=== Example 1: Rotating Loop in a Magnetic Field ===
A loop rotating in a magnetic field <math>\vec{B}</math> with angular velocity <math>\omega</math> has a time-dependent flux:

<math>
\Phi_B(t) = B A \cos(\omega t)
</math>

Then the induced EMF is:

<math>
\mathcal{E} = -\frac{d\Phi_B}{dt} = B A \omega \sin(\omega t)
</math>

== See Also ==
* [[Electromagnetic Induction]]
* [[Lenz's Law]]
* [[Magnetic Flux]]
* [[Maxwell's Equations]]
* [[Electromagnetism]]
* [[Electric Generator]]
* [[Transformer]]