Faraday's Law - Yousef's Notes
Faraday's Law

Faraday's Law

Faraday’s law states that a changing Magnetic Flux induces an Electromotive Force (emf). Namely, the induced emf in a closed loop equals the negative rate of change of magnetic flux through the loop. $$ \mathcal{E}=-\frac{d \Phi_{B}}{d t}=-\frac{d}{dt}\int \vec{{B}} \cdot d \vec{A}=-\frac{d}{dt}\int B d A \cos \phi $$ Generalization of a coil with N turns, $$ \mathcal{E}=-N \frac{d \Phi_{B}}{d t} $$
  • Only a dynamic flux produces an emf, not a static flux.
  • A dynamic flux is not necessarily produced by a dynamic magnetic field. The vector area $\vec A$ could also vary in time!
  • Why the negative sign? It gives right the direction of the induced current.
  • Experimental law that summarizes the experiments of Michael Faraday and Joseph Henry in the beginning of the XIX century.
  • This law is at the heart of any electric generating station, which converts other forms of energy into electric energy(gravitational energy at hydroelectric plants, chemical energy in a coal-fired plant, or nuclear energy in a nuclear power plant)

https://youtu.be/Hh58afwzHfA

#How to find the direction of the emf

  1. Fix direction of vector area $\vec A$.
  2. In combination with the magnetic field, determine the sign of the magnetic flux (through the dot product $\vec{B} \cdot d \vec{A}$)
  3. Determine the sign of the induced emf or current using Faraday’s law: when the flux increases, the induced emf or current decreases and vice versa.
  4. Finally, use the right-hand rule to determine the direction of the induced emf or current. For a positive emf, it is in thee same direction as your curled fingers, and negative otherwise.

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#Induced Electric Field

We can understand the emf induced in a moving conductor based on Lorentz forces, but what drives the charges around the circuit when the conductor is still and there is only a changing magnetic flux?

Let us re-express Faraday’s law using the definition of an emf

$$ \oint_{C} \vec{E} \cdot d \vec{l}=-\frac{d}{d t} \int_{S} \vec{B} \cdot \vec{n} d A $$

There has to be an induced electric field in the conductor caused by a changing magnetic flux!

Remarks:

  • Contrary to what we study for a static fields, this electric field does not fulfill $\oint_{C} \vec{E} \cdot d \vec{l}=0$, hence it is not conservative and the concept of potential is meaningless! Only when $\frac{d \Phi_{B}}{d t}=0$ we recover a conservative field $\vec E$. Nonetheless, the definition $\vec F=q\vec E$ is always correct.