Faraday's Law

Michael Faraday discovered that electromagnetic fields are induced when a magnetic field is changed. He also discovered that a similar effect could be created by using Alternating Current (AC). Whenever the magnetic field in a part of a circuit changed, so did the current. Faraday consolidated this into his law, and ultimately found:

The emf ε induced is the negative change in the magnetic flux Φ​​ per unit time. Any change in the magnetic field or change in orientation of the area of the coil with respect to the magnetic field induces a voltage (emf).

The magnetic flux is a measurement of the amount of magnetic field lines are piercing a given surface. This method is very very similar to the definition found in Gauss’ Law. If we have: ${\Phi }_m={\int }_S\vec{B}\cdot \hat{n}dA$ then the induced emf or the voltage generated by a conductor or coil moving in a magnetic field is: $\mathcal{E}=-\frac{d}{dt}{\int }_S\vec{B}\cdot \hat{n}dA=-\frac{d{\Phi }_m}{dt}$

NOTE: ANY OPEN surface "S" can be used to evaluate Faraday's Law as long as it's bounded by the circuit.
NOTE: THE MINUS SIGN DENOTES THE DIRECTION OF THE EMF!

The SI unit for magnetic flux is called the weber(Wb): $1Wb=1T\cdot m^2$ Many applications of Faraday’s Law has coils of wire with $N$ turns. For this reason, Faraday’s Law can sometimes be written as: $\mathcal{E}=-\frac{d}{dt}(N{\Phi }_m)=-N\frac{d{\Phi }_m}{dt}$ WARNING: TO DETERMINE THE SIGN OF THE EMF, YOU MUST USE Lenz’s Law!!!

Motional EMF

Motional EMF is a subset of Faraday’s Law that pertains to the induced electric field as a result of moving coils or moving particles. The motion of such elements creates a magnetic field, which, in turn, creates its own electric field. In a scenario above, where a rod is moved on a frictionless wire, the magnetic flux is constantly increasing because the area of the loop is also constantly increasing. We can represent the flux of the loop as follows: ${\Phi }_m=\vec{B}lx$ where $lx$ is the area of the loop. In order to find the EMF, we simply derive this expression. The only non-constant number in this scenario is $x$ and we simply derive this with respect to time. We then notice a simple, but important fact: $\frac{dx}{dt}=||\vec{v}||$ we can now rewrite our expression as follows: $\mathcal{E}=Blv$ A similar scenario can be had when you are rotating the loop, you simply have to find a way to express the magnetic flux in terms of the variables in the problem and then derive the non-constant parts.

Induced Electric Fields

The source of the work done by emfs is an induced electric field $\vec{E}$. The work done by moving a unit charge all the way around a circuit is the induced EMF: $\mathcal{E}=\oint \vec{E}\cdot \overrightarrow{dl}$ Thus, we can rewrite Faraday’s Law in terms of the induced electric field as: $\oint \vec{E}\cdot \overrightarrow{dl}=-\frac{d{\Phi }_m}{dt}={\int }_S\vec{B}\cdot \hat{n}dA$ It must be made clear that this potential difference is not because of a difference in charge. This potential difference is caused solely by a changing $\vec{B}$ field that induces a new electric field!! This electric field STILL EXISTS even if you remove the wire, and it circulates in a circle!!

There are two kinds of electric fields:

1. Changing $\vec{B}$ field induces a circular electric field (curling field).
   • Induced fields on a closed loop DO NOT EQUAL 0. $\oint \vec{E}\cdot \overrightarrow{dl}\ne 0$
2. (Net) electric charge creates an electric field (create diverging fields).
   • Electrostatic fields on a closed loop DO EQUAL 0. $\oint \vec{E}\cdot \overrightarrow{dl}=0$

The induced electric field circulates the same direction as the induced current.

Magnetic Damping:

Magnetic Damping is a phenomenon that takes advantage of changing magnetic fields and induced currents.

Consider a metal pendulum swinging through a strong magnetic field, such as in the illustration below In this example, when the plate moves into the magnetic field, it experiences a change in magnetic field. This causes an Eddy current to be formed in order to oppose the change in the magnetic field. The current generated interacts with the magnetic field and creates a force that pushes back on the pendulum.