# [Physics] How eddy current brakes function

electromagnetisminduction

Take the following example:

where a rectangular sheet of metal is entering a constant magnetic field at $v \dfrac{m}{s}$. Due to Faraday's law of induction + Lenz's law, we can state that an eddy current will be generated to oppose the increase of magnetic flux through the sheet of metal, so as to produce a magnetic field coming out of the page (represented by the red dots). Intuitively, I believe that this induced magnetic field should act as a 'brake' on the metal plate, as Lenz's law implies that the induced current should always in some way act against the motion, but I don't see how to calculate this 'retarding' force that would act to reduce the plate's speed?

I will use a very crude model to find the maximum restoring force, whereas the real force will be smaller. Imagine the disk to be made of rectangular loops, such that their horizontal parts are superimposed at the upper and lower edges of the plate, while their vertical parts are alined consecutive to each other and each having "width" $dx$. Then the EMF along each rectangular loop will be $Blv$, where $l$ is the vertical length of the plate, and $v$ the velocity at which the disk is moving into the magnetic field. The current is then $I = Blv/R = Bvt\sigma dx$, where $t$ is the thickness of the plate and $\sigma$ is the conductivity of the material. The restoring force due to each loop is thus $dF = lB^2 vt\sigma dx$, which implies that the maximum force is
$F_{max} = \mathcal{V}vB^2 \sigma$
where $\mathcal{V}$ is the volume of the plate within the magnetic field. This crude model implies that the largest possible force occurs right before the disk is completely within the magnetic field region, and abruptly vanishes upon entering said region.