Magnetic flux density from eddy currents in a circular plate

[Orlando]

How would I calculate the magnetic flux density of the magnetic field generated by eddy currents induced in a circular plate? I decided it would be reasonable to approximate this by considering the flux density of a current loop. However, it is my understanding that eddy currents are induced throughout the entire plate and are not confined to a circular loop which is hollow in the centre. I am not sure if that would ultimately influence the accuracy of the equation but if it does, how could I alter the equation for the flux density of a current loop to include the entire conduction area?

B(current loop)= Mu_0 * I / (2*r) where r=radius of loop.

 

[marcusl]

This is not a problem that can be solved by elementary physics. It involves at least two diificulties:
First, eddy currents are governed by a vector diffusion equation which is considerably more complex even than the scalar diffusion equation that describes heat flow. Second, your geometry is three dimensional, so symmetry arguments and reductions in the number of variables cannot be made. (For example, the three-dimensional magnetic field from your simple-sounding current loop involves complete elliptic integrals of the first and second kinds.) There is only one eddy current calculation that can be considered simple, and that is from a plane wave at normal incidence to an infinite conductive slab. This case can be found in undergraduate texts such as Reitz and Milford. Many graduate texts such as Jackson, in fact, cover no more than this.

One exception is the wonderful E&M book by Smythe, who covers eddy currents in a variety of geometries. (As an aside, Smythe's text has a reputation for difficulty, and is rather feared in the graduate student community .) Beyond that you are into specialty literature. I used to own a text called something like "Eddy Currents in Linear Conducting Media" that solved many challenging geometries, including, I seem to recall, the one you are asking about. I got rid of it maybe 15 or 20 years ago, because it was printed on acidic paper and the pages turned brown and brittle.