# An interesting infinite series

• A While I was was numerically integrating the magnetic field caused by an infinite array of magnetic moments, I observed the interesting limit ( limit (1) in the image). It may seem difficult to prove it mathematically but from the physic point of view, I think it can be proved relatively easily.

If we set i=0 ( no summation on i), the sum seems to have a limit too, but I have not not able find it in a close form ( for example in terms of pi). Any comments?

#### Attachments

Last edited:

Homework Helper
Gold Member
(1) is interesting, and I can almost follow what you are doing, (knowing the magnetic field of a magnetic dipole, and it looks like you are computing the x-component of the magnetic field at the origin of something that is magnetized in the x-direction), but what is the geometry of the array? It looks to me like you may be trying to compute the result for a rectangular block, but the geometry that gets the uniform internal field with the well-known result everywhere including the origin is a sphere.

@Charles Link Yes, you got it right! The array is the result of discretizing an infinitely long rectangular prism whose edges are parallel with x, y, and z axis of Cartesian coordinate system. into cubic cells and. The prism is uniformly magnetized in x direction and I have excludes the field due to the central cell itself. Note that the dimension of the prism is infinite in all three dimensions but the prism length is infinitely longer than the other cross-sectional dimensions. This geometry does not give uniform field everywhere and this limit is for the center if the prism. A rectangular prism is somehow similar to an infinitely long cylinder for which the magnetic field is uniform everywhere. The demagnetizing factor of such cylinder is 0.5 and for the prism too, the field in the center is related to the magnetization by factor 0.5.

Last edited: