How uniform is magnetic field in current loop?

Answer 1

A current loop, by itself, does not produce a very uniform magnetic field. People use a Maxwell coil, Helmholtz coil, or a long solenoid, when they want a relatively uniform magnetic field.

What is the magnetic field in a current loop?

There are two magnetic fields in a current loop. There is the magnetic field caused by the current, such as what is found in a straight wire, and is given by B=ui/2pr where B is the magnetic field; u is the permeability constant; i is the current; p is pi; r is the radial distance from the wire. If the wire is now circular and has a radius R, then one can calculate the magnetic field inside the wire loop. Granted this is complex, but this is the idea.

The second field is perhaps a little bit more practical, but really never discussed. One can solve this problem by assuming a vector A, the current density, then take the curl of vector A, and this is the magnetic field inside the current loop. The question is what is the vector A? The current density vector inside the loop is the product ir/R and a unit vector function representing a circle. This current density is only valid for r less than or equal to R. Here the r is measured from the center of the circle. For r greater than or equal to r, the current density is the product iR/r and the unit vector function representing a circle. This is complicated by the selection of the coordinate system representing the circle. My preferance is to use spherical coordinates, but most books use Cartesian coordinates, and as such the expressions are complicated i.e., r in spherical coordinates is r but in Cartesian coordinates is (x^2+y^2+z^2)^1/2. I hope this gives some insight to the question. I do have a solution in spherical coordinates, but cannot furnish it because of my inability to use greek letters.