OnAxis Field Due to a Current Loop
This simple formula uses the law of Biot Savart, integrated over a circular current loop to obtain the magnetic field at any point along the axis of the loop.  
Current loop in cross section view. 
B is the magnetic field, in teslas, at any point on the axis of a current loop. The direction of the field is perpendicular to the plane of the loop. is the permeability constant (1.26x10^{6} H/m) i is the current in the wire, in amperes. r is the radius of the current loop, in meters. x is the distance, on axis, from the center of the current loop, in meters. 
Special Case: x = 0  
Special
Case: x >> r
Note that this is equivalent to the expression for on axis magnetic field due to a magnetic dipole: where A is the area of the current loop, or 
