Magnetic field from arbitrarily shaped flat coils with filamentary, ribbon, and rectangular cross sections

This report describes the derivation of three groups of equations: (1) Field components from an arbitrarily shaped filament lying in a plane. (2) Field components from an arbitrarily shaped ribbon of infinitesimal thickness with center line lying in a plane. (3) Field components from an arbitrarily shaped bar of rectangular cross section with its center line lying in a plane. In all three cases analytical expressions for the field components were found for an infinitesimal element of the cross section. These expressions are then integrated numerically along the arbitrarily shaped center line of the coil to obtain the three field components. As a check for accuracy the calculated field values of an elliptically shaped coil were compared to an existing analytic expression for a filamentary elliptical coil