magnet formulas

B is the magnetic field, in teslas, at any point on the axis of the solenoid. The direction of the field is parallel to the solenoid axis.

μ_o is the permeability constant (1.26x10^-6 H/m or Tm/A, 1.26x10^-4 Tcm/A or 4.95x10^-5 Tin/A for coils measured in meters, centimeters and inches, respectively)

i is the current in the wire, in amperes.

n is the number of turns of wire per unit length in the solenoid.

r₁ is the inside radius of the solenoid.

r₂ is the outside radius of the solenoid.

x₁ and x₂ are the distances, on axis, from the ends of the solenoid to the magnetic field measurement point.

Note that the units of length may be meters, centimeters or inches (or furlongs for that matter), as long as the correct value of the permeability constant is used.

The field can be expressed in a form that separates the unit system, power and winding configuration from the unit-less geometry of the coil. This introduces the "G Factor":

where G is the unitless geometry factor:

where,

P is the total power consumed by the coil, in watts.

λ is equal to the total conductor cross section area divided by the total coil cross section area, which ranges from 0.6 to 0.8 in typical coils.

ρ is the conductor resistivity, in units of ohms-length. The length units must match those of r₁.

When the magnetic field measurement point is at the center of the solenoid:

or...

j is the bulk current density in the coil cross section, in amps per unit area.

l is the length of the solenoid.

N is the total number of turns of wire in the coil.

The unitless geometry factor G is simply:

Note that G is maximum when α=3 and β=2. A coil built with a given inner diameter and input power will deliver the highest central field strength when these conditions are met.