magnet formulas

On-Axis Field of a Finite Solenoid

This formula uses the formula for the field due to a thin shell solenoid, integrated over a range of radii to obtain the magnetic field at any point on the axis of a finite solenoid.

Solenoid in cross section view

Solenoid in cross section view

General Case

Field inside a solenoid

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.

μo 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 G Factor

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":

G factor form of axial field due to a solenoid

where G is the unitless geometry factor:

G factor for a solenoid


definitions for G

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.

Special Case: x1=(-x2)

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

B at the coil center


B at the coil center, alternate form

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:

G factor for measuring B at coil center

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.

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