Magnetic induction calculator (solenoid)

Magnetic induction B (in tesla) describes the strength of a magnetic field. It determines the Lorentz force on moving charges and is fundamental to motors, transformers and sensors.

It follows from Ampere's law applied to an ideal solenoid. For an infinite coil the interior field is uniform and equals mu0*n*I. Mu0 = 4pi*10^-7 T*m/A is the permeability of free space.

Mu0 is the magnetic permeability of free space, equal to 4pi*10^-7 T*m/A in SI. It describes how strongly an electric current produces a magnetic field in vacuum.

B (magnetic flux density, in tesla) is the total magnetic field. H (magnetic field intensity, in A/m) is the field contribution from free currents alone, excluding magnetisation. In vacuum B = mu0 * H.

Increase the number of turns N, shorten the solenoid length L, or raise the current I. Inserting a ferromagnetic core multiplies B by the material's relative permeability (e.g. iron: ~1000).

Energy density = B^2 / (2*mu0) in J/m^3. It tells you how much energy is stored per unit volume of field. Important in coil and energy-storage design.

B = mu0*n*I is exact for an infinitely long solenoid (length >> diameter). For short coils the result is approximate; Nagaoka's correction or numerical methods are needed for precision.

B is measured in tesla (T). 1 T = 1 V*s/m^2 = 1 Wb/m^2. In practice millitesla (mT) and microtesla (µT) are common. The CGS unit gauss is also used (1 T = 10 000 G).

The vacuum formula does not. However coil resistance rises with temperature, so at constant voltage current I falls and B decreases. For superconducting coils temperature must stay below the critical temperature Tc.

Solenoids and electromagnets are used in MRI scanners (1.5–3 T), particle accelerators, electric motors, solenoid valves and induction heaters. Computing B is a key design step for all of these.