Engineering
Solenoid Magnetic Field & Helmholtz Coil Pair Uniform Field formulas
Calculate axial magnetic flux density inside a long solenoid B = mu_0 * (N / L) * I, midpoint uniform field of a Helmholtz coil pair B = (4/5)^(3/2) * mu_0 * N * I / R, and magnetic energy density.
Ideal air-core coil estimates
The solenoid expression approximates the axial field well inside a long, uniformly wound air-core solenoid, away from its ends. Its length must be large compared with its radius; the supplied worksheet inputs do not establish that geometry. It is not a finite-coil field map or a calibration certificate.
The Helmholtz expression assumes two identical coaxial circular thin coils separated by their common radius, each with N turns and equal currents producing fields in the same axial direction. It gives the midpoint field; the field is approximately uniform near that midpoint, not everywhere. Symmetry makes the first axial derivative zero; separation equal to radius also cancels the second derivative.
B2 is a rounded approximation to vacuum permeability. In the revised SI, vacuum permeability is experimentally determined rather than exactly defined as 4*pi*10^-7. Use meters and amperes for geometry and current; results are millitesla. Ferromagnetic cores, saturation, finite winding thickness and heating are outside this model.
| Electromagnet Geometry | Formula (in millitesla mT) | N = 500, I = 4 A (L=0.25m, R=0.10m) | Zero Current (I = 0 A) |
|---|---|---|---|
| Long Solenoid Interior B | (mu_0 * (N / L) * I) * 1000 | 10.0531 mT | 0.0000 mT |
| Helmholtz Coil Midpoint B | (((4/5)^1.5) * (mu_0*N*I/R)) * 1000 | 17.9835 mT | 0.0000 mT |
| Solenoid Energy Density u_B | (B_tesla ^ 2) / (2 * mu_0) | 40.2124 J/m^3 | 0.0000 J/m^3 |
Long Solenoid Axial Magnetic Flux Density B (mT)
N is the total number of solenoid turns, a positive integer. Signed current sets the axial field direction; zero current gives zero field. Negative field values mean reversed direction, not negative field magnitude.
=IF(COUNT(B2:B3,B4,B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B3<>INT(B3),B4<=0),"Invalid inputs",ROUND((B2 * (B3 / B4) * B5) * 1000, 4)))Helmholtz Twin-Coil Uniform Midpoint Field B (mT)
N is the number of turns in each coil, not the sum for the pair. The coil radius must be positive. The formula does not use the independent solenoid length B4.
=IF(COUNT(B2:B3,B6,B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B3<>INT(B3),B6<=0),"Invalid inputs",ROUND((((4 / 5) ^ 1.5) * (B2 * B3 * B5 / B6)) * 1000, 4)))Magnetic energy density in vacuum (J/m^3)
Use the unrounded solenoid field in tesla, squared and divided by twice the permeability. Reversing the current leaves this nonnegative energy density unchanged.
=IF(COUNT(B2:B3,B4,B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B3<>INT(B3),B4<=0),"Invalid inputs",ROUND(((B2 * (B3 / B4) * B5) ^ 2) / (2 * B2), 4)))Try the example
Use the original values or change them to test your understanding.