Hong Life CalculatorsUnderstanding the magnetic moment calculator — concept and principle
This calculator finds a magnetic moment from three kinds of source: a current loop or coil (μ = N I A), the spin-only moment of an ion with unpaired electrons, or the orbital moment of an electron with quantum number l. It then gives the moment in Bohr magnetons and the maximum torque and energy difference in a chosen field. It is used in physics and inorganic chemistry courses.
How it is calculated
Current loop: μ = N × I × A
Spin-only: μ = √(n (n + 2)) μ_B for n unpaired electrons
Orbital: μ = √(l (l + 1)) μ_B
Maximum torque = μ B; energy difference between opposed and aligned = 2 μ B
- μ_B = eħ ÷ (2mₑ) ≈ 9.274 × 10⁻²⁴ J/T; 1 A·m² = 1 J/T. Area is entered in cm².
The principle
Any circulating charge is a magnetic dipole. For a coil, the moment is simply the current times the enclosed area times the number of turns. At the atomic scale, electrons carry moments from their orbital motion and from their spin, measured in Bohr magnetons. In many transition-metal ions the orbital contribution is largely quenched by the surrounding atoms, so the spin-only formula gives a useful estimate that is compared with measured magnetic susceptibility to count unpaired electrons.
Worked example
Default inputs: current loop, N = 100, I = 0.5 A, area 12 cm², B = 0.2 T.
- μ = 100 × 0.5 × 12 × 10⁻⁴ = 0.06 A·m².
- In Bohr magnetons: 0.06 ÷ 9.274 × 10⁻²⁴ = 6.47 × 10²¹.
- Maximum torque = 0.06 × 0.2 = 0.012 N·m.
- Energy difference = 2 × 0.06 × 0.2 = 0.024 J = 1.498 × 10¹⁷ eV.
| Number of turns N | Magnetic moment μ (A·m²) |
|---|---|
| 25 | 0.015 |
| 50 | 0.03 |
| 75 | 0.045 |
| 100 | 0.06 |
| 125 | 0.075 |
| 150 | 0.09 |
| 200 | 0.12 |
How to read the result
The Bohr-magneton figure shows how enormous a coil's moment is compared with one atom. For ions, the spin-only value rises with the number of unpaired electrons: about 1.73 μ_B for one, 3.87 μ_B for three and 5.92 μ_B for five. Comparing a measured moment with these values estimates how many electrons are unpaired.
Common mistakes
- Entering the area in m² when the field asks for cm².
- Applying the spin-only formula to lanthanide ions, where orbital contributions matter.
- Confusing the moment's magnitude with its component along the field.
Frequently asked questions
What is a Bohr magneton?
The natural unit of electron magnetic moment, about 9.274 × 10⁻²⁴ J/T.
Why is it called "spin-only"?
It ignores the orbital contribution, which is often quenched in transition-metal complexes.
How many unpaired electrons give 4.90 μ_B?
Four, since √(4 × 6) ≈ 4.90.
Does the field change the moment?
No. It changes the torque and the energy, not the moment itself.
Sources: OpenStax, University Physics Volume 3 (2016), section 8.2 "Orbital Magnetic Dipole Moment of the Electron" · OpenStax, University Physics Volume 2 (2016), section 11.5 "Force and Torque on a Current Loop"