Magnetic Dipole: Discovering μ and Its Origin

[enerieire]

does anyone knows where this formula comes from?

μ=½B(R^3)

I am considering a sphere of radius R, with B its magnetic field. Who is μ?

Thanks

 

[Baluncore]

Starting with; μ = ½ B r³, we can apply dimensional analysis to the problem.
B is measured in tesla = N⋅m⁻¹⋅A⁻¹
Multiply B by m³ to get; μ = N⋅m²⋅A⁻¹
We know force; N = kg⋅m⋅s⁻²
So; μ = (kg⋅m⋅s⁻²)⋅(m²⋅A⁻¹)
μ = m³⋅kg⋅s⁻²⋅A⁻¹
Which has exactly the same dimensions as;
See tables; https://en.wikipedia.org/wiki/SI_derived_unit
magnetic moment = weber⋅meter = m³⋅kg⋅s⁻²⋅A⁻¹
So μ is the magnetic moment.

 

[enerieire]

Ok, that's right. But where does it comes from?

Starting from the classical definition for the magnetic moment for a charge distribution

μ=∑qr

I don't find an answer

Thanks

 

[weirdoguy]

Starting from the classical definition for the magnetic moment for a charge distribution

μ=∑qr

Are you sure that is the definition of magnetic moment?

 

[vanhees71]

We need the precise context of which problem you want to solve to help you. If you mean the induced dipole moment of a paramagnetic or diamagnetic medium by applying an external magnetic field, see Jackson, 3rd edition, Sect. 5.11.

 

[Baluncore]

There are many well trodden paths through this field. But the OP equation seems to be one or two steps off the path.
I agree we need more context to find the path again.
Maybe page 2 of this article will help; http://ccmc.gsfc.nasa.gov/RoR_WWW/presentations/Dipole.pdf
I quote:
The first equation of the dipole field in spherical polar coordinates (r,θ,φ)
is; B = 2 M cos θ / r³
where M is the dipole moment, which can be positive or negative.

From eqn (1) we get; M = ½ B r³ / cos θ
But on the dipole axis θ = 0, so; Cos θ = 1.
So; M = ½ B r³

The text box in the bottom corner gives the explanation and;
μ = 4π M / μ_o = M x 10⁷.