# I Magnetic dipole

Tags:
1. Jan 19, 2017

### 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

2. Jan 19, 2017

### 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.

3. Jan 19, 2017

### 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

Thanks

4. Jan 19, 2017

### weirdoguy

Are you sure that is the definition of magnetic moment?

5. Jan 21, 2017

### 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.

6. Jan 21, 2017

### 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.
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 / μo = M x 107.

Last edited: Jan 21, 2017