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New Theories / Re: Where does quantization of energy of electromagnetic radiation come from?
« on: 05/06/2023 17:01:50 »
Hi.
In general Maxwell's equations are used for Magnetostatics. As @paul cotter stated, some are more relevant than others and the Biot-Savart law is useful. You can get the Biot-Svart law out of Maxwell's equations with minimal assumptions.
They won't model the force between permanent magnets and ferromagnetic materials on their own. You need to make a few additions:
(i) Assign a magnetic substance a (vector) Magnetisation M. This depends on the properties of the material. LaTex still isn't working so you can't have any formulas, only words. It is the vector sum of magnetic dipole moments per unit volume of the material.
(ii) You also have to understand what a magnetic dipole is. In general they can always be modelled as being equivalent to a microscopic loop of current. (We have terminology like "bound current" to describe the equivalent current that would create the dipole that exists in a place. Sometimes that can be thought of or identified as an actual current, e.g. an electron whizzing around an atom. Sometimes it can't. Bound current is not usually an actual current but just a way of modelling a magnetic dipole).
(iii) Make assumptions about how magnets and magnetic materials behave. The microscopic explanation doesn't necessarily need to concern us (but there is one, with varying degrees of sophistication and accuracy). For the macroscopic behaviour you only need to know how the Magnetisation, M, of a magnetisable material will vary with an applied field. So, for linear materials we have M = χm H with χm = a constant called the magnetic susceptibility. The H field for a simple linear material is just a linear multiple of the B field, B = μH with μ being the permeability of the material (rather than μ0 as for a vacuum).
I think that's enough additions. With that you will be able to determine the forces between magnets and/or simple linear magnetic materials. Basically all magnets are just a collection of magnetic dipoles and magnetic materials become a collection of dipoles in their Prescence. So the final determination is entirely based on the fields and forces exerted on and by magnetic dipoles.
I think there was a previous post or two that discussed non-linear materials. That's more complicated.
As outlined by @alancalverd , dipoles don't follow an inverse square law for the field strength they produce. At large enough distances, it's an inverse cube law. (To be honest I'm surprised Alancalverd uses a 1/r2 law as an approximation at any range - but very close it isn't a perfect 1/r3 law, which we both agree on)
To further complicate it, in addition to the thing that created the field being a dipole, the thing that is acted on by the field is also a dipole. So it experiences a force that depends on the gradient of the B field instead of being directly proportional to B (we have F = Bm for a simple monopole of magnetic charge m but F = ∇(B.m) for a dipole moment m).
So at large distances a dipole experiences a force from another dipole that falls off ~ 1/r4. This assumes that both dipoles were permanent and unchanging dipoles. If only one had been a permanent magnet and the other is merely a magnetisable material, then (by the relationship M = χmH ) the magnetisable material becomes less magnetised the further away it gets from the permanent magnet. So the force existing between the permanent dipole and the magnetisable material falls off with an even higher power (~ 1/r7 ).
I know formulas are not always displaying well. See https://en.wikipedia.org/wiki/Force_between_magnets#Magnetic_dipole%E2%80%93dipole_interaction
for some discussion of constructing the 1/r4 formula for the force between two magnetic dipoles.
Best Wishes.
Can it describe interaction between two permanent magnets?
How about a magnet and a small ferromagnetic material?
In general Maxwell's equations are used for Magnetostatics. As @paul cotter stated, some are more relevant than others and the Biot-Savart law is useful. You can get the Biot-Svart law out of Maxwell's equations with minimal assumptions.
They won't model the force between permanent magnets and ferromagnetic materials on their own. You need to make a few additions:
(i) Assign a magnetic substance a (vector) Magnetisation M. This depends on the properties of the material. LaTex still isn't working so you can't have any formulas, only words. It is the vector sum of magnetic dipole moments per unit volume of the material.
(ii) You also have to understand what a magnetic dipole is. In general they can always be modelled as being equivalent to a microscopic loop of current. (We have terminology like "bound current" to describe the equivalent current that would create the dipole that exists in a place. Sometimes that can be thought of or identified as an actual current, e.g. an electron whizzing around an atom. Sometimes it can't. Bound current is not usually an actual current but just a way of modelling a magnetic dipole).
(iii) Make assumptions about how magnets and magnetic materials behave. The microscopic explanation doesn't necessarily need to concern us (but there is one, with varying degrees of sophistication and accuracy). For the macroscopic behaviour you only need to know how the Magnetisation, M, of a magnetisable material will vary with an applied field. So, for linear materials we have M = χm H with χm = a constant called the magnetic susceptibility. The H field for a simple linear material is just a linear multiple of the B field, B = μH with μ being the permeability of the material (rather than μ0 as for a vacuum).
I think that's enough additions. With that you will be able to determine the forces between magnets and/or simple linear magnetic materials. Basically all magnets are just a collection of magnetic dipoles and magnetic materials become a collection of dipoles in their Prescence. So the final determination is entirely based on the fields and forces exerted on and by magnetic dipoles.
I think there was a previous post or two that discussed non-linear materials. That's more complicated.
As outlined by @alancalverd , dipoles don't follow an inverse square law for the field strength they produce. At large enough distances, it's an inverse cube law. (To be honest I'm surprised Alancalverd uses a 1/r2 law as an approximation at any range - but very close it isn't a perfect 1/r3 law, which we both agree on)
To further complicate it, in addition to the thing that created the field being a dipole, the thing that is acted on by the field is also a dipole. So it experiences a force that depends on the gradient of the B field instead of being directly proportional to B (we have F = Bm for a simple monopole of magnetic charge m but F = ∇(B.m) for a dipole moment m).
So at large distances a dipole experiences a force from another dipole that falls off ~ 1/r4. This assumes that both dipoles were permanent and unchanging dipoles. If only one had been a permanent magnet and the other is merely a magnetisable material, then (by the relationship M = χmH ) the magnetisable material becomes less magnetised the further away it gets from the permanent magnet. So the force existing between the permanent dipole and the magnetisable material falls off with an even higher power (~ 1/r7 ).
I know formulas are not always displaying well. See https://en.wikipedia.org/wiki/Force_between_magnets#Magnetic_dipole%E2%80%93dipole_interaction
for some discussion of constructing the 1/r4 formula for the force between two magnetic dipoles.
Best Wishes.
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