[alancalverd]

Do Maxwell equations explain electrostatic and magnetostatic interactions?
If they don't, what does?
Is it compatible with Maxwell equations?

No, they are derived from experiments that show that a moving charge creates a magnetic field and a varying magnetic field induces a voltage in a conductor. These are essentially dynamic phenomena.

[alancalverd]

There are some materials for which we just can't - there isn't a simple scalar relating E and D fields  OR  the B and H fields.

....which is why I said for various materials.

We are quite used to dealing with hysteresis and birefringence. You could still model em propagation in nonlinear materials by saying that ε_m  is not necessarily constant, but what usually matters in practice is the overall "black box" transfer function of a dense material, not the detail of what happens inside the box.

[hamdani yusuf (OP)]

Do Maxwell equations explain electrostatic and magnetostatic interactions?
If they don't, what does?
Is it compatible with Maxwell equations?

No, they are derived from experiments that show that a moving charge creates a magnetic field and a varying magnetic field induces a voltage in a conductor. These are essentially dynamic phenomena.

Afaik, electrostatic force follows Coulomb's law. What about magnetostatic force?

[alancalverd]

Not quite an inverse square law because there is no point source of magnetic field, but Wikipedia sets out all the equations you are likely to need, and when we are dealing with the projectile effect of an MRI magnet on a steel oxygen cylinder, hammer head or nut and bolt, an inverse square law is an adequate approximation from 3 meters distance until the projectile becomes supersonic at about 0.5m from the patient.

[paul cotter]

Hamdani, the biot-savart equation is the counterpart to coulomb, describing the magnetostatic.

[hamdani yusuf (OP)]

Hamdani, the biot-savart equation is the counterpart to coulomb, describing the magnetostatic.

Can it describe interaction between two permanent magnets?
How about a magnet and a small ferromagnetic material?

[Eternal Student]

Hi.

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  =  χ_m H   with  χ_m = 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 = χ_mH )  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.

[paul cotter]

Thanks for that ES, I was fairly sure Hamdani would be looking for elaboration and you saved me a lot of work and I very much doubt I could have supplied such a clear and concise description. There is one slight addition I would like to make, purely for the general reader. Coulomb and Biot Savart equations look very similar but there is one important difference: the source of these forces is the charge and current respectively- a differential charge can be any charge added or subtracted but a differential current does not exist as current flow requires continuity and only an integral can be used. I haven't expressed that to my satisfaction, didn't sleep much last night and i'm wooly headed.

[alancalverd]

To be honest I'm surprised Alancalverd uses a 1/r2 law as an approximation at any range

It's good enough to persuade clinical folk not to take ferromagnetic objects inside the magnet cage without boring them with divs and curls. I'm sure you could calculate the precise acceleration of a flat-screen television in a 3T field, but my problem was how to extract the debris without scaring the next patient.

[hamdani yusuf (OP)]

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)

This article shows the derivation.

The net force acting between the dipole and point entity X will be:
F_D = k X x / (R-δ /2)²  - k X x / (R+δ /2)²
we can rewrite the above in the form:
F_D = [kXx/R²] / (1-δ /2R)² - [kXx/R²] / (1+δ /2R)²
For the condition δ <<2R, which was set as one of our assumptions, we are justified to apply the
binomial approximation (1+x)ⁿ
≈ 1+nx, or 1/(1+x)ⁿ
≈ 1-nx, valid for x<< 1. This reduces:
1/(1-δ /2R)² to 1+δ /R, and 1/(1+δ /2R)² to 1-δ /R
The force field equation can therefore be approximated as:
F_D ≈ [kXx/R²](1+δ /R) - [kXx/R²](1-δ /R)
F_D ≈ [kXx/R²](1+δ /R - 1 + δ /R)
F_D ≈ 2kXxδ /R³ or simply F_D ~ 1/R³

https://www.gsjournal.net/h/papers_download.php?id=1833

And the question remains. Classical theories don't tell where the quantization of energy comes from.

[Bored chemist]

I'm sure you could calculate the precise acceleration of a flat-screen television in a 3T field,

Quite possibly zero.

So it experiences a force that depends on the gradient of the B field instead of being directly proportional to B

[Bored chemist]

Classical theories don't tell where the quantization of energy comes from.

In fairness this is true.
It's more or less tautology because, in this context , "classical" means "not quantum".

(Rather than between the Baroque and the Romantic periods)

[hamdani yusuf (OP)]

AFAIK, there are at least 2 distinct quantization in electromagnetic radiation.

First is quantization of radiation energy, as expressed by Planck's energy equation E = n.h.f, which was concluded by Planck to justify his formula known as Planck's law which fits the curve from experimental data of black body radiation. In this equation, the frequency can still have any real (positive) value.

The second one is quantization of atomic radiation frequency, which is observed in spectral line emission. Balmer discovered empirical formula to describe the spectral line emissions of the hydrogen atom. Bohr interpreted it as the evidence for the existence of atomic orbitals.

[hamdani yusuf (OP)]

https://en.wikipedia.org/wiki/Planck_postulate

The Planck postulate (or Planck's postulate), one of the fundamental principles of quantum mechanics, is the postulate that the energy of oscillators in a black body is quantized, and is given by

E = n h ν
where

n is an integer (1, 2, 3, ...),
h is Planck's constant, and
ν  (the Greek letter nu, not the Latin letter v) is the frequency of the oscillator.

The postulate was introduced by Max Planck in his derivation of his law of black body radiation in 1900. This assumption allowed Planck to derive a formula for the entire spectrum of the radiation emitted by a black body. Planck was unable to justify this assumption based on classical physics; he considered quantization as being purely a mathematical trick, rather than (as is now known) a fundamental change in the understanding of the world.[1] In other words, Planck then contemplated virtual oscillators.

In 1905, Albert Einstein adapted the Planck postulate to explain the photoelectric effect, but Einstein proposed that the energy of photons themselves was quantized (with photon energy given by the Planck?Einstein relation), and that quantization was not merely a feature of microscopic oscillators. Planck's postulate was further applied to understanding the Compton effect, and was applied by Niels Bohr to explain the emission spectrum of the hydrogen atom and derive the correct value of the Rydberg constant.

For convenience in typing, I'll just use f for frequency.

Planck's energy equation E = n.h.f

This energy equation is equivalent with power equation, if we take another parameter j as h/t.
P = E/t = n.h/t.f = n.j.f
while h has a unit of Joule.second/cycle, the unit of j is Joule/cycle.
Increasing radiation power can be done be adding more oscillators, which can only be done discretely.

I think this is more intuitive, for following reason. Suppose we have a radiation source so dim that n=1 and f=1 Hz. Minimum value for E=h Joule. But radiation power is still undetermined. If it's radiated in 1 second then the power is h Watt. If it's radiated in 1000 second, then the power is h milliWatt.
The unit of Joule/cycle for j is more intuitive than Joule.second/cycle for h.
If a radiation source oscillates for 1 cycle, then the radiated energy is j Joules. If it oscillates for 1000 cycles, then the radiated energy is 1000.j Joules.

[Bored chemist]

I think this is more intuitive,

Nobody else seems to.

[hamdani yusuf (OP)]

I think this is more intuitive,

Nobody else seems to.

Does it mean I'm wrong? We haven't heard someone else's opinion about this.

I think this is more intuitive, for following reason. Suppose we have a radiation source so dim that n=1 and f=1 Hz. Minimum value for E=h Joule. But radiation power is still undetermined. If it's radiated in 1 second then the power is h Watt. If it's radiated in 1000 second, then the power is h milliWatt.

In currently more common used form of equation, radiation power is not quantized, due to unrestricted time period. Note that quantization of power equation is mathematically equivalent, but it has no flexibility to change the time period.
This type of quantization reminds me of Millikan's oil drop experiment to determine the electric charge of single electron. They can only change in a system by a discrete amount.

[Bored chemist]

In currently more common used form of equation, radiation power is not quantized,

And that's fair enough because power isn't quantised in the way that energy is.
Power isn't even a conserved quantity.

[hamdani yusuf (OP)]

Power isn't even a conserved quantity.

When power changes, where does the difference go? Does it just appear/disappear? Or Is it merely converted into something else?
Planck got the equation from his research on black body radiation, which can be reasonably generalized to electromagnetic radiation. There's no adequate justification to extrapolate it to other type of power or energy, such as gravitational potential energy.

[Bored chemist]

When power changes, where does the difference go?

You tell me.
I charge a battery slowly at a low power overnight, then, in the morning, I use that battery to deliver a lot of power briefly to start my car.

Where did the "extra" power come from?

Essentially any time we talk about "energy storage" we are using it to change power.

[Bored chemist]

There's no adequate justification to extrapolate it to other type of power or energy, such as gravitational potential energy.

Do you know that gravity affects photon energy?