What can Maxwell’s Equations tell us about permanent magnets/ how are permanent magnets and electromagnets…

It makes sense that Maxwell’s equations tell us that there are no monopoles, but can the equations tell us anything else about the magnetic fields of permanent magnets on their own, i.e. without interactions with a wire/current, or how such fields arise?

I only have a facile understanding of Maxwell’s equations and I was wondering if someone who knows more than me can elaborate a bit. Permanent magnets and electromagnets must be intimately related somehow, but it seems a lot of the introductory literature emphasizes their differences.

Difference between semi-classical Maxwell Boltzmann Statistics and Boson Statistics

Since semi-classical MB assumes the indistinguishability of particles and Boson Einstein statistics similarly treats degenerate states as indistinguishable states.

What is their difference when calculating the canonical partition function?

The action of Einstein Maxwell system for arbitrary dimensions

The question is as mentioned in the title. To write the action for the Einstein-Maxwell system in arbitrary dimension.

Is it possible just to add them (The Lagrangian for gravity and for electromagnetism) linearly to the Lagrangian in order to get the action?

I have not been able to find a lot of resources regarding this on the internet, if you have something relevant please comment.

How should MTW’s derivation of the Maxwell-Faraday formula be interpreted

In the following derivation, I am not sure exactly how the components
of the final vector equation are established. I suspect this is a
situation analogous to the vector addition of infinitesimal rotations.
The discussion is in reference to section 3.4 beginning on page 80
of Gravitation by Misner, Thorne and Wheeler.

The electromagnetic field tensor is determined by writing the Lorentz
force law as a derivative of 3-momentum with respect to proper time,
and writing the derivative of energy with respect to proper time as
the time component of the particle’s momentum 4-vector. That is, the electromagnetic field tensor is defined without appeal to Maxwell’s equations.

The objective is to derive the Maxwell-Farady formula, beginning with
the invariant vanishing of the divergence of the magnetic field, and
the transformation laws for the electric and magnetic fields. We consider
an infinitesimal Lorentz boost $vec{beta}$ in the positive
$X$ direction. The following form of the magnetic part of the electromagnetic
field transformation is established by applying the Lorentz transformation
to the electromagnetic field tensor and equating terms

gammaleft(B_{y}+beta E_{z}right)\
gammaleft(B_{z}-beta E_{y}right)

Since our boost is infinitesimal, we may write $gamma=1$. The transformation
of the partial derivatives with respect to space coordinates for an
infinitesimal boost in the $X$ direction are

frac{partial}{partialoverline{x}}=frac{partial}{partial x}+betafrac{partial}{partial t};frac{partial}{partialoverline{y}}=frac{partial}{partial y};frac{partial}{partialoverline{z}}=frac{partial}{partial z}.

In terms of the barred system, the divergence law of the magnetic
field takes the form

overline{nabla}cdotoverline{mathfrak{B}}=0=frac{partial B_{overline{x}}}{partialoverline{x}}+frac{partial B_{overline{y}}}{partialoverline{y}}+frac{partial B_{overline{z}}}{partialoverline{z}}.

We rewrite this by replacing the barred terms with their unbarred

0=frac{partial B_{x}}{partial x}+betafrac{partial B_{x}}{partial t}+frac{partial B_{y}+beta E_{z}}{partial y}+frac{partial B_{z}-beta E_{y}}{partial z}.

We group factors of $beta$

0=frac{partial B_{x}}{partial x}+frac{partial B_{y}}{partial y}+frac{partial B_{z}}{partial z}+betaleft(frac{partial B_{x}}{partial t}+frac{partial E_{z}}{partial y}-frac{partial E_{y}}{partial z}right),

then apply the magnetic field divergence law in its unbarred form

nablacdotmathfrak{B}=0=frac{partial B_{x}}{partial x}+frac{partial B_{y}}{partial y}+frac{partial B_{z}}{partial z},

to arrive at the condition

frac{partial B_{x}}{partial t}+frac{partial E_{z}}{partial y}-frac{partial E_{y}}{partial z}=0.

Had the velocity of the transformation been directed in the $y-$
or $z-$ directions, a similar equation would have been obtained for
$partial B_{y}/partial t$ or $partial B_{z}/partial t$. In the
language of three-dimensional vectors these three equations reduce
to the one equation
frac{partialmathfrak{B}}{partial t}+nablatimesmathfrak{E}=mathfrak{0}.

If the boosts under consideration were relativistic in magnitude,
I don’t believe we could consider the results as constituting independent
vector components. My uncertainty in how to interpret the quoted statement
is that it implies that we are obtaining our results one at a time,
but combining them as if they all exist simultaneously. Would someone
please help me understand this?

How is the third case obeying integral form of Maxwell’s second equation?

Let $m$ denote pole strength. In the diagrams:

(1) Sky blue: Closed Gaussian surface
(2) Red: North pole of magnet
(3) Green: South pole of magnet
(4) Yellow: Part of magnet cutting Gaussian surface

Case 1: When both poles lie inside Gaussian surface

enter image description here

$$iint_S vec{B}.vec{dS}=4 pi m+4 pi (-m)=0$$

Case 2: When one pole lies inside Gaussian surface and other outside

enter image description here

$$iint_S vec{B}.vec{dS}= 4 pileft( iint_S vec{H}.vec{dS}+iint_S vec{M}.vec{dS} right)=4 pi m+4 pi (-m)=0$$

Case 3: When one pole lies on the Gaussian surface and other inside

enter image description here

Due to the inverse square nature of magnetic field intensity , $4 pi vec{H}$ at a point on the north pole due to that pole must be infinite. Therefore flux due to positive pole must be infinite. On the other hand flux due to negative pole is finite ($-4 pi m$). Hence net flux must be infinite.

But it should be zero. Where have I gone wrong?