Electromagnetism Physics Problems: Get Expert Help with Plainmath

How many neutral points can be obtained in a given plane perpendicular to the length of the two parallel wires conducting current in the same direction? Explain, Neglect earth's magnetic field of a bar magnet NS

What is tangent law of magnetism?

Railgun magnetic force does no work?
Q1) The Lorentz magnetic force on a moving electron is always perpendicular to its motion; so no work is done. Yet, why is work done in railgun operation.
Q2) For every action, there must always be a reaction. The recoil force of the railgun will be transmitted to the structure holding the rails. But what is the actual mechanism that transfers the force of reaction?

I have a confusion regarding how magnetic field propagate in a medium (ferromagnets or air etc..). It is knows when a ferromagnet is exposed to an external magnetic field, the magnetic domain of the ferromagnet will be aligned with the external magnetic field. My question is this alignment of domain assure the propagation of the external magnetic field in the ferromagnet? The same question for the air. Do the magnetic field propagate in the air because the magnetic domains of the air are aligned?

Why are there more diamagnetic substances in nature than there are paramagnetic or ferromagnetic substances?

I know that there is some relationship between electric current and magnetism, but I am having trouble pinning down the exact relationship. Is electric current a necessary condition for the existence of magnetic field? And is electric current a sufficient condition for the existence of magnetic field?
In other words, is it true that "there is electric current if and only if there is a magnetic field"?

Magnetic force between 2 moving charges
If 2 positive charges are moving in the same direction with a velocity v relative to the ground frame, will they experience a magnetic force? I’ve read several times that the velocity in the Lorentz force law is relative to the ground frame, not to the object creating the magnetic field, so I feel like there should be a magnetic force.
Moreover, assuming there is a force, I tried computing the velocity at which the electric and magnetic force cancel each other, and got the speed of light. Is this just a coincidence, or is there any proper reasoning behind this?

How can i derive the dynamic of a relativistic charged particle in a uniform magnetic field $B=(0,0,B)$?

I am trying to get the relation between developing a Ginzburg-Landau theory, let's say for a ferromagnet with magnetization field $\stackrel{\to <!--\; \to \; -->}{m}=\stackrel{\to <!--\; \to \; -->}{m}(\stackrel{\to <!--\; \to \; -->}{r})$, and the formal expansion of the free energy density $\mathcal{F}=\mathcal{F}(\stackrel{\to <!--\; \to \; -->}{m})$ in terms of a Taylor series.
Considering an isotropic ferromagnet, the lowest-order terms in our Ginzburg-Landau theory should be given by
$\mathcal{F}=\frac{r}{2}{\stackrel{\to <!--\; \to \; -->}{m}}^2+\frac{U}{4}{\left({\stackrel{\to <!--\; \to \; -->}{m}}^2\right)}^2+\frac{J}{2}[{\left({\mathrm{\partial <!--\; \partial \; -->}}_x\stackrel{\to <!--\; \to \; -->}{m}\right)}^2+{\left({\mathrm{\partial <!--\; \partial \; -->}}_y\stackrel{\to <!--\; \to \; -->}{m}\right)}^2+{\left({\mathrm{\partial <!--\; \partial \; -->}}_z\stackrel{\to <!--\; \to \; -->}{m}\right)}^2]$
with r<0 and U,J>0.
However, when I think of a Taylor expansion of $\mathcal{F}(\stackrel{\to <!--\; \to \; -->}{m})$ around the origin
$\mathcal{F}={\mathcal{F}}_0+{\stackrel{\to <!--\; \to \; -->}{m}}^T\cdot <!--\; \cdot \; -->D\mathcal{F}+\frac{1}{2}{\stackrel{\to <!--\; \to \; -->}{m}}^T\cdot <!--\; \cdot \; -->D^2\mathcal{F}\cdot <!--\; \cdot \; -->\stackrel{\to <!--\; \to \; -->}{m}+\dots <!--\; \dots \; -->$
this is giving me terms of all the individual powers in $\stackrel{\to <!--\; \to \; -->}{m}$ , which are either zero or identified with the r- and U-term, but no gradient terms for the J-term? How to motivate these through a Taylor expansion?

Let us suppose that there is a block of a ferromagnetic material inside a very long ideal solenoid, the magnetic field lines inside are in a straight line perpendicular to the surface of the block at both the ends.Why does the magnetic field inside the block increases? It is known form the boundary conditions of magnetic fields that the perpendicular or the normal component of the magnetic field $\stackrel{\to <!--\; \to \; -->}{B}$ doesn't change when it goes from one medium to other.So, what's happening here?

How to derive ferromagnet Lagrangian from the Heisenberg model: $H=-<!--\; -\; -->J_{ij}{S}_i{S}_j$ ? I understand how to obtain potential energy term, but it is not clear how to get "kinetic energy".
$L=\int <!--\; \int \; -->d^{2}x\frac{\mathrm{\rho <!--\; \rho \; -->}}{2}[\frac{({\mathrm{\partial <!--\; \partial \; -->}}_t\mathrm{\theta <!--\; \theta \; -->}{)}^2}{v^2}-<!--\; -\; -->(\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\theta <!--\; \theta \; -->}{)}^2]$

Why is magnetic field vector perpendicular to magnetic force vector?
So recently in physics class, we learned about the magnetism right hand rules.
One of them states that the index finger points in the direction of the velocity of a particle, the middle finger points in the direction of the magnetic field, and the thumb points in the direction of the magnetic force.
I'm curious why the magnetic field vector is perpendicular to the magnetic force vector.

The inverse square law for an electric field is:
$E=\frac{Q}{4\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\epsilon <!--\; \epsilon \; -->}}_0{r}^2}$
Here:
$\frac{Q}{{\mathrm{\epsilon <!--\; \epsilon \; -->}}_0}$
is the source strength of the charge. It is the point charge divided by the vacuum permittivity or electric constant, I would like very much to know what is meant by source strength as I can't find it anywhere on the internet. Coming to the point an electric field is also described as:
$Ed=\frac{Fd}{Q}=\mathrm{\Delta <!--\; \Delta \; -->}V$
This would mean that an electric field can act only over a certain distance. But according to the Inverse Square Law, the denominator is the surface area of a sphere and we can extend this radius to infinity and still have a value for the electric field. Does this mean that any electric field extends to infinity but its intensity diminishes with increasing length? If that is so, then an electric field is capable of applying infinite energy on any charged particle since from the above mentioned equation, if the distance over which the electric field acts is infinite, then the work done on any charged particle by the field is infinite, therefore the energy supplied by an electric field is infinite. This clashes directly with energy-mass conservation laws. Maybe I don't understand this concept properly, I was hoping someone would help me understand this better.

Magnetic force between two point charges
I tried to derive the magnetic force between two point-charges for iterative computation. Starting out with Lorentz force and Biot–Savart law for a point charge.
$\stackrel{\to <!--\; \to \; -->}{F}=q_2(-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{B})$
$\stackrel{\to <!--\; \to \; -->}{B}=(\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}v\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}x)(\frac{q_1}{||\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{x}|{|}^3}\frac{{\mathrm{\mu <!--\; \mu \; -->}}_0}{4\mathrm{\pi <!--\; \pi \; -->}})$
And got this for the answer by direct subtitution:
$\stackrel{\to <!--\; \to \; -->}{F}=q_2(-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->(\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}v\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}x)(\frac{q_1}{||\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{x}|{|}^3}\frac{{\mathrm{\mu <!--\; \mu \; -->}}_0}{4\mathrm{\pi <!--\; \pi \; -->}}))$
It does not seem to be right for several reasons.
1.It seems as if electron would be attracted to the nucleus by both magnetic and electrostatic force. Considering hydrogen atom.
2.It is possible to show that between non-moving particle and a non-moving wire with current in it should exist magnetic force. (Magnetic force acts between charge carriers in the wire and the point-charge. Non-moving particles in the wire do not have influence on the magnetic force)
Where have I gone wrong and how to find the correct expression for the magnetic force between two point-charges? Does the equation hold if $\mathrm{\Delta <!--\; \Delta \; -->}v<<c$? If this equation proves to be wrong, what would be the correct approach?

State Gauss’ Law for Magnetism. What significant thing does it tell us about the universe?

If an electric field $\mathbf{E}$ exists at a point in free space then the energy density in that point is $\frac{1}{2}{\mathrm{ϵ<!--\; ϵ\; -->}}_0|\mathbf{E}{|}^2$
When a field with same magnitude $|\mathbf{E}|$ exists in a material with relative permettivity k, the energy density becomes $\frac{1}{2}{\mathrm{ϵ<!--\; ϵ\; -->}}_{0}k|\mathbf{E}{|}^2$, i.e. it increases k times.
A similar analysis for energy density in magnetic field $\frac{1}{2{\mathrm{\mu <!--\; \mu \; -->}}_0}|\mathbf{B}{|}^2$ states that: if the field $\mathbf{B}$ exists in a material with relative permeability k (consider a ferromagnetic material, for example) then $\mathrm{\mu <!--\; \mu \; -->}=k{\mathrm{\mu <!--\; \mu \; -->}}_0$ and thus the energy density decreases k times (?). But this does not seem right because inductors with ferromagnetic cores store more energy.

I was wondering what is the difference between a spin-fluid Heisenberg Magnet and spin-glass Heisenberg Magnet. As far as I understand in a spin-glass the spins are randomly oriented compared to a ferromagnet material but I don't quite understand what we mean by spin-fluid phase. And also in the article I am reading its mentioned that spin-fluid phase is found to be generically "gapless", but I am not sure what we mean by phase being gapless and how is it related to the spin phase of the material?

Gauss's Law for magnetism is
$\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->B=0$
This allows us to write the magentic field B as the curl of another field the magnetic vector potential, A.
$B=\mathrm{\nabla <!--\; \nabla \; -->}\times <!--\; \times \; -->A$
This adhers to $\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->(\mathrm{\nabla <!--\; \nabla \; -->}\times <!--\; \times \; -->A)=0$
However, if a monopole does exist then we have
$\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->B={\mathrm{\rho <!--\; \rho \; -->}}_m$
Where ${\mathrm{\rho <!--\; \rho \; -->}}_m$ is some magnetic charge density however with a magentic vector potential this violates the equation, $\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->(\mathrm{\nabla <!--\; \nabla \; -->}\times <!--\; \times \; -->A)\ne <!--\; \ne \; -->0$
Does that mean if magnetic monopoles does exist, that the magnetic field can no longer be defined by a magnetic vector potential? In which case how was dirac able to still define the magnetic field by a magnetic vector potentials?

A metal conductor of length 1 m rotates vertically about one of its ends at angular velocity 5 rad/s . If the horizontal component of earth’s magnetism is $2\times <!--\; \times \; -->{10}^{-<!--\; -\; -->5}T$, then voltage developed between the two ends of the conductor is ?

Consider one-dimensional ferromagnet namely N spin-1/2 objects placed around a circle with the Hamiltonian
$\mathcal{H}=-<!--\; -\; -->\mathcal{J}\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}{\stackrel{\to <!--\; \to \; -->}{\mathcal{S}}}_n\cdot <!--\; \cdot \; -->{\stackrel{\to <!--\; \to \; -->}{\mathcal{S}}}_{n+1}$
where we assume the periodic boundary condition ${\stackrel{\to <!--\; \to \; -->}{\mathcal{S}}}_{N+1}\equiv <!--\; \equiv \; -->{\stackrel{\to <!--\; \to \; -->}{\mathcal{S}}}_1$ and $\mathcal{J}>0$
I'm trying to show that total spin ket is a good quantum number that is they commute with $\mathcal{H}$ and finding out the energy corresponding to
$|{\mathrm{\psi <!--\; \psi \; -->}}_0⟩<!--\; ⟩\; -->=|\uparrow <!--\; \uparrow \; -->{⟩<!--\; ⟩\; -->}_1\otimes <!--\; \otimes \; -->|\uparrow <!--\; \uparrow \; -->{⟩<!--\; ⟩\; -->}_2\otimes <!--\; \otimes \; -->\cdots <!--\; \cdots \; -->\otimes <!--\; \otimes \; -->|\uparrow <!--\; \uparrow \; -->{⟩<!--\; ⟩\; -->}_N$
By definition:
${\mathcal{S}}^2={\left(\underset{n}{\sum <!--\; \sum \; -->}{\stackrel{\to <!--\; \to \; -->}{\mathcal{S}}}_n\right)}^2=\underset{n}{\sum <!--\; \sum \; -->}{\mathcal{S}}_n^2+\underset{i,j}{\sum <!--\; \sum \; -->}{\stackrel{\mathcal{\to <!--\; \to \; -->}}{\mathcal{S}}}_i\cdot <!--\; \cdot \; -->{\stackrel{\mathcal{\to <!--\; \to \; -->}}{\mathcal{S}}}_j$
The second term has our Hamiltonian but there are other terms also. I don't understand, How do I proceed from here?