What do we mean with magnetic monopole and dipole? I can not find a
way to relate magnetic monopoles and dipoles with electric ones. I do
not understand their outcomes.
Luckily, there exists a truly amazing one-to-one correspondence between magnetism and electricity.
Monopole in magnetism is analogous to charge in electrostatics/electricity. Just like we have two types of charges (+ve & -ve), we have two types of monopoles (north & south). The north magnetic pole is also known as the positive pole and the south magnetic pole is known as the negative pole.
We refer to the size/magnitude of the charge as charge itself, but for a monopole, we refer to its size/magnitude as pole strength.
Electric charge is denoted by the letter 'q', the pole strength of a magnetic pole is denoted by the letter 'm'.
Electric charges produce electric fields. The magnetostatics analog is the magnetic field.
We make use of electric field lines to represent the electric field visually. By convention, electric field lines start from a positive charge and terminate at a negative charge. So is the case with magnetic poles as well. Magnetic field lines originate from the north pole (+ve pole) and terminate at the south pole (-ve pole).
The strength of the magnetic field produced by a monopole is given by
$$\vec{B} = (\frac{\mu_o}{4\pi})\frac{m}{r^3}\vec{r}$$
You probably noticed that the equation is identical to Coulomb's law except for the fact that we have magnetic pole strengths instead of the magnitude of charge.
As of 2016, we aren't sure if magnetic monopoles exist (we haven't found one yet). When we try to make a monopole by slicing a bar magnet exactly in the middle, we end up creating two new bar magnets where each magnet has both north and south pole.
Well, you might ask how do I know about the behavior of monopoles if monopoles don't exist. Whatever I said above is just a hypothesis and the hypothesis is consistent with the reality.
When two magnetic poles of equal pole strength are kept close to each other, we call it a dipole. A bar magnet is an example of a dipole.
We have an electric dipole moment equivalent in magnetostatics. We call it magnetic dipole moment (usually represented by '$\vec{M}$').
$$M = md$$
where d is the distance between the two poles and m is the pole strength of the poles. The direction of the magnetic dipole momentum is from negative pole to the positive pole (In the case of an electric dipole, it is from the negative charge to the positive charge).
The field along the axis of the magnetic dipole is given by,
$$B = 2(\frac{\mu_o}{4\pi})\frac{M}{r^3}$$
and the field along the equatorial line of the dipole is given by
$$B = (\frac{\mu_o}{4\pi})\frac{M}{r^3}$$
If you open your textbook and look for the formulae of the electrostatics analogue of the dipole, you will find that the formulae are a perfect match.
The formulae given above can also be derived from the magnetic monopole hypothesis.
The similarities don't end here.
Torque on an electric dipole in an external uniform electric field is given by
$$\tau = \vec{p}\times\vec{E}$$
Torque on a magnetic dipole in an external uniform magnetic field is given by
$$\tau = \vec{M}\times\vec{B}$$
You shouldn't be surprised if you find more exact matches. The magnetic monopole behaves just like electric charge and the formulae are identical. The derivations for magnetic dipoles are identical to derivations for the electric dipoles so you should end up with the same formulae.
A circulating electric current behaves like a magnetic dipole.
If you derive the formula for magnitude of magnetic field along the axis of the circular coil, you should get something similar to the following,
$$B = \frac{\mu_o NiR^2}{2(R^2 + x^2)^{\frac{3}{2}}}$$
where $N$ is the number of turns in the coil, $i$ is the current passing through the wire, $R$ is the radius of the coil and $x$ is the distance along the axis of the coil.
If we use the approximation $x>>R$, we get,
$$B = \frac{\mu_oNiR^2}{2x^3}$$
Rearranging further, we get
$$B = 2(\frac{\mu_o}{4\pi})\frac{(iN\pi R^2)}{x^3}$$
If you look carefully, you will notice that the formula looks very similar to the formula which gives the strength of the field along the axis of a magnetic dipole.
We can define magnetic dipole moment as
$$M = iN(\pi R^2)$$
The direction of the magnetic moment is the direction of the magnetic field at the center of the coil.
Now the formula looks pretty much identical to the formula given earlier for the strength of the field along the axis of a magnetic dipole.
In general, magnetic moment for any closed loop circuit can be defined as
$$\vec{M} = iN\vec{A}$$
where $\vec{A}$ is the unit normal vector for the loop.
- Also,what is their role in Gauss' law for magnetism (the net magnetic
flux through a closed surface is zero)?
I read that the magnetic dipoles are essential for the meaning of this law. Why?
If magnetic monopoles existed,then why would the law not be valid?
- Lastly,why do we say that the magnetic field is divergeless?
One of the consequences of monopoles not existing is that magnetic field lines are always closed. Consider any dipole, a field line which emanates from the north pole must end up back at the south pole.
The number of field lines that leave the surface is equal to the number of field lines entering the surface. Hence, there is no net flux entering or leaving the surface.
$$\oint{\vec{B}.d\vec{A}} = 0$$
In the Gauss Law for electrostatics, we consider the volume charge density or the net charge enclosed within the gaussian surface. The same goes for Gauss Law for magnetism.
$$\oint{\vec{B}.d\vec{A}} = \mu_o m_{enclosed}$$
since there can never be any isolated monopole, $m_{enclosed}$ is always zero.
If monopoles existed, then $m_{enclosed}$ needn't be zero. Hence, the Gauss law would be invalid.
Some physicists try to find a magnetic monopole (something Dirac tried
to explain i think). So if they actually find one, what does that
mean?
If we ever find a monopole, it would imply that Gauss law for magnetism would be invalid. We will need to amend the law to make allowance for monopoles. (the amended version of the law was presented in the answer to your previous question)
Discovery of magnetic monopoles would imply the existence of magnetic currents. We would have a new chapter in our physics textbooks. Possibly a new engineering branch might appear (magnetronics?).
For a classic magnet(N-S) we know that we have charges inside moving
in a circular motion and they althogether form that magnetic field of
the magnet. So how does finding a magnetic monopole change this? What
does it mean for those small currents?
Every electron has two types of magnetic dipole moment, namely spin magnetic dipole moment and orbital magnetic dipole moment.
There are several electrons in an atom and most electrons cancel out each other's magnetic moment. When all the electrons cancel out neatly, the total magnetic moment of the atom is zero. The substances where the total magnetic moment of the constituent atoms is zero are known as diamagnetic substances.
There are many substances with unpaired electrons. This gives rise to a net magnetic moment for each atom. When these atoms collectively align in the same direction. This kind of collection of atoms is known as a domain. When domains align to give a net magnetic moment to a substance, the substance is said to be ferromagnetic.
Discovery of a magnetic monopole shouldn't affect the existing theories. However, it might force us to investigate further at the fundamental level.
Best Answer
Since the magnetic monopole acquires a significant amount of energy, you cannot just treat it like a (magnetic) test charge that moves passively on and on in the supposedly constant solenoidal magnetic field. Only the whole system encompassing the solenoidal magnetic field, the electric current that creates it, the moving monopole and finally the solenoidal electric field that will be generated by it, must obey energy conservation.
The situation is easier to understand, if you take an infinite straight line current as the generator of the original magnetic field. Then the magnetic flux density at a distance $r$ from the line current has constant magnitude and its field lines are strictly circular.
Now place a cylindrical shell of magnetic monopoles around the electric line current at radius $r$. Then these monopoles will start moving circularly around the line. By symmetry of Maxwell's equations with magnetic monopoles, these circularly moving monopoles are similar to an infinitely long "coil" in electrical engineering, and hence, will generate a homogeneous electric field inside of them, which is then, of course, directed in parallel to the electric line current that generated the original magnetic field. Hence, this additional electric field will decelerate the line current and weaken the magnetic field, which in turn leads to attenuated acceleration of the magnetic monopole shell.
After all, potential energy of the original magnetic field and kinetic energy of the initially moving charges will be converted into potential energy of a newly generated electric field and kinetic energy of the moving magnetic monopoles. Total energy will stay the same, however.
When you take this simple example to your non-ideal magnetic dipole field and a magnetic point charge instead of a shell, only the details get more complicated, while the principles stay the same.
Basically, this is all pretty similar to Lenz' rule in classical electrodynamics, only adding magnetic monopoles: the effect of induction (when applied to magnetic monopoles) counteracts its cause and so makes sure that energy is conserved.
Additional note: be aware of the fact that classical electrodynamics (i.e. without magnetic monopoles) also allows closed electric field lines. Although these only occur during radiation, they principally exist for a certain duration. If your argument were true, you could also apply it to closed electric field lines, and so energy conservation would be violated already for Maxwell's theory. But of course, energy is also conserved there, again because of the "test charge" acting back on the "external field".
Yet another note: Actually, energy conservation is a surprisingly non-trivial topic in classical electrodynamics. While in classical mechanics, you are always assuming a complete Lagrangian from whose explicit time-independence you can derive energy conservation, there is no such complete Lagrangian for classical electrodynamics with massive charges. This is known as the problem of radiation damping, and it is one of the things that motivated the development of quantum mechanics. In quantum electrodynamics radiation damping can be treated consistently. But until the discovery of quantum electrodynamics, physicists had to assume that energy was "just conserved" although they could not strictly prove it from theory (at least not in the presence of charged masses). On the other hand, if there are no charges/masses, energy conservation in classical electrodynamics is trivial, but then again, what use is a "dry" theory of vacuum fields that cannot act on something material?