Demystifying the connection between magnetic and electric fields

Question

Part (1):

In the classical theory of electromagnesitm, as given by Maxwell, we know that by just looking at the four famous equations:

1. An electric field has a source: there are charged particles (non-zero divergeance)
2. Whereas there's no such equivalence for the magnetic field, i.e., there are no known magnetic charges.
3. And roughly that, the change in time of either field generates the other.

Despite the difference in 1. and 2., it is nonetheless known that electric and magnetic fields are just different views of the same physical thing. That is, by considering the relative motion of charges, in different frames, we observe a magnetic field being generated or equivalently a static electric field.

Although I understand the reasoning behind this, as we are simply switching frames (once being at rest w.r.t. to the charge, once being in motion relative to it), it remains still a very confusing picture.

• To help clarify matters, are we saying that based on the classical Maxwellian theory of electromagnetism, magnetism has no fundamental physical meaning, instead it's all about the behaviour of charged particles?

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Part (2):

On the other hand, in our modern theories, of QM and QFT, we quickly learn about a new fundamental physical property other than the charge, namely the spin, and how it is at the very core of everything in magnetism. Taking simple toy models such as all the Ising variants, we explain all sorts of magnetic behaviours (ferromagnetism, paramagnetism,...and phase transitions between them) based on the understanding of how spins interact, how they can be locked in blocks of same orientation, how they respond to an external field, and so on.

Moreover, unlike the concept of charge, spin extends to photons as well, where mathematically we assign half-integer spins to fermions (electrons e.g.) and integer ones to bosons (photons). Compared to the starting discussion of the classical theory, the contrasting feature here is the fact that magnetism deals with the spin properties of a system and not the charges, meaning that there does not seem to be a dual equivalence anymore between magnetic fields and electric fields based on QM.

• Is there a way to meaningfully connect these two pictures? I.e., that of the classical theory of electromagnetism to the modern understanding of charges, spins in QM? For example, we know that part (1) is a macroscopic theory, so as a consistency check, is it possible to retrieve the results therein, but starting from the modern picture? (i.e., the collective behaviour of fermions)

• In our modern understanding of the electromagnetic theory, considering relativistic and quantum mechanical corrections, do we still treat magnetic fields and electric fields as different views of the same thing?

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This has been my attempt at claryfing what is confusing me, hopefully the questions are not too vague as they stand, please let me know if any additional details and clarification are required. Although this post is not necessarily a literature recommendation one, any books or papers that you think will help me better understand this whole matter, are perfectly welcome.

Answer 1

The main difference between electricity and magnetism is, as you say, that there are no magnetic charges. The field generated by a localized source can be expanded in a multipole series. This tells us immediately that while the electric field is sourced by charges ("electric monopoles"), the dominant source of magnetic field is a magnetic dipole moment.

In classical electromagnetism, there is only one source of magnetic dipole moment, that is, a localized distribution of electric currents. Quantum mechanics enters the game by telling us that there is additional magnetic moment, associated with the spin of charged particles. This then leads to different physical phenomena, associated with the orbital magnetic moment (diamagnetism) and the spin magnetic moment (para- and ferromagnetism). Why the latter is much stronger than the former is a different question, and I will not go into details.

Altogether, the electromagnetism of atoms is described by the same old Maxwell's theory as one uses for macroscopic phenomena. The only novelty is the presence of a new source of magnetic fields, coming from spin. There is no additional source of electric fields, i.e. no new "quantum charge", on the quantum level.

By the way, it is often stated that the magnetic field can be understood as a relativistic effect. Although you do not say this so explicitly, your formulation "magnetism has no fundamental physical meaning, instead it's all about the behaviour of charged particles" goes in the same direction. Such statements should be taken with a grain of salt. It is only true for very special situations (such as a single charge moving at a constant velocity) that the magnetic field can be completely generated from electric field and charge by switching to a different reference frame. For more complicated charge and current distributions, there will be some magnetic field in any frame. On the other hand, it is true that as a consequence of relativity and gauge invariance, the fundamental object in Maxwell's theory is the electromagnetic field strength tensor, which requires the existence of both the electric and the magnetic field. In that sense, one can say that magnetism is a necessary consequence of the relativistic theory of electric charges.

Answer 2

The other answers have given a very thorough description of the physics involved, but I wonder if it is worth taking a step back for a broader view.

If I understand your question correctly you are concerned that the fundamental component of the magnetic field is a dipole, so we have the dichotomy that electric fields are generated by monopoles while magnetic fields are generated by dipoles.

But a magnetic dipole is not really a fundamental object. Classically a magnetic dipole is generated by charges flowing in a circle - the obvious example is an electromagnet. So magnetic dipoles originate from electric charges. Life gets more complicated when we move to the spin of elementary particles because there is no simple classical analogy for spin. It does not mean the charge is rotating in the sense that a macroscopic object rotates. Nevertheless, the magnetic dipole generated by a fermion is directly related to the angular momentum of that fermion just as the macroscopic magnetic dipole generated by a current loop is related to the angular momentum of the charges flowing in the loop.

In this context I would argue that it is misleading to treat the magnetic dipole of an elementary particle as a fundamental building block for the magnetic field. Charge is still the origin of the field. For example uncharged elementary particles like the photon, Z and Higgs do not generate a magnetic dipole. (Composite neutral particles like the neutron can have a magnetic dipole moment, but this is because they contain charged particles).

Answer 3

Consider a static point charge $q$, we know that this charge will induce an electric field $$\vec{E} = \frac{q \vec{r}}{4 \pi \epsilon_0 r^3}$$ Now consider the same situation from a frame moving with (nonrelativistic) velocity $\vec{v}$. In this frame the charge is moving and thus corresponds to a point-like current $\vec{j} = -q \vec{v}$. This, in the non-relativistic limit, does not change the electric field but adds a magnetic field due to the Biot-Savart law $$\vec{B} = -\frac{\mu_0}{4 \pi} \frac{q}{r^3} \vec{v} \times \vec{r}$$

In other words, the point charge looks like a charge plus a spatial current in a different frame. Furthermore, an electric field looks like an electric field plus an additional magnetic field. This means that, in fact, the two equations of electromagnetism connected by relativistic transformations must be (assuming stationarity of fields) $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \leftrightsquigarrow \nabla \times \vec{B} = \mu_0 \vec{j}$$ If the magnetic fields fall off at infinity (which they should in a real physical situation), there will be no magnetic fields without the presence of currents. The electric field lines may diverge out of a charge, but the magnetic field lines need a current to "wind around".

I.e., even in a completely non-relativistic setting we find electric and magnetic fields being mixed because a charge from a moving frame looks like a current. When we consider special relativity we find also that motion reduces our perception of the magnitude of the charge and thus the electric field. The electric and magnetic field thus act as communicating vessels - reduce one and increase the other.

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If you want to know how this "duality" works out in special relativity, I give a brief description here. There are three spatial components of a current, and only a single component of the charge density. But then again, there are three spatial dimensions and one time dimension in our glorious four-dimensional space-time. One may then easily guess that the charge density and spatial current organize into a single "four-current" with the components $$\mathbf{j} = (\rho c, j_x,j_y,j_z)$$ The field strengths $\vec{E}, \vec{B}$ then organize into a space-time, $4 \times 4$, anti-symmetric matrix called the Faraday tensor $$\mathbf{F} = \begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\ & 0 & -B_z & B_y \\ \vdots & \vdots & 0 & -B_x \\ & & \cdots & 0 \end{pmatrix}$$ It is then easy to see that the "source" Maxwell equations are characterized simply as $\partial_\mu F^{\mu\nu} = \mu_0 j^\nu$, where the indices $\mu,\nu$ run through all space-time components (now this also includes non-stationary situations). A similarly simple law holds for the other "source-free" set of equations. Both the four-current and the Faraday tensor then transform as real tensors in Minkowski space-time according to the prescriptions of special relativity.

There exist frames where locally some of the components of the Faraday tensor or the four-current vanish, but the transformations push their components around, like a fluid in communicating vessels. This is the point of the connection between electric and magnetic field and the notion that they are different aspects of the same, electromagnetic field. Then again, if the current is purely spatial, you will not make a purely static charge by changing into any frame and there will thus always be a magnetic field in every physical frame in this case. This means that magnetic fields certainly aren't any nonphysical ghost or something similar.

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As for spin, you may simply understand it as a tiny "quantum loop" of current due to the fact that the particle is a spinning charge. This is more or less the point of view which can be taken for most electromagnetic interactions of particles with spin.

Quantum theory does not really change the point of view on electromagnetism in this sense. If anything, it makes the marriage between electricity and magnetism even tighter by quantizing the electromagnetic field into photons, tiny electromagnetic waves where "electric" comes with "magnetic" in one snug package which cannot be unwound.