# How does a magnet cause magnetic force and its magnitude/direction?

Magnetic fields are created due to electron spin in the magnet. But how exactly does an electron "spinning" create forces around the magnet? Also, the magnetic force on a charge moving in the magnetic field is $qvB\sin\theta$, and its direction is perpendicular to the field and velocity of the charge. Why are these two facts true?

I read the explanation for how magnetic force arises next to an electric current (How do moving charges produce magnetic fields?), but does the special relativity argument apply to magnets?

• youtube.com/watch?v=3D2RaDVkylY "When you ask a 'why', you have to be in some framework that you allow something to be true". That's FAR FROM the final say on this (which is why I'm not posting it as an answer), so don't let the video discourage you, but it's important to keep in mind. – user12029 Jul 18 '15 at 4:40
• I'll throw in some keywords: electron shells, d-shell metals, Pauli exclusion principle, Orbit orientation, vector product, current loop - micromagnetic models. You asked a lot in a single question. Try to figure magnetism on the level of a current loop first, then try to see electron orbits as current loops. If you want a logical explanation for force, look at energy. Force is always acting in the direction which reduces system energy, look into magnetic coenergy and energy. You are mixing too many high and low level explanations, take a step back and rethink your approach :-) – WalyKu Jul 27 '15 at 12:14
• The SE question you cite has a well-upvoted, selected answer that includes this sentence "... magnetism is nothing more than electrostatics combined with special relativity. " This way of phrasing things is not correct and can lead to misunderstanding. That line of reasoning stems from an over-generalization of a presentation in Purcell's textbook. Read the comments below that answer. – garyp Jul 29 '17 at 12:24

As was pointed out in the comments, physics answers questions within a given framework, modeled with mathematics and accepting as extra axioms laws/postulates to pick up from the mathematical solutions the ones that describe data and are predictive in new situations.

Electricity and magnetism were observed from ancient times, the word "electron" comes from the greek word for amber, because rubbing it caused electrostatic phenomena, and the word "magnet" comes from the Asia Minor region of Magnesia , where the first stones attracting and repulsing each other were found.

The theory developed slowly, with laws which described the behavior of charges and magnets, and took off with Maxwell and his equations which is classical electrodynamics.

With classical electrodynamics the behavior of electromagnets can be predicted without entering into the quantum mechanical framework.

Permanent magnets need an explanation using the spins of the quantum mechanical building blocks of matter. This framework depends on more postulates, in order to model mathematically and predict data.

The how a permanent magnet is created can be explained by how the tiny magnetic moments of electrons and nuclei had been aligned by the magnetic field of the earth during its creation, and as the metal crystallized the alignment became permanent, the tiny dipoles adding up into a large magnetic dipole. This process can be reproduced in the lab and the existing theories, classical and quantum, are very accurate in predicting the behavior.

But how exactly does an electron "spinning" create forces around the magnet?

The quantum mechanical model is a probabilistic model, the assignment of spins to electrons ( part of the axiomatic framework) and the use of the mathematics developed for electromagnetism lead to the result of magnetic forces.

Also, the magnetic force on a charge moving in the magnetic field is qvBsinθ, and its direction is perpendicular to the field and velocity of the charge. Why are these two facts true?

Because the electrodynamic models, classical and quantum mechanical, fit the observations. The two frameworks blend smoothly into each other, the classical always emerges from the quantum mechanical, but the quantum mechanical details are not needed at macroscopic levels where h_bar can be assumed to be zero.

The link you give is another mathematical transformation within the framework of quantum electrodynamics that explains the consistency of the models. It is an extra complication obscuring that electricity and magnetism are observed input effects that have been modeled successfully mathematically.

• Dear anna v, can you please tell me how those original stones, and the electrons and nuclei inside them got their magnetic dipole moment aligned? From Earth's magnetic field? – Árpád Szendrei Jul 17 '18 at 1:18
• @ÁrpádSzendrei this is a question for a geologist, but I would guess so. While the whole thing was liquid much higher magentic fields could be induced, which oriented the basic dipoles. Even now when the magnetic field of the earth is very weak, if you measure old iron radiators which have remained in place for decades, you will find there is a field aligned with the earth;s field, or so I have read someplace – anna v Jul 17 '18 at 3:19

In physics there are chains of reasoning. These chains seek to uncover the causes of given effects, and to "unpack" the complex mechanism or structure which combines together to produce given results. Such a chain of reasoning has to reach to things and structures which are deemed "simpler", and then on to things "simpler still" but eventually it has to stop at something where we have no further explanation to offer. All we can say is "this set of ideas is logical and elegant, and fits the observed behaviour". In the case of electric and magnetic effects this basic set of ideas is called quantum electrodynamics, which itself is part of the structure of a theory called "the standard model" and there are hints that this model is itself part of a rather complex set of ideas called string theory.

To answer basic questions in physics, therefore, all we can do is point to aspects of these basic descriptions and say "well this is part of the basic structure; that is what the universe is like. The best we can do is describe it in detail and show how it all hangs together neatly."

Questions about how charges produce fields, and how fields produce forces on charges, are like this. They are asking about aspects of the behaviour which are close to the fundamental structure of ideas.

In the case of a magnetic interaction, a moving charged particle, or one carrying a property called magnetic dipole moment, gives rise to a magnetic field near to it, and this magnetic field cannot suddenly disappear, so the field near the particle joins smoothly to the field further away, and so on, until eventually the field encounters some other charged particle, and exerts forces on it. So the right way to think of this is not to focus only on the particles, but also allow that the fields themselves are part of the physical setup. It is a bit like the way a ship at sea can influence another ship by causing waves or a wake, or just making the water flow differently. In the ship analogy the water is not just a by-stander; it is part of the system. In a similar way, a magnetic field is part of the magnetic system.

But I have not yet said how a charged particle can give rise to a field. It is harder to do that. The best answer turns out to be "it just does", but we can say more about it by offering precise mathematical statements.

In the case of electric and magnetic effects the basic idea is that there is mutual interaction between electromagnetic fields and charged particles. We cannot answer the question "why is there this mutual interaction". It is just a basic property. It is the meaning of the term "electric charge". If the interaction happens then we say there is electric charge present. If there is no interaction then we say there is no electric charge. The theory then tells us further properties of this charge, especially the interesting ones that it cannot suddenly appear out of nowhere, nor disappear (charge conservation); it exists along with mass; the amount of charge does not change when you observe it from different inertial reference frames. Also, the total combination of electric and magnetic fields around a charge depends on how the charge is moving. What we can say is that this is not just a hotch-potch of ideas, like a collection of random articles found in the street. Rather, it all remains consistent with a remarkably small collection of mathematical equations, called the Maxwell equations and the Lorentz force equation (when quantum effects don't need to be considered in detail), or the equations and methods of quantum electrodynamics which underpin the Maxwell and Lorentz equations. These equations can themselves be obtained by a general method called the Lagrangian method, applied to a fairly simple starting point, so this makes us judge that the overall framework can be called "simple" in the sense of "basic". (It is not simple in the sense of "easy to learn"; the mathematical methods are quite advanced.)

So, to conclude, to ask "why does a charge produce an electric and magnetic field?" is like asking "why is 2 the square root of 4?" The answer to the mathematical question is "because it just is", but we could add that $$2 \times 2 = 4$$. Similarly, a charge produces a field because that is the nature of what charge is; it is the name we give to a basic aspect of the physics. But to fill it out we can offer equations showing how the connection between charge and field looks in various cases, such as the equation $${\bf E} = \frac{q}{4 \pi \epsilon_0 r^2} \hat{\bf r}$$ for the electric field around a static point charge. The equation for the magnetic field gives a non-zero result either when the charge is in motion, or when the particle in question has magnetic dipole moment.

Ultimately the right way to think of magnetic and electric fields is not to regard them as separate, nor as if one causes the other, but rather they are different aspects of a single field called the electromagnetic field. This is bit like the way we can describe vectors in terms of one set of components or another, except here the field vectors are themselves components of a mathematical object called a tensor.

Magnets are made from metals. Metals have crystalline structures and this means that the atoms somehow are fixed. Under the influence of an external magnetic field the magnetic dipole moments of the atoms get somehow aligned and sometimes could stay aligned after removal of the external field. The magnetic dipole moment in atoms arises from the magnetic dipole moments of the electrons, the protons and neutrons too.

Now about moving particles. In this case the mentioned from you intrinsic spin comes into game. The intrinsic spin and the magnetic dipole moment of an electron are linked unambiguously. Say, the magnetic dipole moment is parallel to the spinning axis for electrons, then it is antiparallel for positrons. When move electrons in a bended wire due to gyroscopic effect the intrinsic spins get aligned and by this the magnetic dipole moments get aligned too. A coil under current flow will be a magnet.