# The apparent arbitrariness in Electromagnetism

In studying the world, physicists found that, in order to completely understand what today we call "Electromagnetism", it was necessary to do three things: find an equation which describes the electric field under all circumstances, find an equation which describes the magnetic field under all circumstances, and find an equation which establishes the way these fields act upon matter.

To achieve this, they did three things:

1. Invented the concepts of divergence and curl.
2. Invented Helmholtz's Theorem, which shows that knowing the divergence and curl of a field, you can find the aforementioned equations which completely describe the field.
3. Found the divergence and curl of electric and magnetic fields.

My two questions are:

Isn't this way of describing things rather arbitrary? I mean, the concepts of divergence and curl were kind of taken out of the blue. It's sort of as if physicists said: "let's invent these two properties of fields, calculate them for electric and magnetic fields, and describe these fields using these concepts we just made up."

1. Is it possible to describe the electric and magnetic fields in different ways, which employ completely different concepts?

2. Where can I refer to in order to understand the historical development of the concepts of divergence and curl?

I'm trying to understand what is the line of thought which led from empirical experiments and data towards these seemingly arbitrary concepts.

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mathman
Divergence and curl are mathematical concepts related to vector fields (a mathematical concept). They are used in physics since they work.

Matterwave
Gold Member
Actually, the Aharanov-Bohm effect suggests that the Vector potential (and possibly the scalar potential...I'm kind of rusty on this) are more "real" than just merely mathematical constructs to help us evaluate the EM field. This effect suggests that even in regions of space where there is no EM field (e.g. outside of a solenoid), the vector potential CAN have a direct effect on the trajectory of particles.

1) Curl and Divergence were not "invented" to understand electromagnetism. Here is a little history.

William Rowan Hamilton (1805-1865) may have used the del (also called nabla) as a general purpose symbol or abbreviation for whatever operator he wanted to introduce at any time. In 1837 Hamilton used the del in Trans. R. Irish Acad. XVII. 236 (1837). See also Transactions of the Royal Irish Academy, 18 (1839), pp. 171-259.

The concept of the curl first occurs in an 1839 paper by James MacCullagh entitled “An essay towards a dynamical theory of crystalline reflexion and refraction” presented to the Royal Irish Academy on December 9, 1839, and published in Transactions of the Royal Irish Academy, vol 21.

James Clerke Maxwell added the displacement current term to Ampère's circuital law and this enabled him to derive the electromagnetic wave equation (actually eight equations) in his 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is anelectromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in 1887. The term Maxwell's equations originally nowdays applies to modified versions of four of these equations that were grouped together in 1884 by Oliver Heaviside, concurrently with similar work by Willard Gibbsand Heinrich Hertz. These equations were also known variously as the Hertz-Heaviside equations and the Maxwell-Hertz equations, and are sometimes still known as the Maxwell–Heaviside equations.

So del and curl were shorthand ways of signifying mathematical operations, and were not "invented" to understand electromagnetism.

Bob S

look at Gauss' law for example.

It states, that the flux of $$\vec E$$ through an arbitrary closed surface, that framed the volume $$\mathcal{V}$$, is always

$$\oint \limits_{\mathcal{S}} \mathrm d\vec S \cdot \vec E(\vec r) \sim Q$$

That can be proven empirically! For example with a Van de Graaff generator (which can be the source of a large electric field). It is possible to calculate the charge Q distributed on the metal sphere of the generator. In addition you can measure the flux of $$\vec E$$ easily, because of the spherical symmetry you only have to measure $$\vec E$$ in a fixed distance from the center of the metal sphere.

Experiments shows, that for an arbitrary surface Gauss' law is exact. Since the equation above is related to Gauss' divergence theorem, you can write

$$\int \limits_{\mathcal{V}} \mathrm d^3r \, \vec \nabla \cdot \vec E(\vec r) = \oint \limits_{\mathcal{S}} \mathrm d\vec S \cdot \vec E(\vec r) \sim Q = \int \limits_\mathcal{V} \mathrm d^3r \rho_q(\vec r)$$

so

$$\int \limits_{\mathcal{V}} \mathrm d^3r \, \vec \nabla \cdot \vec E(\vec r) \sim \int \limits_\mathcal{V} \mathrm d^3r \rho_q(\vec r)$$

Cause the integral is the same on each side, that yields

$$\vec \nabla \cdot \vec E(\vec r) \sim \rho_q(\vec r)$$

So, if you can show, that the flux of $$\vec E$$ through a closed surface is always proportional to the total charge $$Q$$ inside the volume $$\mathcal V$$, you can deduce that

$$\vec \nabla \cdot \vec E(\vec r) \sim \rho_q(\vec r)$$
is valid! So it's not arbitrary and it lasted a long time till Maxwell's equations were formulated in the commonly known modern form.

Thank you all for your replies and for the historical references. I think perhaps I haven't expressed myself well. I understand that Maxwell's Equations are true, in the sense that they describe accurately what can be seen in experiments. But you have to agree with me that there are several ways, mathematically, to describe the electromagnetic field. There's the Vector Calculus approach we use, there's also the Geometric Algebra approach, which condenses all four Maxwell's Equations into one, and I'm sure other approaches can be though of, which don't necessarily use the concepts of divergence and curl. It's because of this that I say the "divergence and curl" approach is arbitrary. The questions are:

1. Do you know of other mathematical descriptions (other than the ones I mentioned) which are perhaps simpler and more elegant?

2. Out of all the ways of describing the fields that exist, why did physicists go with "divergence and curl", and not different approaches?

Matterwave
Gold Member
If multiple mathematical models exist, then the simplest one should be used most of the time. The other models can help when doing specific problems.

For example, Newtonian mechanics and Lagrangian mechanics describe the world through two very different mathematical models. Newtonian mechanics is used MOST of the time because it's the simplest to use conceptually and mathematically. One doesn't need to learn calculus of variations to use Newtonian mechanics. However, some problems (especially ones with constraints) are easier to do using Lagrangian mechanics. In those cases, one uses Lagrangian mechanics. (It is also easier to generalize Lagrangian mechanics to QED, or w/e)

In this sense, it is not arbitrary which model we stick with.

Andy Resnick
<snip>The questions are:

1. Do you know of other mathematical descriptions (other than the ones I mentioned) which are perhaps simpler and more elegant?

2. Out of all the ways of describing the fields that exist, why did physicists go with "divergence and curl", and not different approaches?

1) yes- differential forms. Physicists are behind the mathematicians, is the answer for #2.

Something else, perhaps a bit off-topic: Helmholtz's Theorem states that knowing the divergence and curl of a field, one can find an equation of the form field=something which completely describes the field. Can someone link me to these equations for the electric and magnetic fields? Why don't we see them more often?

Stingray
The definitions of divergence and curl are not arbitrary. The divergence differentiates a vector field and returns a scalar field. The curl returns a vector field. Other than multiplication by an overall constant, there are essentially no other ways to do these things using just the geometric structure of (3D) Euclidean space.

I mean arbitrary in the sense that we could have done it differently, which is true, since we could have built the theory through geometric algebra, or, as Andy puts it, through differential forms. I think Andy pretty much answered the questions: we could have described the fields differently, in a simpler manner, through more advanced mathematics, and the reason we didn't is because at the time the theory was being developed the mathematics didn't exist. The reason we still don't is because of tradition. (I have seen some users of this forum advocate in favor of teaching Electromagnetism through geometric algebra, which seems to be very elegant and intuitive.)

Just to re-enforce my other question: Helmholtz's Theorem states that knowing the divergence and curl of a field, one can find an equation of the form field=something which completely describes the field. Can someone link me to these equations for the electric and magnetic fields? Why don't we see them more often?

Stingray
You can't completely determine the field from knowledge of the divergence and and curl. For example, you can add pure radiation to any solution of Maxwell's equations and it will still be a solution. If you extend things out to cover all of $R^3$, it won't fall off at infinity. In short, you need to specify boundary conditions.

The Helmholtz theorem just states that you can write a general vector field in terms of a scalar and vector potential. An equation for the curl only affects the vector potential, and an equation for the divergence involves only the scalar potential. That decoupling is useful, but does not completely solve the problem in the way you seem to be suggesting.

Born2bwire
Gold Member
You can't completely determine the field from knowledge of the divergence and and curl. For example, you can add pure radiation to any solution of Maxwell's equations and it will still be a solution. If you extend things out to cover all of $R^3$, it won't fall off at infinity. In short, you need to specify boundary conditions.

The Helmholtz theorem just states that you can write a general vector field in terms of a scalar and vector potential. An equation for the curl only affects the vector potential, and an equation for the divergence involves only the scalar potential. That decoupling is useful, but does not completely solve the problem in the way you seem to be suggesting.

This is a very good point, the uniqueness theorem requires that we specify boundary conditions. For example, in electrostatics we use Poisson's equation to relate the scalar potential to the electric field. If we have a zero electric field, without boundary conditions we can arbitrarily set the scalar potential to be any static number like 10 V, 0 V, -pi V, etc.

Either way, I would not agree that we rarely see Helmholtz decomposition in electromagnetics. The use of the scalar potential is ubiquitous in electrodynamics, particularly statics. In addition, in quantum electrodynamics, the scalar and vector potentials are the true primitives of the electromagnetic fields. Instead of working with operators for the electric and magnetic fields, the potentials are the operators and the fields are the observables. As Matterwave previously stated, this relationship can be explicitly seen in the Aharonov–Bohm effect.

I would just reiterate what has been previously stated. One can express classical electromagnetics via different mathematical forms. The two most common are the integral and differential forms. These are not the only forms, as Bob_S mentioned Maxwell originally had 8 equations using, I believe, quarternions. However, the current vector calculus form is readily accessible, compact, and a very powerful tool. Classical electromagnetics has been pretty much a complete theory, at least in its primitive equations, for over 125 years now. Over the many years that it has been taught, studied, and applied, the current form has risen as the preferential formulation by no mere accident.

I see. Thank you all for your replies, I think I understand this subject much better now.

PS.: But even though I mentioned Helmholtz's Theorem, I don't really understand it well and I can't find a good, clear source to read about it online. Perhaps I'll have to start a topic about it. Wikipedia ain't doing it this time!

One doesn't need to learn calculus of variations to use Newtonian mechanics..
When Newton was given the Brachistochrone problem in 1696

http://mathworld.wolfram.com/BrachistochroneProblem.html

he solved it in one day, and it is often said that he invented the calculus of variations to do it. So some Newtonian mechanics problems do require calculus of variations.

Bob S