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I used to get confused when I first learned the electromagnetic field is said to have a momentum. A few years later, I came across an argument for why 0.999... = 1: When faced with the task of defining what a recurring decimal means, we as a community decided to make the definition consistent with the rules of algebra of "normal" numbers, paying the cost of this prima facie counter-intuitive result.

When faced with a similar task in electrodynamics, we apparently chose to keep the theory consistent with the conservation laws of mechanics, at the cost of assigning energy and momentum to seemingly "non-corporeal" entities called fields.

Why was this choice made? Why salvage conservation laws that seem "wrong"?

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  • $\begingroup$ If you know that light is an electromagnetic wave, isn't it pretty obvious that electric and magnetic fields must carry energy? $\endgroup$ – Ben Crowell Apr 13 '16 at 4:51
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    $\begingroup$ This question is a bit over-dramatic. $\endgroup$ – David Hammen Apr 13 '16 at 13:07
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Keep in mind that "intuitions" are personal, 0.999... = 1 may be counterintuitive to you but natural to others, they also strongly depend on historical and cultural context. In 19th century first the wave optics and then Maxwell's electrodynamics strongly suggested existence of a material carrier for light and electromagnetic waves, the ether. Young conducted experiments which seemingly allowed one to "see and touch" the ether, he even constructed a "ripple tank" to demonstrate the analogy between water waves and light, see How did Young perform his double slit experiment? Faraday conducted experiments with magnetic fields demonstrating how his "lines of force" literally materialize when using metal filings, see his On the Physical Character of the Lines of Magnetic Force.

Even for gravity action at a distance was considered undesirable, so much so that a material intermediary was sought to explain it. Laplace in 1799 even researched a modification of Newtonian gravity with finite propagation speed, and Mossotti in 1830 proposed a hypothesis where gravity is the imbalance in the electromagnetic attraction and repulsion, see What 19th century developments contributed to the General theory of Relativity? Many 19th century mathematicians (Cauchy, Navier, Green, Stokes, etc.) were constructing elastic theories of ether, it was thought to be a special solid, and waves in it to be a case of elastic waves. In that context physicists did not see fields as "incorporeal", elastic waves carry momentum and energy in a rather literal sense of the words, they can knock you off your feet in a particular direction. Hertz's 1887 experiments demonstrated that electromagnetic waves were as corporeal, they could also do mechanical work.

If the ether was just another mechanical system then it was only natural that it should obey mechanical conservation laws, and Hamiltonian reformulation of dynamics turned momentum and energy into fundamental notions of mechanics. Even after such a rationale was removed at the beginning of 20th century others took its place, the Noether theorem demonstrated a fundamental relation between conservation laws and symmetries, so conservation of momentum and energy came to be viewed as expressing fundamental symmetries of space and time.

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Regarding 0.999... = 1, that's a just another representation of 1. Best advice: Get over it! There are lots of representations of $e$, for example $\sum_{n=0}^{\infty} \frac 1 {n!}$, or $\lim_{n->0} (1+n)^{1/n}$, and a host of others.

There are also lots of representations of 1 besides 0.999....

A couple of very simple tricks in mathematics are to add zero to an expression or multiply an expression by one (but zero is represented as $a-a$ rather than $0$, where $a$ is an expression of choice, or one is represented as $a/a$ rather than $1$, where $a$ is an expression known to be non-zero) and then rearrange and simplify. The mathematical world would be a much sadder place if there was only one representation of $\pi$, $e$, $0$, or $1$.

Regarding electromagnetic fields carrying energy and momentum, what seems wrong about that? It should be intuitively obvious rather than confusing that electromagnetic fields carry energy. You can feel the heat of the Sun warming your skin thanks to the fact that the electromagnetic field carries energy. We can convert this energy to electricity, and use that to power the motors in the appliances in your house. Those motors would not work if electromagnetic field did not carry linear and angular momentum.

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1) We know since Einstein that mass contains or is energy. So the energy of a field has mass. If the field moves then we get a moving mass, $mv$ which is defined as momentum. This is not a choice that has been made but a fact that has been explored. It is of practical importance for instance in case of satellites the orbits of which are measurably changed by light-pressure, i.e., the momentum of photons (in addition to the "solar wind" consisting of particles with rest mass).

2) I am always happy to see that good students don't easily swallow the equation $0.999... = 1$. In fact it is not only counterintuitive but strictly speaking wrong and should not be accepted without the explanation (usually only supplied by good teachers) that according to common agreement by $0.999...$ the limit 1 of the sequence $0.9 + 0.09 + 0.009 + ...$ has to be understood like we write $3.1415... = \pi$, knowing that $\pi$ cannot be represented by any string of digits and that $3.1415... \rightarrow \pi$ would be correct.

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