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There are interesting arrangements of magnets that can strengthen the field in some places while weakening it in others. This is discussed in Can magnetic fields be redirected and focused at one point? and examples are the solenoid, the Halbach array, and an iron cone.

However, these all appear to be limited in their range. In contrast, light can be focused at a distant point, for example with a magnifying lens. A laser beam can be focused such that its reflection on the Moon can be detected. We can communicate with Voyager 1 at 130 AU.

Is it possible to focus a magnetic field such that it would be able to, say, move a nail from a kilometer away? If not, what is the fundamental difference between light and magnetic field that is responsible for this limitation?

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Daniel Darabos
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Probably not. Here's a simple argument.

First, Gauss' law implies the magnetic field lines are closed. Thus the concentration of field lines would look something like the image below. However, then we could pick a closed rectangle-like loop as in the image, but with curved edges (in the image, the edges are straight). The edges would be chosen so that the first, $HI$, is always along a field line in the dense region, $IJ$ is always perpendicular to magnetic field, $JK$, is always in opposite direction to the field in the non-dense region and $KH$ is again perpendicular to magnetic field. However applying Ampere's law implies that the average magnetic field along $HI$ should be the same or weaker than that of $JK$ (the field is curved "outside", as implied by localization, which makes $JK$ in practice shorter than $HI$), which is a contradiction. Ampere's Law

Note that for electromagnetic waves my derivation is not valid, as Ampere's circuital law actually depends on the time-derivative of electric flux through $HIJK$.

kristjan
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Through my studies with electromagnet configurations and Permanent Magnet Configurations I can answer two of your questions.

You ask us “Is it possible to focus a magnetic field such that it would be able to, say, move a nail from a kilometer away? If not, what is the fundamental difference between light and magnetic field that is responsible for this limitation?”

I can answer both questions.

So you ask “Is it possible to focus a magnetic field such that it would be able to, say, move a nail from a kilometer away?”

The answer to that is simply no, but in a way a magnetic field can be ‘focused’ so to speak in such a way that it is almost uniform. For example, if you had a solenoid stretching a ft long and was put over an iron core the magnetic field would be ‘focused’ but in proper terms it would be almost perfectly uniform. But to further answer your specific question. No, to this day and age a magnetic field can not be focused to move a nail a kilometer away.

Okay next question:

“If not, what is the fundamental difference between light and magnetic field that is responsible for this limitation?”

According to the top scientists there isn’t an answer to describe the difference between light and magnetic fields. Besides the fact that magnetic fields are known as invisible photons and light is not invisible.

Scientist are still trying to figure out the correlation between light and magnetic fields and the differences between them.

Scientist Smith YT
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