# Can a magnetic field be focused at a distance?

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?

• You could say that you're focusing magnetic fields when you focus light, since it's just an electromagnetic oscillation. ;) – oink Jan 31 '15 at 21:49
• Now that you mention it, I'm not sure why light does not attract iron objects. It's probably for the best though. – Daniel Darabos Jan 31 '15 at 21:51
• The oscillations are so rapid that even the individual electrons are just jerked around a bit. If it's jerked around enough the electron jumps up an energy level and negates the energy in the oscillation... leading to electron excitation, yay! (I think this explanation is flawed, though, because what if you have an awfully long wavelength, one that has a human-scale frequency?) – oink Feb 2 '15 at 4:11

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. 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 $HIJK$.

• Thanks! I'll have to read up on these laws you mention... – Daniel Darabos Feb 4 '15 at 7:26

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?”