# Can magnetic fields be redirected and focused at one point?

I know that magnetic fields can be redirected, but... given a situation where you have static magnetic field over a large area, and you want to quickly change the magnetic field strength. Is it feasible to redirect nearby fields and curve the field towards a single point, thus increasing flux density and strength?

the answer with the cone is great and all but are there any alternatives?

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Yes, it is possible to guide magnetic field lines using a shaped magnetic material. Just as field lines concentrate when entering the south pole of a magnet from a large area, an external magnetic field can be "gathered" using, for example, a cone-shaped piece of iron. The cone can be positioned such that the static field spread over a large area enters the wide end of the cone. The iron confines the field and will guide it to the tip of the cone, where it will emerge with a much higher density and, therefore, a much higher magnetic field strength.

This will, of course, reduce the field to the sides of the cone, since this method won't increase the total magnetic field present in the region. The field lines that used to occupy that space are now simply confined inside the cone.

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does any metal work for this cone? Or would things like mumetal work best? Also I'm assuming that the change in magnetic field is proportional(if bigger end is 2x size of smaller end of cone, I should expect double field strength through the cone). –  mugetsu Nov 18 '11 at 21:50
@mugetsu Any magnetic material will work (you need something that will interact with the magnetic field, which is, by definition, a magnetic material). And yes, I would expect, ignoring edge effects and assuming a uniform initial field, that your increase in field strength will be proportional to the ratio of the areas of the two ends of the cone. –  Mitchell Nov 19 '11 at 1:11

This is basically what a solenoid does. You have multiple current rings, and "within" the solenoid the magnetic field loops are concentrated whereas outside they are very weak and actually divergent in the limiting case. A much more interesting questions is if one could design a solenoid or solenoid like structure which "minimizes" the magnetic fields and currents "within" the sources (wire loops) while maximizing it in the region "outside" of the "sources" (interior of the solenoid without any currents or sources). This would have practical implications, since there are limits to how much current and magnetic fields materials which hold the currents can tolerate before they breakdown. It would be great if one could generate very large fields outside of the sources to say confine a magnetic fusion plasma, without breaking down structure containing the generating currents. It is a much more difficult problem because you have to treat the field within the conductors themselves. I have thought about it fruitlessly for a while and would love to find someone who might have worked on this more. Perhaps it could not easily be solved analytically, but just like they are doing with antennas these says, perhaps, since the equations are already there, the deux ex machina of genetic algorithms might be useful if one could define all the parameters. Also, there is perhaps a completely different approach than the solenoid one, that is a dynamical electromagnetic field. Since these can be self-propagating in the vaccuum, and one could theoretically focus a magnetic field outside of a source. Technically, there would be far-field (radiation) in such cases, but not all cases. For example, Schott in 1933 discovered non-radiating solutions for spherically charged objects rotating at relativistic velocities. To my knowledge, no one has designed an object which could do this without such high velocities, but these kinds of problems have been solved before by a more clever design.

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Suppose a beam of particles, each with momentum $p=\gamma mv_0$ parallel to the axis of a cylinder (of length $L$ and radius $R$ with uniform current $I$) and each with positive charge $q$, impinges on the cylinder's end from the left. I will neglect the slowing down and scattering of the beam particles by the material of the cylinder, and I will assume the cylinder is much shorter than the focal length (for a thin lens'' approximation).

Letting $\hat{z}$ point along the cylinder's axis, the magnetic field produced by this current is $B(\bar{r})=\frac{\mu_0I}{2\pi R^2}(-y\hat{x}+x\hat{y})$. It produces a force $\bar{F}=q\bar{v}\times\bar{B}=\gamma m\dot{\bar{v}}$ on each particle, where $\bar{v}=\dot{x}\hat{x}+\dot{y}\hat{y}+\dot{z}\hat{z}$ has initial condition $v_0\hat{z}$. Note that $v_0=\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}$ because magnetic fields cannot do work. By a thin lens approximation, we can assume $\dot{x},\dot{y}<< v_0$, so that $\dot{z}=v_0\sqrt{1-(\frac{\dot{x}}{v_0})^2-(\frac{\dot{y}}{v_0})^2}\approx v_0$. Taking the $\hat{x}$-component of the force equation gives $\ddot{x}=-\alpha\dot{z}x=-\alpha v_0x$ (where $\alpha\equiv \frac{\mu_0qI}{2\pi\gamma mR^2}$) with oscillatory solution $x(t)=x_0\cos(\sqrt{\alpha v_0}t)$. Now when the particles exit the cylinder, the focal length is $f=v_0t$ where $t$ is the time it takes to reach the focal point. To linear order, $t=\frac{x(\triangle t)}{|\dot{x}(\triangle t)|}$ where $\triangle t=\frac{L}{v_0}$ is the differential time of the focusing. Then $t=\frac{x_0\cos(\sqrt{\alpha v_0}\triangle t)}{\sqrt{\alpha v_0}x_0\sin(\sqrt{\alpha v_0}\triangle t)}\approx \frac{x_0}{\alpha v_0x_0\triangle t}$, giving focal length $f=\frac{2\pi \gamma mv_0R^2}{\mu_0qIL}$. As the $\hat{y}$-component is identical, we see that the particles focus at a point (at the focal length $f$ from the exiting-end of the cylinder).

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This is a ''magnetic lens'' –  Chris Gerig Nov 19 '11 at 0:59
so you are bending particles? I need to bend magnetic fields, not deflect particles –  mugetsu Nov 19 '11 at 4:26