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This is theory for a science fiction scenario. For the sake of speculation, suppose that power is not an issue.

I've done reading on eddy currents, and I know they can exert drag forces on objects moving parallel to the source of the field. What I want to know is if they can exert the same force on an object moving directly towards the source of the field.

Wikipedia for those interested: https://en.wikipedia.org/?title=Eddy_current

The theoretical situation would be a sort of magnetic shielding that uses powerful (UNLMITED POWWEEEEERRRRR) electromagnets embedded in the hull of a flagship to deplete the kinetic energy of incoming conductive munitions. I don't need speculation of whether this would be necessary, or advantages/disadvantages, I just want to know if it's possible.

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    $\begingroup$ Of course. Take the magnetic field at the surface of a neutron star and you are in business. 12 or even 25T from a physical research magnet won't do it, though. I could imagine that an optimized undulating magnetic field might actually be used to stop something the size of a bullet within the distance of a couple of meters, though, but I will let somebody else do the math. Having said that, technically this is not a solution. There really is no solution to defend against rail guns with tungsten penetrators - why do you think the Navy wants them? $\endgroup$ – CuriousOne Jun 18 '15 at 0:17
  • $\begingroup$ @CuriousOne 25T Ye Gods! I'm a bit out of touch - so I googled "highest magnetic field achieved". I had to laugh - one of the entries, from Science Daily, featured a (coincidental unrelated) ad for a car key tracking device. Its caption, under the reporting of a 90T pulsed field, was "Never Lose Your Keys Again": carkeys next to a 25T magnet evokes quite an interesting mind picture. $\endgroup$ – WetSavannaAnimal aka Rod Vance Jun 18 '15 at 0:27
  • $\begingroup$ @WetSavannaAnimalakaRodVance: I used to work in a lab with 8T and 12T magnets. Anybody who was seen with a screwdriver near the magnets was given an earful... for good reason. The impact of such a metallic object near the bore would have destroyed or damaged the cryostat and endangered people. $\endgroup$ – CuriousOne Jun 18 '15 at 0:31
  • $\begingroup$ @CuriousOne I can imagine. I was involved in the signal processing for 1T to 2T medical imaging devices. Fatalities, thankfully seldom because people quickly learnt from them, nonetheless made for a gruesome folklore woven out of accounts of early deaths: oxygen bottles in the next room, and, sadly, patient deaths from prostheses - especially tiny little clips in the brain to staunch brain aneurysms that people mistakenly thought would be no problem. $\endgroup$ – WetSavannaAnimal aka Rod Vance Jun 18 '15 at 0:42
  • $\begingroup$ @WetSavannaAnimalakaRodVance: Folklore is always good to make people cautious, whether it is true, or not. I have seen only one accident where someone lost track of a tool and it actually impacted the magnet. It made a very loud noise, everybody came running and a red-faced physicist saw his PhD chances decay exponentially... luckily nothing happened and even the magnet was OK, but it taught everybody a lesson and the minimum safety distance for tools was voluntarily raised. Most importantly, the cryostat didn't start bleeding. $\endgroup$ – CuriousOne Jun 18 '15 at 0:50
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You have absurd electromagnets on your ship. They are powered. They produce (somehow) a large magnetic field gradient outside your ship.

Lenz's Law means every conductive thing passing through this absurd field will pick up a magnetic eddy current and slow down. Very resistive things will not generate large currents/fields and therefore not slow down as much as conductive things. Conductive things will dissipate much more power and therefore are more likely to melt or catch on fire. In terms of whether any particular bullet can be stopped/melted/vaporized depends on what that bullet is made of(and its shape). I could imagine glass bullets or what have you might, if traveling very fast indeed, exceed the dielectric breakdown voltages and become much more conductive.

So: given a large enough magnetic field, you can melt or break down most things. Does this help you at all? Maybe not. A glass bullet that is traveling fast enough to induce an eddy current that ionizes it into a conductive plasma is not made much less deadly by turning it from a bit of glass to a rapidly expanding cloud of conductive plasma(which will interact with the field in complicated ways I don't really know much about. Possibly the field will 'bottle' the plasma. Possibly it won't. Possibly it will accelerate it into the magnets like a tiny aurora borealis.). Small conductive objects might be thrown at the shielding simply to vent as much power as possible from the fields through the object, turning them into tiny EMP devices that also explode conventionally.

(At first there was a bit here about how your flagship would heat the water it floated on, and possibly simply sink to the bottom of the sea in a cloud of superheated steam. I realized you probably didn't mean ocean flagship, but it's a fun mental image anyway.)

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The general answer is no. The eddy currents are always exerting drag on objects moving perpendicular to the magnetic field, because they come from Faraday's Law of Induction. That's simply a limit of physics. The magnetic field is perpendicular to what we tend to think of as "the magnetic source," in laymans terms, so the only way to apply drag is to move parallel to the source.

However, if you are really playing games with whether it is possible, consider that electric fields emitted by a ship are not as simple as those generated by a simple horseshoe magnet. There is some curvature to the field, and flying through that curvature would generate magnetic flux, which causes inductive effects.

However, there would have to be a pole, at which the curvature would vanish. Fortunately, you could probably make this go straight up. After all, if someone did manage to put a metal projectile directly above your ship, there would be no way to stop the impact, because the effects diminish as velocity goes down.

There's also a funny side effect that flagship would be unable to move while using this field, because the motion of the propeller would probably generate enough heat to melt the propeller... it's in the same field. The field could also not be deployed all that quickly without buckling the walls of the battleship under the strain of the changing magnetic fields (motion of metal in a field and change of a field on stationary metal are identical in many cases in E-M).

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If you are able to put a charge on the incoming object, then the effect of the magnetic field (or electrostatic field) would be enhanced and it could be used to deflect, or slow down the incoming object.

Strong pulsed magnetic fields, pulsed lasers, and obviously charged plasma can cause an object to gain a specific charge.

Or you could simply create an ionised layer of gas at stratospheric pressure perhaps 20 KM thick which could be swung by magnetic fields into the path of the oncoming objects. The objects kinetic energy would cause it to destroy itself in most cases.

On a mass to protection basis, gas is more effective than armour, but obviously typically needs to be at least 1000 times thicker, reasonably practical in space.

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