# How much power would it take to stop a bullet with a magnet?

If a 7.5 gram copper-jacketed lead bullet (say, a 9x19mm Parabellum) was travelling at 360 m/s, how much power would it take to diamagnetically stop it in the space of one meter?

This question comes from discussion about this question on the SF&F SE, about the scene in the X-Men movie where Magneto stops a bullet before it hits someone in the face. I'm aware that a strong enough magnet to cause this effect is going to have a whole heap of other side effects. I'm mainly interested in the ideal case of simply stopping a bullet, rather than the complication of doing it within an inch of someone's face.

• Do you really mean to ask about power (a technical term meaning energy expended per unit time), or did you intend to ask about the size of the magnet that would be needed? – Nathaniel May 30 '14 at 4:42
• I did mean power. Knowing the size of the magnet would be interesting, but I'm more trying to figure out just how feasible (as much as that word means anything when talking about superheroes) it is for Magneto to stop that bullet. – DuckTapeal May 30 '14 at 4:50
• I'm not sure that power is the best figure for determining that, though. Power is energy per unit time, and there's no obvious reason why physics would limit the rate at which Magneto can expend energy using his power. (I mean, it's all made up anyway, but if the total energy was too big then there would be a much better argument to be made.) – Nathaniel May 30 '14 at 5:06
• – mmesser314 May 30 '14 at 5:22
• Are you sure you mean power? Wouldn't it be more interesting to know the magnetic field strength required (in Tesla). Then you could compare it with the most powerful magnets known (somewhere in the range 10 - 100T depending on how you define magnet). – John Rennie May 30 '14 at 8:39

The kinetic energy of the bullet is $\frac12 mv^2 \approx 1$ kilojoule.

If the deceleration is continuous over $x=1$ meter, energy conservation gives us an acceleration $a = v_\text{initial}^2/2x \approx 65,000\,\mathrm{m/s^2} \approx 6600\,g$, and the stopping time is $t = v_\text{initial}/a \approx 5.6$ milliseconds.

Spreading the bullet's energy over the stopping time gives an average power of 175 kilowatts. If you make some hand-waving assumptions that the mechanism for stopping the bullet is inefficient you might multiply this power by a factor of 10–100.

This is a lot of power! But the time interval is very brief. And it's certainly not prima facie unphysical—after all, the gunpowder explosion that launched the bullet involved the same energy transfer and an acceleration length of much less than a meter.

After some thinking, and a silly mistake, I can make an order-of-magnitude estimate of the magnetic field that would have to be involved.

I would expect that the main effect involved in rapidly stopping a bullet would not be diamagnetism, a small effect where the magnetic field strength inside a "non-magnetic" material is changed in its fourth or fifth decimal place (and thus the energy density of the field $E\propto B^2$ is changed in its eighth or tenth decimal place).

The predominant factor on introduction of a strong magnetic field to a bullet would be eddy currents in the material. Wikipedia gives me a formula for energy loss due to eddy currents in a material, $$P = \frac{\pi^2 B^2 d^2 f^2}{6k\rho D}$$ where $P$ is the power in watts per kilogram, $B$ is the peak field, $d$ is the thickness of the conductor, $f$ is the frequency, $k$ is a dimensionless constant which depends on the geometry, $\rho$ is the resistivity, and $D$ is the mass density. Wiki gives $k=1$ for a thin plane and $k=2$ for a thin wire, so I wild-guess $k=3$ for a zero-dimensional bullet. Using values for the lead core of the bullet, we find the rate of field change \begin{align*} (Bf)^2 &= \mathrm{ \frac{18}{\pi^2} \frac{2\times10^{-7}\,\Omega\,m \cdot 10^4\,kg/m^3}{(10^{-2}\,m)^2} \frac{2\times10^5\,W}{8\times10^{-3}\,kg} } = \mathrm{10^{9} \frac{N^2}{C^2\,m^2} }\\ {}\\ Bf &= \mathrm{ \pi\times10^4\,T/s } \end{align*}

The simplest assumption about the frequency is that the field is being ramped up to its maximum while the bullet stops, so we've seen a quarter-oscillation and $1/f = 20\,\mathrm{ms}$. This gives us a peak field of 600 tesla, which is large, but not absurdly large.

On the other hand, if Magneto is actually an FM radio broadcaster at 100 MHz, he'd need only a field of $$\mathrm{ \frac{ \pi\times10^4\,T/s }{ 10^8\,Hz } = \pi\times10^{-4}\,T. }$$ I don't think that radio engineers ordinarily think about local peak magnetic field strengths, but this isn't outrageous either. My college NPR station has a 100 kW transmitter. However, their antenna isn't shaped correctly to put that entire power into a one-cc volume.

• OK, now what magnetic field strength will it take to interact with the Cu slug(bullet)? – Carl Witthoft May 30 '14 at 11:57
• It's been a while since I've seen X1, but I'm pretty sure that the gun fired was less than a meter away from the cop's head. – Kyle Kanos May 30 '14 at 13:19
• @Carl Fine, be that way – rob May 31 '14 at 3:10
• There is just one thing, the power given for as the Eddy current losses isn't of kinetic nature(maybe to a small degree, depends on the model). It would mostly be physical deformation and heating of the bullet. The physical energy needed to stop the bullet could be calculated by the Lorentz law. You would need two perpendicular fields. One would induce the a current in the copper, the other would exert a force on the conductor. The problem is that you won't have a return path for the current if you assume a linear dimensionless bullet. Throw in diamagnetism for a fun night. – WalyKu May 31 '14 at 23:00
• @Kurtovic Hmmmm. I was thinking of the demonstration where a strong magnet falls very slowly through a copper tube, or where the rotation of a conducting wheel is stopped by a magnetic field. But in that case all of the $\partial B/\partial t$ is due to the motion of the magnet and conductor relative to each other. That'd be the case with my first example, a bullet entering a region of strong field. A bullet interacting with a radio wave might not feel any less $\partial B/\partial t$ by slowing down, so maybe it wouldn't. – rob Jun 1 '14 at 16:15