Whoa there!  Your magnetic field fails to satisfy the Maxwell equation div B = 0.   --  Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, June 4, 2015 1:56 PM Subject: [math-fun] Can a strong magnetic field create electron-positron pairs? Here is an argument, which seems absolutely indisputable, that an INHOMOGENOUS magnetic field can and will create pairs.  Let the field be of form (Bx,By,Bz) = (0, 0, K*x) incidentally arising from vector potential (Ax,Ay,Az) = (0, K*x*x/2, 0). We know from the Stern-Gerlach effect that an electron (positron also) in such a field will be sucked in either the +x or -x direction depending on its spin. And the force sucking it will be   F = mu_e * K. And this is experimentally confirmed, not mere theorizing. Therefore, if the distance L we can go in the x direction obeys   L*F > m*c^2 then the total energy from all that sucking is enough to create an electron.  A pair could be created at x=0, then suck distance L, and then have regained enough energy to pay for the pair creation. Therefore, pairs will be created. And if a distance  L <= hbar/(m*c) suffices, then this creation process should be very fast. I don't see how anybody can possibly argue with this. Now any real-world magnetic field is NOT uniform, and it must go to 0 someplace.  Therefore, any real magnetic field with enough strength, specifically if max strength B>m*c^2/mu_e, will generate pairs at a nonzero rate. Then the only question, it seems to me, is "what is that rate?" And how does the rate depend on the specific geometry of the field, for example on grad|B|?  Optimally, we would express the pair generation rate (pairs/second) in terms of some triple integral of some function (or functional) of the magnetic field. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

On 6/4/2015 1:56 PM, Warren D Smith wrote:

Here is an argument, which seems absolutely indisputable, that an INHOMOGENOUS magnetic field can and will create pairs. Let the field be of form (Bx,By,Bz) = (0, 0, K*x) incidentally arising from vector potential (Ax,Ay,Az) = (0, K*x*x/2, 0).

We know from the Stern-Gerlach effect that an electron (positron also) in such a field will be sucked in either the +x or -x direction depending on its spin. And the force sucking it will be F = mu_e * K. And this is experimentally confirmed, not mere theorizing.

Therefore, if the distance L we can go in the x direction obeys L*F > m*c^2 then the total energy from all that sucking is enough to create an electron. A pair could be created at x=0, then suck distance L, and then have regained enough energy to pay for the pair creation. Therefore, pairs will be created.

And if a distance L <= hbar/(m*c) suffices, then this creation process should be very fast.

I don't see how anybody can possibly argue with this.

First, the linear Bz field is only an approximation of the field in a SG device. You can't really have a quadratically increasing potential out to infinity. Second, if you did all you've shown is that the electron could be accelerated to high energy by the magnetic field acting on it's magnetic moment. But an electron won't decay all by itself just because it has lots of kinetic energy. The kinetic energy is frame dependent. A suitable Lorentz transformation to the electron's rest frame reduces that energy to zero. Brent Meeker

Now any real-world magnetic field is NOT uniform, and it must go to 0 someplace. Therefore, any real magnetic field with enough strength, specifically if max strength B>m*c^2/mu_e, will generate pairs at a nonzero rate.

Then the only question, it seems to me, is "what is that rate?" And how does the rate depend on the specific geometry of the field, for example on grad|B|? Optimally, we would express the pair generation rate (pairs/second) in terms of some triple integral of some function (or functional) of the magnetic field.