Magnetic field in empty space?

[brian0918]

I was given this problem to think about from my professor. It's not for any class. It went something like this:

Imagine there is some magnetic field in empty space (tightly-packed lines curling around in a circle) at t=0. What happens to the field as t -> infinity.

I figured Maxwell's equations might help, but they all talk about a current being involved. From what he said, though, it seems like the field lines are just there, in empty space. How did they get there, though, and what would sustain them?


Thanks.

 

[CharlesP]

Can you draw this thing?

 

[brian0918]

This is all he drew. (see attachment)

 

[Gonzolo]

You should be able to tell where a current would be. What's t? Distance from the center?

 

[sinyud]

There doesn't need to be a current. Just use the wave equation and have this configuration be your initial condition.

wave equation is
Laplacian[H] - epsilon*mu*SecondDerivate w/ respect to time of H=0

I think the solution for the x component of H is given by

Hx = A Sin[Dot[k,r]- wt] + B Cos[Dot[k,r]-wt] where k is the wave_vector and w is the angular frequency. But don't take my word for it.
The Hy and Hz components should have similar form. For a further description, search for solutions to the Helmholtz equation. Or just write a quick numerical simulation (Should take 45 minutes but only is useful in two dimensions, If you have mathematica, I can write one and send it to you).


As for where the B field came from, perhaps there was an electron positron annahilation, or some one sent a time reversed wave in order to create this B field.

BTW anyone know how to get symbols I'm getting tired of writing
Laplacian[...]

 

[brian0918]

t is the time.

 

[sinyud]

And nothing would sustain this Bfield, it would propagate away radially.

 

[Tide]

And, I might add, the details are nontrivial! :-)

 

[brian0918]

He seemed to think I could figure out what happens as t-> infinity using Maxwell's equations. Any ideas?

 

[Tide]

I'll have to think some more about it. It's easy to derive a wave equation from Maxwell's equations as Sinyud suggested and in the absence of any sources for the magnetic field you might be able to demonstrate that the resultant waves will eventually propagate away. If you want details you'll have to use, say, a Laplace transform rather than Fourier analysis to account for the initial conditions but I suspect your teacher really isn't looking for that level of detail.

 

[brian0918]

There's a good chance he is. This is for my senior project. It's on particle propagation in large-scale astrophysical magnetic fields.

 

[sinyud]

I always wondered what would happen in such a case.

From a classical perspective, I would imagine that the intensity of the field would approach zero as t-> infinity. However, from a quantum perspective, a wave is made up of a finite number of photons; In such a case perhaps the photons would become so diluted over space, that the photons might never occupy certain regions of space.

Can you email me your project when you finish? I'm very curious to know what the answer is. My email address is sz2123@columbia.edu.

 

[mathfeel]

Unless you detect the photon, then it becomes localized. I am assuming in this problem you let it just runs by iteself. So not sure if quantum is the problem here...

Can this problem be physically achieved this way: has a constant current running to turn on the circling B field. Then turn off the current at t=0 and asks how does the B field evolves.

I haven't done any calculation, but just out of intuition I think it goes like this. The "signal" that the current has been turned off will travel at speed of light as electromagnetic wave so within the cylinder of radius ct, there will be no field. then there will be a tiny region of radii different c*dt in which there is electromagntic radiation, where dt corrects sponds to the time it takes to shutdown the current. Outside then, there is still a curlly field...