Maxwell's Equations: Find Magnetic Field from Curl

[Savant13]

I'm working with Maxwell's equations, and I have found the curl of a magnetic field at all points. How can I figure out what the magnetic field is at those points?

 

[Savant13]

Should I be asking the differential equations section?

 

[Meir Achuz]

It involves solving a partial differential equation with boundary conditions.

 

[lalbatros]

http://en.wikipedia.org/wiki/Biot-Savart_law

 

[Savant13]

In this case, the magnetic field is being created by an electric dipole consisting of two point particles of equal mass and opposite charge in mutual orbit, not a current, so the Biot-Savart law doesn't apply

 

[weichi]

Posted the following in your other thread:

"There *are* expressions for this. You might be able to find them in something like Boas or Arfken and Weber.

If you are familiar with differential forms, many (most? all?) proofs of the converse of Poincare's lemma also give the expressions that you want. See, e.g, Flanders."

Also, note that the field you are trying to calculate is radiating radiation, so you might want to look at something like Jackson ch 9. Problem 9.1 discusses approaches for solving this type of problem.

 

[Ben Niehoff]

In this case, the magnetic field is being created by an electric dipole consisting of two point particles of equal mass and opposite charge in mutual orbit, not a current, so the Biot-Savart law doesn't apply

Of course there is a current. Use the continuity equation:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \vec J = 0$$

Also, as was mentioned, if you are trying to calculate the radiation fields, there is a shortcut. See Jackson.

Also, sometimes (but not always) these things are easier to do in k-space, rather than using curl and grad and such.

 

[weichi]

Ben is of course correct - there *is* a current here. It would be a good exercise to calculate it! Note that this is, in a sense, a "baby" version of the problem you are asking about, but in this case you need to find a vector field who's *divergence* you know.

But once you calculate the current, Biot-Savart isn't going to help you. Do you see why?

 

[Savant13]

Ben is of course correct - there *is* a current here. It would be a good exercise to calculate it! Note that this is, in a sense, a "baby" version of the problem you are asking about, but in this case you need to find a vector field who's *divergence* you know.

But once you calculate the current, Biot-Savart isn't going to help you. Do you see why?

Is it because the current is not constant?

 

[Savant13]

Is there any good way to do this that doesn't involve tensors?

 

[weichi]

Is it because the current is not constant?

Yes, exactly! The direction (and location) of the current is always changing. Biot-Savart only applies to steady currents.

Not sure what you mean about tensors, I don't see a use of tensors here.

If you are stuck, why not post what you have so far? Both your solution approach and your result for curl B. (Either on this thread or a new one.) There might be a better way to go about obtaining the solution.

 

[Savant13]

k-Space was mentioned, and I found that k-space requires tensors

 

[weichi]

I don't see why working in k-space would require tensors. I also don't think working in k-space would be helpful for this particular problem, but I admit I haven't given it a great deal of thought.

 

[Savant13]

Ben is of course correct - there *is* a current here. It would be a good exercise to calculate it! Note that this is, in a sense, a "baby" version of the problem you are asking about, but in this case you need to find a vector field who's *divergence* you know.

But once you calculate the current, Biot-Savart isn't going to help you. Do you see why?

How would one find the equation based on its divergence?

The divergence is this case is the partial derivative with respect to time of the divergence of the time-varying electric field. So basically what is happening is you take the divergence of the electric field, take the partial derivative of that, and then undo the divergence. I'm not sure if that makes it any easier.

 

[Savant13]

I think I know how I can do this.

Is it possible for a vector field to be perpendicular to its divergence at a point?

 

[Born2bwire]

I think I know how I can do this.

Is it possible for a vector field to be perpendicular to its divergence at a point?

How can it be perpendicular to its divergence? Divergence results in a scalar.

 

[Savant13]

I'm not sure what I was thinking there, haven't been getting a lot of sleep lately.