Maxwell's Equations: Find Magnetic Field from Curl

In summary: I was thinking about the curl. But now that I think about it, the curl of a vector field would also result in a vector, not a scalar. So I guess my question doesn't make sense. Sorry about that.In summary, the conversation revolved around finding the magnetic field at points using Maxwell's equations and the curl of the magnetic field. It was discussed that the Biot-Savart law may not apply in this case due to the changing nature of the current. Other approaches such as using k-space were mentioned, but it was unclear if tensors would be required. It was also suggested to find the equation based on its divergence, but there was uncertainty about whether a vector field could be perpendicular to its divergence.
  • #1
Savant13
85
1
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?
 
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  • #2
Should I be asking the differential equations section?
 
  • #3
It involves solving a partial differential equation with boundary conditions.
 
  • #5
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
 
  • #6
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.
 
  • #7
Savant13 said:
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:

[tex]\frac{\partial \rho}{\partial t} + \nabla \cdot \vec J = 0[/tex]

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.
 
  • #8
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?
 
  • #9
weichi said:
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?
 
  • #10
Is there any good way to do this that doesn't involve tensors?
 
  • #11
Savant13 said:
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.
 
  • #12
k-Space was mentioned, and I found that k-space requires tensors
 
  • #13
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.
 
  • #14
weichi said:
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.
 
  • #15
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?
 
  • #16
Savant13 said:
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.
 
  • #17
I'm not sure what I was thinking there, haven't been getting a lot of sleep lately.
 

1. What are Maxwell's Equations?

Maxwell's Equations are a set of four fundamental equations in electromagnetism that describe the relationship between electric and magnetic fields, electric charges, and electric currents.

2. What is the significance of the "curl" in Maxwell's Equations?

The "curl" in Maxwell's Equations refers to the mathematical operation that describes the rotation or circulation of a vector field. In the context of electromagnetism, it is used to calculate the magnetic field from the electric field and vice versa.

3. How do you find the magnetic field from the curl in Maxwell's Equations?

The magnetic field can be found by taking the curl of the electric field. This operation involves taking the partial derivatives of the electric field with respect to each spatial coordinate and then combining them in a specific way, as described in the curl equation.

4. What are the units of the magnetic field in Maxwell's Equations?

The units of the magnetic field in Maxwell's Equations are typically measured in teslas (T) in the SI system. In other systems, it may be measured in gauss (G) or oersteds (Oe).

5. How are Maxwell's Equations used in real-world applications?

Maxwell's Equations are used in a wide range of real-world applications, including the design and operation of electrical and electronic devices, telecommunications, and the study of electromagnetic radiation. They are also essential for understanding and predicting the behavior of electromagnetic fields and waves in various systems and environments.

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