Maxwell's equation and Helmholtz's Theorem

In summary, the conversation revolves around finding the magnetic field around a moving point particle. The person is trying to use Helmholtz's theorem, but is unsure about how to deal with the singularities at the origin and examined point while integrating. The other person suggests using the Biot-Savart law instead, as the charge density for a single charge is not continuous. The original person's question about setting up the integration to work around the singularities remains unanswered.
  • #1
Savant13
85
1
I am trying to find the magnetic field around a moving point particle. I have already found the curl. The only step remaining is to use Helmholtz's theorem. I am using http://farside.ph.utexas.edu/teaching/em/lectures/node37.html" . I am going to use equation 300, but I am not sure what to do about the singularities (at the origin and the examined point) while integrating.
 
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  • #2
If the limits of integration are at the singularity and C(r') has a leading order constant term then there is nothing you can do. However, I'm not sure how applicable that equation is though. My understanding is that C(r') would be current, which you do not have, and what are your limit of integration? You want the magnetic field due to a point charge, why not just use the Biot-Savart Law?

EDIT: This is assuming non-relativistic velocities. Under relativistic velocities I think you're better off doing the Lorentz transformations.
 
  • #3
current is charge density times the velocity of the generating particle, so there is current. The electric field is changing in the reference frame I am using, which also generates magnetic field.

I do not have limits of integration because I do not know how to set up the limits of integration for this problem. That is what I am asking. I am also not sure how to integrate with respect to the r' vector. Since it is a triple integral, do I just integrate with respect to each of its components? Wouldn't that require the order in which I integrate them to be irrelevant?
 
  • #4
You can't express a single charge as a continuous current, what is your charge density, just a delta function. Use the Biot-Savart law.
 
  • #5
Actually you can, but it is not a continuous current. Charge density = charge / r^3.

Besides, the Biot-Savart law requires a continuous current.

My question remains unanswered. How can I set up my integration to work around the singularities?
 
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  • #6
Savant13 said:
Actually you can, but it is not a continuous current. Charge density = charge / r^3.

Besides, the Biot-Savart law requires a continuous current.

My question remains unanswered. How can I set up my integration to work around the singularities?

You can't. Like I said, your charge distribution is a point source, you can use the Biot Savart law.
 

1. What are Maxwell's equations?

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields in space. They were developed by physicist James Clerk Maxwell in the 19th century and are essential to understanding electromagnetism.

2. What do Maxwell's equations describe?

Maxwell's equations describe the relationship between electric and magnetic fields, their sources, and how they interact with each other. They also describe how these fields can generate and propagate electromagnetic waves.

3. What is Helmholtz's theorem?

Helmholtz's theorem is a mathematical theorem that states that any continuous, differentiable vector field in three-dimensional space can be decomposed into a sum of two vector fields: an irrotational field and a solenoidal field. This theorem is often used in the study of electromagnetism.

4. How is Helmholtz's theorem related to Maxwell's equations?

Helmholtz's theorem is closely related to Maxwell's equations because it helps to describe the behavior of electric and magnetic fields in space. Specifically, it allows us to break down a complex electromagnetic field into simpler components, making it easier to analyze and understand.

5. What are the practical applications of Maxwell's equations and Helmholtz's theorem?

Maxwell's equations and Helmholtz's theorem have numerous practical applications. They are used in the design of electronic devices, such as computers and cell phones, as well as in the development of technologies like radar and MRI machines. These equations also play a crucial role in understanding and predicting the behavior of light and other electromagnetic waves.

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