Maxwell's equation and Helmholtz's Theorem

[Savant13]

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.

 

[Born2bwire]

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.

 

[Savant13]

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?

 

[Born2bwire]

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.

 

[Savant13]

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?

 

[Born2bwire]

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.