Maxwell's Equation: Magnetic B Field from Moving Charges

[martin f]

Under what circumstances does a moving electric charge fail to produce a magnetic field?and what is the equation for finding the magnetic B field produced by multiple charges moving in a vacuum?

 

[Darwin123]

Under what circumstances does a moving electric charge fail to produce a magnetic field?and what is the equation for finding the magnetic B field produced by multiple charges moving in a vacuum?

A moving electric charge always generates a magnetic field. The Laws of Biot and Savart describe how the magnetic field is generated by the electric charge. Although the expressions for this law often are written in terms of electric current, there are expressions appropriate for point charges.
Here is a link that may help.
http://en.wikipedia.org/wiki/Biot–Savart_law
“In the case of a point charged particle q moving at a constant velocity v, then Maxwell's equations give the following expression for the electric field and magnetic field:[
<connect to link to see the equations in reference>


When v2 ≪ c2, the electric field and magnetic field can be approximated as
<connect to link to see the equations in reference>

These equations are called the "Biot–Savart law for a point charge"[6] due to its closely analogous form to the "standard" Biot–Savart law given previously. These equations were first derived by Oliver Heaviside in 1888.