Magnetic field of a moving point charge

[Cemre]

Hi,

I'm writing a small simulator where two objects apply force on each other inversely proportional to square of distance between them. ( like gravty )

in terms of gravity, I think this is the only force that I have to take into consideration.

but I'm confused about electrical and magnetic force.

When the point charges are not moving, the force between them is only electrical ( F = k*q1*q2/d^2 )

but as they start to move, and because a moving charge is "current", one will start to generate a magnetic field and the other will be effected from this field. ( correct ? biot-savart ) in addition to electrical field.

the particles may have some non-zero initial velocity and the particles may not have aimed towards each other.

Generally:
how do I calculate the magnetic field of a moving point charge? ( and finally apply it to the other by F = qVxB where B is the magnetic field generated by other particle )

in addition to attractive electrical force, can opposite charged particles apply "repulsive" magnetic force to each other?

Can someone clarify this to me?

Thanks.

PS: how do i calculate the "current" related to the motion of a charged particle?
does accelerated motion or constant velocity motion result in different currents?
in general: are electrons moving at constant velocity within a wire or are they accelerated from one pole of battery to the other?

 

[Born2bwire]

The Lienerd-Wiechert potentials are exact generalized fields due to a point charge. You can use the resulting field equations to find the emitted fields from each particle and then sum up the fields at the locations of the particles to find the fields for the Lorentz force that would act on the appropriate particle.

That part is easy, what is less easy is how to do this on the fly for a dynamic system. Without knowing the exact trajectory of your particles a priori, you cannot calculate the full Lienerd-Wiechert potentials for the duration of your experiment. Instead, you will have to use something like a finite difference time domain approach. You will need to find an appropriate integrator, like an Euler or Verlet (better) integrator.

You can find the Lienerd-Wiechert potentials and their resultant fields in many textbooks like Griffiths. Do note that often they reference the equations using the retarded position vectors and not the instantaneous field vectors, this makes a difference in the calculations.

 

[Bob S]

It is also possible to use the Lorentz-transformation equations to get the magnetic field for a single moving charged particle . See
http://pdg.lbl.gov/2009/reviews/rpp2009-rev-electromag-relations.pdf
The last four equations are the Lorentz-transforms. The last equation gives the transverse magnetic field B' (in the primed frame) from the transverse E component of a charged particle at rest in the unprimed frame. You will need to break down the radial E field of the charged particle into the longitudinal (E cos θ) and transverse component (E sin θ).

It is also useful to think about the Poynting vector of the field of a moving charged particle. If the Poynting vector is not pointing straight forward, then the particle is (usually) losing energy. The particle does not "drag" the field lines along (like a bow wave) in free space. If the Poynting vector is pointing outward, the energy loss may be radiation (if the particle is accelerating) or more commonly inducing (eddy) currents in nearby conducting materials.

Bob S