Scalar Potentials and Electromagnetic Current

[Philosophaie]

For a Dipole and a Torad (or a Solenoid) I need to find the scalar Potential,phi, Charge Density,rho, and then 4-Electromagnetic Current,J(rho*c,j) where A and J are 4-vectors and a and j are 3-vectors.

-grad^2(phi) + 1/c^2*d/dt(phi) = rho/epsillon0

where grad(A(phi/c,a)) = -1/c^2*d/dt(phi)

-grad^2(A(phi/c,a)) + 1/c^2*d/dt(A(phi/c,a)) = mic0*J(rho*c,j)

I need a little help deriving rho, phi and J for a copper solenoid of diameter, d.
This is not a homework problem this is for my personal use.

 

[NFuller]

Is there any reason you want to solve everything in terms of Lorenz invariant equations? The 3-magnetic field of a dipole can be found by looking at the magnetic potenial $\mathbf{A}$ of a dipole and the field from a solenoid can be found using the Biot-Savart law.

 

[Philosophaie]

The magnetic fields of the Earth (Neglecting the Sun Fields, Solar Wind, etc.) can be calculated by using the magnetic moment. The magnetic moment is located on the z-axis relative to the x- plane of the equator. What is the magnitude of this magnetic moment so the others can be calculated? Also could this be calcated for the other planets and the sun (neglecting its periodic nature).

 

[NFuller]

The magnetic fields of the Earth (Neglecting the Sun Fields, Solar Wind, etc.) can be calculated by using the magnetic moment. The magnetic moment is located on the z-axis relative to the x- plane of the equator. What is the magnitude of this magnetic moment so the others can be calculated? Also could this be calcated for the other planets and the sun (neglecting its periodic nature).

Not quite. The magnetic field around the Earth and other planets is rather complicated and does not match that of a simple dipole even when ignoring external influences. Here are some magnetic maps showing different elements of the Earth's magnetic field.
https://www.ngdc.noaa.gov/geomag/WMM/image.shtml

 

[Philosophaie]

The Gaussian model of Earth give a better approximation but is very difficult to formulate. Are there other mathematical models to give better approxs?