After many years, I needed to review SI electromagnetic units. My main source was section 4 of the Appendix on Units and Dimensions in Jackson's Classical Electrodynamics (, page 782 - see the bottom of this page for the citation). To begin, the exact value of the speed of light is (, page 3) as follows:
A shown in the middle of (, page 176), "In SI units, newton per square ampere () or henry per meter (H/m)." Table 4 (, page 783) indicates that the henry is the SI unit of inductance, and the just cited quote establishes that inductance has units of energy (joule) divided by the square of current (ampere) as shown below. From the descriptive paragraph in Table 3 (, page 782), and from the discussion near the top of (, page 780), we see that and this defines as shown below. Tables 3 and 4 (, pages 782 and 783) show that the units of are farad per meter with farad being the unit of capacitance. Thus the farad is equal to the square of charge (ampere second) divided by energy as shown below.
Here are the exact values of the constants with their units:
The function bPHYSrθφ defined below is equation (5.41) (, page 183) for the magnetic induction far from a circular current loop. The dipoleMoment has been given arbitrarily a convenient value. Table 4 (, page 783) gives the Tesla as the unit of magnetic induction. On (, page 176) we see that " has the dimensions of newtons per ampere-meter", " has the dimensions of [newtons (force) over coulombs (charge)]", and " times a speed has the same dimensions as ." From these facts we conclude that the Tesla is given by the expression shown below - in agreement with the units produced by the function.
The numerical values of the constants are computed below, in agreement with Table 3 (, page 782).
 J. D. Jackson, Classical Electrodynamics, 3rd Edition, John Wiley & Sons, 1999