In this page, we use the System-Independent Electromagnetic Equations; electric charge, charge density and current density are denoted as qe, ρe and Je; magnetic charge, charge density and current density are denoted as qm, ρm and Jm.
The Maxwell's equations and Lorentz force with magnetic monopoles can be derived by Duality Transformation with an angle of 90°:
Note that the coefficient αLc makes B the same dimension as E.
Now let's introduce a new coefficient αM into Gauss's law for magnetism:
which can be derived by (6.1) and Gauss's law for electric charge:
Therefore, the transformation between electric and magnetic charge densities should be:
Analogously, we have the transformation between electric and magnetic current densities:
By applying (6.1) and (6.5) on Ampère-Maxwell equation:
we get the Maxwell-Faraday equation with magnetic current density: [1]
We also have the transformation between electric and magnetic charges:
By applying (6.1) and (6.8) on Lorentz force:
we get the Lorentz force for magnetism: [1]
Let's combine (6.9) and (6.10) into one equation:
The dimension and unit of magnetic charge qm can be defined arbitrary, because αM can be given arbitrary dimension and value. However, we should set proper αM to make equations simple in different unit systems.
In Gaussian and natural units where B has the same dimension as E, we can just set αM = 1 to make qm the same dimension as qe.
In Gaussian units:
In natural units:
In SI units, there are some conventions for αM:
- αM = 1 makes Gauss's law for magnetism and Maxwell-Faraday equation simple, where the unit of the magnetic charge is weber;
- αM = µ0 makes Lorentz force simple, where the unit of magnetic charge is ampere-metre;
- αM = µ0c makes magnetic charge the same dimension as electric charge, where the unit of magnetic charge is coulomb.
Weber convention:
Ampere-metre convention:
Coulomb convention:
The conversions between above 5 units of magnetic charge can be derived by comparing 5 unit systems or conventions of the B terms in (6.10):
By applying and
[2] on the above equations, we get: [3]
Note that α has the relative uncertainty of 0.15 ppb, so the uncertainties of the above conversions follow this diagram:
Finally, we get the unit conversions: [4]
The Dirac's Quantization condition can be expressed as: [5]
Then the elementary magnetic charge g can be expressed as:
In different unit systems or conventions: [4]
In Gaussian (6.14g), natural (6.14n) units and coulomb (6.14c) convention: