File:MagnetismFromLengthContraction.png

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Summary

Description
English: This shows how the length-contracted Coulomb force is the cause of magnetic fields, and also the cause of magnetic forces on moving charges. Both of these result from (and act on) current-elements qv=IΔs, which include moving charges (that may also exert electrostatic forces) and currents in neutral wires (that exert no electrostatic forces).

If the positive charge q in the figure moves to the right, it feels an upward Coulomb repulsion because to that moving charge protons (drawn red in the magnified circle) of atoms in the neutral wire below appear closer together due to length contraction than do the uncontracted electrons (drawn blue in the magnified circle) moving along with q.
Date
Source Own work
Author P. Fraundorf

E&M units and their analogs

Here m is meter, s is second, kg is kilogram, N is Newton, J is Joule, C is Coulomb, V is Volt & A is Ampere.
In the CGS system, electric charge is measured in statCoulombs, esu, or Benjamin Franklins.
This is a derived unit [g1/2cm3/2s-1], and so not dimensionally the same as the fundamental SI units discussed here.
quantity gravity electrostatics magnetism
active agent mass m in [kg] charge q in [C] current elements δqv=Iδs=jδx3 in [Cm/s]
force field force/mass g in [m/s2]
since F=mg where δg=Gδmr/r3		 force/charge E in [N/C]≡[V/m]
since F=qE where δE=kδqr/r3		 force/currentelement B in [Tesla]
since F=qv×B where δB=(k/c2)δqv×r/r3
flux area×force/mass in [m3/s2] area×force/charge ΦE in [Vm] area×force/currentelement ΦB in [Weber]
potential energy/mass φ in [m2/s2]
since UG=mφ where δφ=Gδm/r		 energy/charge V in [J/C]≡[Volt]
since UE=qV where δV=kδq/r		 energy/currentelement[1] A in [Vs/m]
since UB=-qv•A where δA=(k/c2)δqv/r
current mass/time in [kg/s] charge/time I in [C/s]≡[Ampere] currentelement/time IB in [Am/s]
capacitance mass2/energy in [kg(s/m)2] charge2/energy C in [C/V]≡[Farad]
as in VC = Q/C where C=κεoA/d
resistance energy×time/mass2 in [m2/(kg×s)] energy×time/charge2 R in [V/A]≡[Ohm]
as in VR = IR where R=ρL/A
inductance energy(time/mass)2 in [m2/kg] energy(time/charge)2 L in [Vs/A]≡[Henry]
as in VL= -LdI/dt where L=μoN2A/h

Note that we don't need magnetic units at all if we think of magnetic fields interacting with (and arisng from) relativistic charge deficits qv/c = (I/c)Δs instead of from current elements qv = IΔs i.e. if we use for our magnetic field description Bc in [Volts/meter] in place of B in [Tesla]. However, separate units are not a bad idea because current elements (which arise from charge q moving at velocity v as well as from relativistic imbalances due to length contraction in a neutral wire of directed-length Δs carrying current I) interact very differently than do static charges. These separate unit systems help us to minimize confusion when keeping track of both electrostatic and magnetic interactions, the latter of which would go away if spacetime constant c got really big.

For example fixed magnetic fields only act on moving charges. Moreover they cannot add to the kinetic energy of such charges. Hence large magnetic fields are often less dangerous than large electrostatic fields (unless of course you have a ferromagnetic plate in your head during an MRI scan, since magnets can add kinetic energy to other magnets). Abrupt collapses of a magnetic field, with the "voltages from nowhere" that they induce, of course might also be dangerous.

Footnotes

  1. ↑ E. J. Konopinski (1978) "What the electromagnetic vector potential describes", Amer. J. Phys. 46:5 p499-502 link.

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