We show that if Maxwell’s equations are expressed in a form independent of specific units, at least three Galilean limits can be extracted. The electric and magnetic limits can be regarded as nonrelativistic limits because they are obtained using the condition |v|c and restrictions on the magnitudes of the sources and fields. The third limit is called the instantaneous limit and is introduced by letting c. The electric and instantaneous limits have the same form, but their interpretation is different because the instantaneous limit cannot be considered as a nonrelativistic limit. We emphasize the double role that the speed of light c plays in Maxwell’s equations.

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4.
The nonrelativistic limit |v|c has been improperly identified with the instantaneous limit c. For example, in
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We also mention that c has been improperly called “the nonrelativistic limit” in, for example,
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9.
If B/t=0 is assumed, the electric limit becomes too restrictive because we would have magnetic fields constant in time, B(x,t)=B(x,t0), and electric fields linear in time, E(x,t)=E(x,t0)+tĖ(x,t0), where t0 is a fixed time and the dot means time differentiation.
10.
Equations (23) imply the continuity equation J+ρ/t=0, which is also Galilean invariant because of the relation J+ρ/t=J+ρ/t, where we have used (vρ)=(v)ρ.
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14.
If E/t=0 is assumed in the magnetic limit, we would have electric fields constant in time, E(x,t)=E(x,t0), and magnetic fields linear in time, B(x,t)=B(x,t0)+tḂ(x,t0). The magnetic limit would be too restrictive.
15.
Equation (29d) implies that J=0, which is Galilean invariant because J=J. Note that ρ/t and ×J are not related by the continuity equation in the magnetic limit. Consider, for example, the charge density ρ=P and the current density J=(1/γ)×M, where P(x,t) and M(x,t) are the polarization and magnetization vectors (J does not include the polarization term P/t because |P|α|M|/(βc) in the magnetic limit). In this case we have ρ/t0 and J=0.
16.
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17.
See, for example,
D. J.
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,
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, 3rd ed. (
Prentice-Hall
,
Englewood Cliffs, NJ
,
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.
18.
The electric and magnetic limits are approximate limits. The former should be written as Eαρ, B0, ×E0, and ×B(β/α)E/tβJ, and the latter as Eαρ, B0, ×E+[α/(βc2)]B/t0, and ×BβJ. In contrast, the instantaneous limit is exact: E=αρ, ×E=0, B=0, and ×B(β/α)E/t=βJ. Thus there exists a subtle difference between the electric and instantaneous limits.
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