C
Maxwell’s Equations: Choice of Units
In high-energy physics, it is not the convention to use the rationalized MKS
system of units when treating Maxwell’s equations. Since the discussion is
always limited to field equations in vacuo, it is usually felt desirable to adopt
a system of units in which these equations take their simplest possible form
– in particular, one such that the constants ∈ 0 and μ 0 , employed in the MKS
system, do not appear. These two constants enter, of course, via the force
laws of Coulomb and Amp` ere, respectively. These laws relate a mechanical
quantity (force) to electrical ones (charge and current). The introduction of
∈ 0 in Coulomb’s law
q 1 q 2 r
F =
(C.1)
4π∈ 0 r 3
enables one to choose arbitrarily one of the electrical units and assign to it
a dimension independent of those entering into mechanics (mass, length and
time). If, for example, we use the coulomb as the basic electrical quantity
(as in the MKS system), ∈ 0 has dimension (coulomb)
2 [T]
2 /[M][L]
3 . Thus
the common practical units (volt, amp` ere, coulomb, etc) can be employed
in applications to both fields and circuits. However, for our purposes this
advantage is irrelevant, since we are only concerned with the field equations,
not with practical circuits. In our case, we prefer to define the electrical units
in terms of mechanical ones in such a way as to reduce the field equations to
their simplest form. The field equation corresponding to (C.1) is
∇ · E = ρ/∈ 0
(Gauss’ law: MKS)
(C.2)
and this may obviously be simplified if we choose the unit of charge such that ∈ 0
becomes unity. Such a system, in which CGS units are used for the mechanical
quantities, is a variant of the electrostatic part of the ‘Gaussian CGS’ system.
The original Gaussian system set ∈ 0 → 1/4π, thereby simplifying the force
law (C.1), but introducing a compensating 4π into the field equation (C.2).
The field equation is, in fact, primary, and the 4π is a geometrical factor
appropriate only to the specific case of three dimensions, so that it should
not appear in a field equation of general validity. The system in which ∈ 0 in
(C.2) may be replaced by unity is called the ‘rationalized Gaussian CGS’ or
‘Heaviside–Lorentz’ system:
∇ · E = ρ
(Gauss’ law; Heaviside–Lorentz).
(C.3)
369
Maxwell’s Equations: Choice of Units
In high-energy physics, it is not the convention to use the rationalized MKS
system of units when treating Maxwell’s equations. Since the discussion is
always limited to field equations in vacuo, it is usually felt desirable to adopt
a system of units in which these equations take their simplest possible form
– in particular, one such that the constants ∈ 0 and μ 0 , employed in the MKS
system, do not appear. These two constants enter, of course, via the force
laws of Coulomb and Amp` ere, respectively. These laws relate a mechanical
quantity (force) to electrical ones (charge and current). The introduction of
∈ 0 in Coulomb’s law
q 1 q 2 r
F =
(C.1)
4π∈ 0 r 3
enables one to choose arbitrarily one of the electrical units and assign to it
a dimension independent of those entering into mechanics (mass, length and
time). If, for example, we use the coulomb as the basic electrical quantity
(as in the MKS system), ∈ 0 has dimension (coulomb)
2 [T]
2 /[M][L]
3 . Thus
the common practical units (volt, amp` ere, coulomb, etc) can be employed
in applications to both fields and circuits. However, for our purposes this
advantage is irrelevant, since we are only concerned with the field equations,
not with practical circuits. In our case, we prefer to define the electrical units
in terms of mechanical ones in such a way as to reduce the field equations to
their simplest form. The field equation corresponding to (C.1) is
∇ · E = ρ/∈ 0
(Gauss’ law: MKS)
(C.2)
and this may obviously be simplified if we choose the unit of charge such that ∈ 0
becomes unity. Such a system, in which CGS units are used for the mechanical
quantities, is a variant of the electrostatic part of the ‘Gaussian CGS’ system.
The original Gaussian system set ∈ 0 → 1/4π, thereby simplifying the force
law (C.1), but introducing a compensating 4π into the field equation (C.2).
The field equation is, in fact, primary, and the 4π is a geometrical factor
appropriate only to the specific case of three dimensions, so that it should
not appear in a field equation of general validity. The system in which ∈ 0 in
(C.2) may be replaced by unity is called the ‘rationalized Gaussian CGS’ or
‘Heaviside–Lorentz’ system:
∇ · E = ρ
(Gauss’ law; Heaviside–Lorentz).
(C.3)
369