© 2016 by Fernando Caracena
The discussion started in a previous post, Maxwell's Equations presented in MKS (meter- kilogram-second, or SI units) continues here, but in Gaussian units. We adapt the previous work, still as a classical theory, but configured in such a way that it fits transparently into the Theory of Special Relativity. Perhaps a future post will extend the theory of classical electrodynamics into a form covered by the quantum theory.
Classical Electrodynamics (CED)
A Question of Appropriate Units
Although electromagnetic (EM) theory written in the MKS (meter, kilogram, second) system (or SI system) of units is convenient for designing radio antennas and investigating large scale effects and motions of charged or current carrying conductors, it is not convenient for discussing the dynamics of charged particles travelling near the speed of light. Such particles satisfy the constraints of the theory of Special Relativity, and so does the electromagnetic (EM) field. In the SI system of units the units of measure that are taken as fundamental are mass, length and time; however, relativity combines the measure of length and times in terms of a unit taken to be fundamental, the speed of light in a vacuum (c = ~ 3.0x108 m/s). A problem then appears in EM theory, it it does not allow well enough for the fundamental constant c to appear explicitly in the equations. In the MKS system of units, c appears only implicitly in a pair of constants, ε0 and μ0, making it a bit clumsy to handle the Lorentz transformations of the various electromagnetic field components.
The Gaussian System of units is just a step away in physics for defining all units in terms of fundamental universal constants, as in the natural system of units where the fundamental constants are given the value of unity: c(speed of light) = ħ (Planck's constant)= G (Gravitational constant) =1. Dimensionless constants also appear in the natural system of units, such as the fine structure constant, α ≈ 1/137, which called the coupling constant, appears as a expansion parameter in expressions in quantum electrodynamics (QED).
Particle physicists prefer to use the Gaussian system of units in CED calculations than the MKS system, because of the nice way it fits into the theoretical structure of relativity. It is not more convenient just because it uses the smaller units. The cgs (centimeter, gram, second) units are also macroscopic, and as such do not come very much closer to the sizes of elementary units. The Gaussian system, which uses cgs units, is configured properly to handle the speed of light explicitly and to eliminate unnecessary units from the equations.
The inverse square law, or Coulomb force between two objects having charges q1 and q2, serves an example of the simplified relations:
F1 = q1q2 (r1 – r2)/ |r1 – r2|3 . (1)
In the post on E&M—Lines of Force The electric field strength is defined as follows:
E= ur *constant*Q/(surface area of the sphere), (EMLF2a)
E = ur *constant*Q/(4 π r2), (EMLF2b)
where in the MKS system of units the constant is 1/ε0. [See Maxwell's Equations, (1a):
E(x,y,z,t) = ur*Q/(4 π ε0 r2) ].
In the Gaussian system, the units of charge are defined directly in terms of the units of mass [M], time [T], and length [L]:
[Q]=([F] [L]2)1/2 =([M][L]3/[T]2) 1/2 =[M]1/2 [L]3/2/[T] ;
and the elementary unit of charge works out to be
e = √(αħc) .
If the electric field (E ) of a charge is located at the origin (Q at r=0), the force that that charge produces ( dF) on a test charge (dq) divided by the value of that test charge defines the electric field strength,
E(r) = Q dq ur /|r|2/dq ,
E(r) = Q ur /|r|2 , (2)
where ur is the unit vector that points from the center of Q toward dq.
The Gaussian system of units eliminates the constants μ0 and ε0, by assigning them a value of unity, and introduces the speed of light explicitly into the equations. Further, source terms include the factor 4π, which reflects the geometry of surface integrals. The Maxwell Equations for a vacuum in Gaussian units are:
∇ •E(r,t) = 4π ρ(r,t) ; (3a)
∇ X B(r,t) = 4π J(r,t)/c+ ∂ E(r,t)/∂ct ; (3b)
∇ X E(r,t) =-∂ B(r,t)/∂ct ; (3c)
∇ •B(r,t) =0 ; (3d)
where the electric current density is given by,
J(r,t) = (c ρ(r,t) v(r,t)). (3e)
All velocities appearing the the Maxwell Equations now are defined as fractions of the velocity of light.
For a comparison of the Maxwell Equations in the various systems of units see Appendix on Units and Dimensions in J. D. Jackson' s, "Classical Electrodynamics," ©1962 John Wiley and Sons, Inc., New York • London. 641pp.
The Electric Potential
When there is an electrostatic field, the electric potential can be written as the gradient of some scalar function
E(r,t) = -∇φ(r,t) . (4a)
Equation (4) substituted into (3a) results in the following equation:
∇2φ(r,t) = -4π ρ(r,t), (5)
which is known as Poisson's Equation. The electric potential is tied to the distribution of charge in space, which is its source.
The Vector Potential
Equation (3a) is interpreted as the electrical potential' s having its source in charge density. By contrast, (3d) indicates that there is no magnetic charge or monopole. This property can be portrayed automatically by having B satisfy the following equation:
B = ∇ x A(r,t) , (6)
for which (3d) is an identity, A being known as the vector potential.
The potentials φ(r,t) and A(r,t) now contain all the information required to define the electric (E) and magnetic (B) field strengths. Is it possible to come up with a relativistic framework using the potentials in a way that reproduces Maxwell's Equations?
Relativity and Classical Electrodynamics
Through tinkering with Maxwell's Equations theoretical physicists discovered that they could put them effectively into a tensor format that could be handled by Special Relativity. This formulation relied on combining the scalar (φ) and vector potentials (A) into a 4-vector:
Aμ =(φ, A), (7)
where A0= φis the fourth component and the Latin subscripted values of A are the normal 3-vector components, A1, A2, and A3. The sign convention for the metric tensor is the same as used in the post, Special Relativity II—standard notation:
A reader unfamiliar with this notation should read the above sited post for more details.
One more piece of shorthand notation should be discussed here. The symbols for four dimensional, partial differentiation are written as folows:
∂μ =∂/xμ . (8b)
Note, that the definitions in (8) give the differential operators opposite signs in the spatial components to those used for ordinary 4-vectors. For example, consider the ordinary scalor product of two 4-vectors:
AμAμ =Ao 2 – A•A, (9a)
where repeated indices are automatically summed over (Einstein's summation convention). Not that the last term on the rhs of (9a) is the ordinary 3-vector, scalar product. The 4-divergence of a four-vector (A) , however, does not have the sign reversal of (9a):
∂μAμ = c-1∂Ao/∂t + ∇•A, (9b)
where the fourth component of coordinates is
x0 = ct. (9c)
The Faraday Tensor
Having gone through some mathematical preliminaries above, we shall now define the Electromagnetic field tensor, which is called the Faraday tensor,
Fμν = ∂μAν - ∂νAμ . (10a)
Notice that the Faraday tensor is antisymmetric in the exchange of subscripts
Fμν = - Fνμ . (10b)
Fi0 = ∂iA0 - ∂0Ai
Fi0 = -[∂/(ct)A+∇φ]i
Fi0 =Ei. (10c)
The electromagnetic field strengths are often given in the literature in terms of the components of the Faraday tensor as in (10c) for the electric field strength and derived below, for the magnetic field strength:
In standard 3-vector notation, the spatial components of the Faraday tensor are written as follows:
Fij = -(∂/∂xi Aj-∂/∂xj Ai)
Fij = -( δilδim - δimδil ) ∂/∂xl Am
Fij = -εklmεkij(∂/∂xl Am )
Fij = -εkij Bk , (10d)
where εkij is the Levi-Cevita 3-dimensional tensor, which satifies the following identity:
εkij εklm = δilδim - δimδil . Note that the above result agrees with that given in the reference to the Faraday tensor, except that the velocity of light does not divide into the components of the electric field vector:
Note that in the Gaussian system of units, the field strengths (E and B) have identical units.
We have replaced the electric and magnetic field strengths with a relativistic tensor that is defined in terms of the vector and scalar potentials. Future posts will develop this theme into a basis for a complete description of classical electrodynamics.