© 2013 by Fernando Caracena
The discovery of electromagnetic waves was prompted by features of electromagnetic theory that came out of the mathematical formulation of Faraday' s intuitive concepts by James Clerk Maxwell.
Let us begin by stating Maxwell's Equations for electric and magnetic fields in the vacuum, which are yet coupled to electric charges and currents
∇ •E(r,t) = ρ(r,t)/ε0 , (1)
∇ •B(r,t) =0 , (2)
∇ X E(r,t) =-∂ B(r,t)/∂t . (3)
∇ X B(r,t) = μ0 J(r,t)+μ0 ε0 ∂ E(r,t)/∂t , (4)
Where r is the position vector, which is equivalent in the arguments of function to the coordinates of a point, x,y,z. The two constants μ0 and ε0 that appear in these equations pertain to the vacuum. Not that their values vary in physical media because of the collective charge effects in matter. The values of these constants in the vacuum are partially defined and experimentally measured as follows
where the Farad, the unit of the electrical capacitance of a capacitor is Coulombs/Volt, the Coulomb being the unit of charge and the Volt, the unit of electrical potential in SI units. The Ampere is the unit of electric current [a Coulomb per second].
In the previous blog on Maxwell’s Equations, we showed that these equations imply the conservation of electric charge through the equation of continuity, which can be derived from (1) and (4). All of the contents of these equations, except the last term on the RHS of (4) were based entirely on observations made by Michael Faraday, which he expressed intuitively through graphical devices, such as lines of force. Maxwell added the last term on the RHS of (4)perhaps for the sake of symmetry between the electric field (E) and the magnetic field (B), and, because he realized that that would automatically imply a conservation of charge. The symmetry involved here would be exact if an exchange of the symbols E and B in the four equations would result in the same set of equations. The addition of a magnetic charge (Qm) , magnetic charge density (ρm) and current Jm , would render the symmetry complete, except in (3) and (4) where the the two terms enter antisymmetrically. And this asymmetry cannot be made to go away. The negative sign on the RHS of (3) is based on observation, and the positive sign in the last term on the RHS of (4) is necessary to make the equation of continuity come out correctly.
Paul A.M. Dirac the English physicist, who won the Nobel Prize for his prediction of the positron, the electron's antiparticle, wrote a paper that investigated what would happen if magnetic monopoles (or magnetic charge) existed. So far, non have been observed, so that the idea of magnetic monopoles remains speculative.
The addition of a term to a set of equations based on symmetry or perhaps "gut feeling" is the kind of thing that theorists often do in physics to explore various possibilities about the behavior of nature. Perhaps the antisymmetry in (3) and (4) is somehow connected with the lack of magnetic charge, or implies something about it that is different from electric charge.
Conservation of charge, however, is not the only thing introduced into electromagnetic theory by the extra term added by Maxwell on the RHS of (4). This extra term allows the existence of electromagnetic waves that travel at a characteristic velocity of
c=1/√(μ0 ε0), (5)
which turns out to be the speed of light in a vacuum, c ≈ 3 x 108 m/s.
Here we show that both E and B satisfy a wave equation. To show this for the electric field strength, take the curl of both sides of (3)
∇ X (∇ X E(r,t)) =-∂ ∇ X B(r,t)/∂t (6a)
and substitute for ∇ X B using (4)
∇ X (∇ X E(r,t)) =-∂/∂t [ μ0 J(r,t)+μ0 ε0 ∂ E(r,t)/∂t] , (6b)
From the blog on vector algebra (VA), we have the vector identity,
A x B x C = (A•C) B – (A•B) C , (VA.7a)
which we can use on the LHS of (5b) to get
∇ ∇•E(r,t)–∇2 E(r,t) = -∂/∂t [ μ0 J(r,t)+μ0 ε0 ∂ E(r,t)/∂t] . (6c)
Rearranging terms in (6c) results in the wave equation with a source term on the right,
[∇2 -1/c2 ∂2/∂t2 ] E(r,t) = μ0 ∂/∂t J(r,t) + ∇ ∇•E(r,t), (6d)
where the product, μ0 ε0 , has been replaced by 1/c2 using (5).
The final term on the RHS of (6d) is eliminated by using (1),
[∇2 -1/c2 ∂2/∂t2 ] E(r,t) = μ0 ∂/∂t J(r,t) + ∇ ρ(r,t)/ε0 . (6f)
Outside the source region, where J(r,t) =0 and ρ(r,t) =0, (6f) implies that there are freely propagating electric wave with a phase velocity of c ≈ 3 x 108 m/s , the speed of light,
[∇2 -1/c2 ∂2/∂t2 ] E(r,t) =0. (7)
In a similar way, beginning by taking the curl of both sides of (3), we get
[∇2 -1/c2 ∂2/∂t2 ] B(r,t) = μ0 ∇ X J(r,t) . (8)
And correspondingly for space devoid of electric charges, we have
[∇2 -1/c2 ∂2/∂t2 ] B(r,t) =0 . (9)
The Poynting Vector
Consider the vector formed from the cross product of the electric and magnetic field vectors, E X B. A scalar results from taking the divergence of this vector, which expands in the following way,
∇•E X B = E• ∇ X B - B• ∇ X E, (10a)
which by substitution of (3) and (4) becomes,
∇•E X B = -∂B/∂t•B -∂E/∂t•E - μ0 J•E . (10b)
A dimensional analysis of the term J•E shows that it has the dimensions of energy flux:
[J][E]=Joules per cubic meter per second.
Therefore, dividing every term in (10b) by μ0 reduces it to the same units,
∇•E X B /μ0 = -(∂B/∂t•B -∂E/∂t•E)/μ0 - J•E . (10c)
The first term on the RHS of (10c) is simplified by recognizing that it is a time derivative of another quantity,
∇•E X B /μ0 = -∂/∂t(B•B +E•E)/2 μ0 - J•E . (10c)
The first term on the LHS of (10c) is the divergence of what is called the Poynting vector, which is defined as follows:
S ≡ E X B /μ0 . (11a)
The first term presented on the RHS of (10c) is the time derivative of an energy density,
u =½ (B2 +E2 )/μ0 . (11b)
Written out in terms of the above defined quantities, (10c) reduces to
∂u/∂t + ∇•S = - J•E , (10d)
which we recognize as an equation of continuity for electromagnetic energy that has a source term representing either a source of sink on the RHS of (10d). In the absence of electric currents to generate or absorb electromagnetic energy, (10d) becomes,
∂u/∂t + ∇•S = 0, (10e)
which account for the conservation of electromagnetic energy when it is in flux in empty space. This is shown by using the divergence theorem from a previous blog on advanced calculus (AC)
∫∫∫ dvol div V = ∫∫closed dSrf n • V, (AC.4), which applied to (10e) yields the following:
∂/∂t∫∫∫ dvol u + ∫∫closed dSrf n •S = 0, (10f)
which means that the accumulation or diminishing of energy within a fixed volume is exactly accounted for the flux of energy entering or leaving the surface bounding that volume, respectively.