The Wave Equation and Its Solution
With Maxwell's Equations in mind we can now ask whether and in what form electric and/or magnetic fields can exist in the absence of electric charges and electric currents. Those conditions would occur in a pure vacuum. In that case Maxwell's Equations take the form
If we combine Equations 3 and 4 in a way that eliminates one of the fields from the description, we should get an equation that will answer the question for us. So take the curl of Equation 3, take the time derivative of Equation 4, and add the results together to get
But we know from vector calculus that . And we know that in this case. So we have
We could also take the curl of Equation 4, take the time derivative of Equation 3 and multiply it by, then subtract the second result from the first to obtain, after a little algebraic manipulation,
Comparing Equations 6 and 7 lets us infer that if free-standing electric and magnetic fields exist, they must conform to the same mathematical description. We also infer that whatever function satisfies either equation must describe a field both spread and varying over distance and time; that is, to take Equation 6 as an example, we must have
That fact comes from the need to use a differentiable function to represent the field in order to satisfy the most basic requirement of Equation 6.
Continuing to take Equation 6 as our example, let's consider, for our convenience, an electric field that varies only in the x-direction. Equation 6 then becomes
Next we infer that Ex=0. By our assumption that the field only varies with displacement in the x-direction we know that , which means, because , that . Thus we now know that Ex must have the same value everywhere in space and that value cannot change with the elapse of time. Even if we could contrive a way to make the value change everywhere by the same amount at the same time, in inertial frames moving parallel to the x-direction by any amount of speed the temporal offset of the Lorentz Transformation would convert temporal change into spatial change in the x-direction, which we cannot have for Ex. So Ex must have the same value from the instant of the Universe's origination and maintain it in an expanding space. But we associate an energy density with an electric field, so a constant Ex in an ever-expanding space gives us a gross violation of the law of conservation of energy unless and only unless Ex=0.
So now Equation 9 becomes
We can separate that into two independent equations and solve one of them, knowing that our solution will apply as well to the other equation. So we want to solve
Because we use Cartesian coordinates, the unit vector j represents a constant, so we can simply divide it out of the equation and get
Let's take another look at the expression on the right side of that equation. We know that Maxwell augmented Ampere's law in order to ensure that a description of a charging capacitor properly accounts for the magnetic field that pervades its vicinity; he added a term representing an electric displacement current,
in vacuum. That current, like an actual motion of electric charge, generates a magnetic field. If we do something to change the displacement current, then its magnetic field will change and induce an electric field that opposes the change in the displacement current in accordance with the version of Faraday's law of electromagnetic induction known as Lenz's law. So we have
which combines with Equation 13 to give us
in which K represents an induction constant.
But we have taken as our goal the determination of whether and in what form an electric field can exist in the absence of electric charges and electric currents. Toward that goal we make and then see whether the solution of Equation 15 can come true to Reality. Better yet, we can write Equation 12 as two equations that we can solve:
To solve those equations we need a function that does not change under differentiation except for being multiplied by a constant. We know that an exponential fits that description, so we have as the solution of Equation 16
in which A represents the to-be-determined amplitude of the field, f(x) represents a constant relative to the elapse of time, and as shown in the appendix. We also have as the solution of Equation 17
in which g(t) represents a constant relative to displacement in the x-direction. Equating those two solutions, as we must do, gives us
We know that the exponential of an imaginary argument represents a sinusoid, whose values lie in the range between +1 and -1. That means that the field has a maximum strength Ey0 that we can measure and that conforms to the statement AB = Ey0. It also means that when or increments by some integer multiple of 2π the pattern repeats. For convenience we define
in which λ represents a characteristic length in the field. We also define
in which P represents a characteristic time in the field. We also know that we have no difference between the product of multiplying two exponentials and adding their arguments, so we can rewrite Equation 20 as
Looking at the argument of the exponential, we see that it remains constant as time elapses if the x-coordinate decreases at a rate equal to ν/k. Thus, any point in the field that has a certain value of the field strength appears to move in the negative x-direction at the speed of light. Because of its sinusoidal shape, the field appears to propagate itself through space as a wave. And because a changing electric field generates a magnetic field and a changing magnetic field generates an electric field, that wave propagates as an electromagnetic wave.
The Lorentz Transformation of the Wave Equation
We deduced Equation 9 implicitly in one specific inertial frame of reference. That equation represents a manipulation of the electric field measured with respect to the coordinates used in the differentials. An observer occupying a different inertial frame would measure a different electric field with respect to different coordinates. But, in accordance with the principle of Relativity, that observer must use an equation of the same form as Equation 9 to describe the relationship between the unsupported electric field and the coordinates. That may seem a relatively trivial proposition, but we actually gain something important if we examine how it plays out in more detail.
Let's assume that in our frame that other observer, whom we call the prime observer, moves in the negative x-direction at some speed v. In two dimensions, then, we have the Lorentz Transformation
If we organize our measurements into some differentiable function, then the primed observers can do the same with their measurements of the same phenomenon and Equations 24 and 25 would help us in converting one function into the other. But we must also transform the differentiation operators using the relation
so we have
Those equations give us the differential operator of Equation 9 in the form
We know that the result cannot contain any cross terms, so we must be able to subtract them out. That fact necessitates that
We make that subtraction and then, making appropriate substitutions from Equation 30, gather like terms on each side of the equality sign; thus, we get
But , so we have at last
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