C13T (Electromagnetic Theory)
Topic – EM Wave Propagation in Unbounded Media (Part – 1)
Introduction:
While studying electricity and magnetism, one soon becomes aware that a number of relationships are described by vector cross-products or right-hand rules. In other words, an occurrence of one sort produces a related, perpendicularly directed response. Of immediate interest is the fact that a time- varying electric ( ) field generates a magnetic ( ) field, which is everywhere perpendicular to the direction in which the electric field changes. In the same way, a time-varying field generates an field, which is everywhere perpendicular to the direction in which the magnetic field changes.
Consequently, we might anticipate the general transverse nature of the electric as well as magnetic fields in an electromagnetic disturbance.
Let us now consider a charge that is somehow caused to accelerate from rest.
When the charge is at rest, it has associated with it a constant radial electric field extending in all directions presumably to infinity. At the instant the charge begins to move, the electric field is altered in the vicinity of the charge, and this alteration propagates out into space at some finite speed. The time-varying electric field induces a magnetic field by means of Maxwell’s Equations. If the charge’s velocity is constant, the rate-of-change of the electric field is steady, and the resulting magnetic field is constant. But here the charge is accelerating.
Therefore
is itself not constant, so the induced magnetic field is time- dependent. The time-varying magnetic field generates an electric field, obtained from Maxwell’s Equations and the process continues, with electric and magnetic coupled in the form of a pulse. As one field changes, it generates a new field that extends a bit farther, and the pulse moves out from one point to
We can draw an overly mechanistic but rather picturesque analogy, if we imagine the electric field lines as a dense radial distribution of strings. When somehow plucked, each string is distorted, forming a kink that travels outward from the source. All these kinks combine at any instant to yield a three- dimensional expanding pulse in the continuum of the electric field. The electric and magnetic fields can more appropriately be considered as two aspects of a single physical phenomenon, the electromagnetic field (or EM field, in short), whose source is a moving charge. The disturbance, once it has been generated in the electromagnetic field, is an untied wave that moves beyond its source and independently of it. Bound together as a single entity, the time-varying electric and magnetic fields regenerate each other in an endless cycle.
EM Wave Propagation in Vacuum:
In regions of free space or vacuum where there is no charge ( ) or current ( ), all the four Maxwell’s equations can be written as
They constitute a set of coupled, first-order partial differential equations for and . They can be decoupled by applying the curl to the 2nd and 4th equations.
Therefore, we get
Using elementary vector algebra, we write , which will be reduced to (since ).
So, we get
or
In a similar process and using the fact that , we obtain
We now have separate equations for and , but they are of second order, that’s the price we paid for decoupling them. In vacuum then, each Cartesian component of and satisfies the three dimensional wave equation, given by
where is the speed of the corresponding wave. So Maxwell’s Equations imply that empty space supports the propagation of electromagnetic waves, travelling at a speed given by
ms-1
which happens to be precisely the velocity of light, denoted as . The implication is astounding, perhaps light is an electromagnetic wave.
Transverse Nature of EM Waves:
We now confine our attention to sinusoidal waves of frequency . Since different frequencies in the visible range correspond to different colours, such waves are called monochromatic. Let us suppose that the waves are travelling in the direction and have no or dependence, these are called plane waves, because the fields are uniform over every plane perpendicular to the direction of propagation (shown in Fig. 1). We are interested then, in fields of the form
where and are the complex amplitudes and .
Fig. 1
Now, the wave equations for and were derived from Maxwell’s Equations.
However, whereas every solution to Maxwell’s Equations in free space must obey the wave equation, the converse is not true, which means Maxwell’s Equations impose extra constraints on and . In particular, since and , it follows that
which give us . So, the electric and magnetic fields do not have any components along the direction of propagation, rather they are perpendicular to the direction of propagation. That is, electromagnetic waves are transverse in nature.
Moreover, Faraday’s Law
implies a relation between the electric and magnetic amplitudes
and
.
So, we write and , which can be generalized as
Evidently, and are in phase and mutually perpendicular, their real amplitudes are related by
.
Fig. 2
If the electric field points in the direction, then the magnetic field points in the direction, we can write
and
This is the paradigm for a monochromatic plane wave (Fig. 2). The wave as a whole is said to be polarized in the direction (by convention, we use the direction of to specify the polarization of an electromagnetic wave). There is nothing special about the direction, of course—we can easily generalize to monochromatic plane waves travelling in an arbitrary direction. The notation is facilitated by the introduction of the propagation vector (or wave vector) , pointing in the direction of propagation, whose magnitude is the wave number . So, in general, it can be written as
where is the polarization vector. Because is transverse in nature, we have
Refractive Index, Wave Impedance. It can be shown that for a linear dielectric medium (which is isotropic and homogeneous), EM wave travels with a speed
where and are the electric permittivity and magnetic permeability of that dielectric medium respectively.
Now, refractive index ( ) of such a medium is defined as , which is the ratio between the speed of EM wave in vacuum to that in that medium.
Therefore, we write .
Wave impedance ( ) or characteristic impedance of a particular medium is given by . It is the characteristic of any medium that can embrace the transmission of an EM wave irrespective of whether it is associated with a power source at one terminal and a load at the other end or not. For the case of vacuum, the wave impedance is given by Ω.
EM Wave Propagation in Conductors:
In the case of vacuum, it has been argued that the free charge density and the free current density are zero, and everything that followed was predicated on that assumption. Such a restriction is perfectly reasonable when we are talking about wave propagation through a vacuum or through insulating materials such as glass or pure water. But in the case of conductors we do not independently
control the flow of charge, and in general is certainly not zero. In fact, according to Ohm’s Law, the free current density in a conductor is proportional to the electric field, given by . is known as the conductivity of the conducting medium.
With this, Maxwell’s Equations for linear media assume the form
where and are the electric permittivity and magnetic permeability of the medium respectively.
Relaxation Time. Now, the continuity equation for free charge gives us
Combining this equation with Ohm’s Law for a homogeneous medium, one can write
The solution of above equation is simple, and is given by . Thus any initial free charge exponentially dissipates in a characteristic time scale given by . This time scale is known as the relaxation time of the conductor. This reflects the familiar fact that if we put some free charge on a conductor, it will flow out to the edges. The relaxation time affords a measure of how good a conductor is. For a perfect conductor, and therefore
, as expected. For a so-called good conductor, is much less than the other relevant times given in the problem (in oscillatory systems, that means ) and for a poor conductor, is greater than the characteristic times in the problem ( ). But we are not interested in this transient behaviour.
So, from now on we will assume and we have
Applying curl to the 2nd and 4th equations, we obtain modified wave equations for and given as
These two equations will still accept plane-wave like solutions given by
but the only difference is that, this time the wave number is complex and can be expressed as the sum of real and imaginary parts, . Moreover, plugging the plane wave solutions into the equations we obtain
.
Comparing this with previously assumed separately for real and imaginary parts, we obtain
and
.
Skin Depth. Using the fact , we can re-write the plane wave solutions as and . It means, the imaginary part of (which is ) results in an exponential attenuation of the wave along direction. The characteristic length scale, called the skin depth ( ) is denoted as . It is the distance inside the conductor the wave takes to reduce the amplitude by a factor of and therefore, is a measure of how far the wave penetrates into the conductor. For a perfect conductor, giving and . A perfect conductor therefore doesn’t allow the EM wave to penetrate inside.
Meanwhile, the real part of (which is ) determines the wavelength ( ), the propagation speed ( ) and the refractive index ( ) of the conducting medium.
, and .
The attenuated plane waves satisfy the modified wave equation for any and . But Maxwell’s Equations impose further constraints, which serve to determine the relative amplitudes, phases and polarizations of and . As seen before, there are no components, since the fields are transverse. We may as well orient our axes so that is polarized along the x direction.
. Then from the 3rd Equation of Maxwell, we obtain
.
Once again, the decaying electric and magnetic fields are mutually perpendicular as seen in Fig. 3.
Fig. 3
Like any complex number, can be expressed in terms of its modulus and phase as with as the modulus and as the phase angle. Evidently the electric and magnetic fields are no longer in same phase and we get . Meanwhile, the real amplitudes of and are related by the relation
.
This concludes part 1 of this e-report.
The discussion will be continuing in the part 2 of this e-report.
Reference(s):
Introduction to Electrodynamics, D.J. Griffiths, Pearson Optics, Eugene Hecht, Pearson Education
(All the figures have been collected from the above mentioned references)
