TASK III: Analysis Electromagnetic Radiation with Maxwell Theory
Anjar Sari (1017016300991)
Physic Education V

ANALYSIS ELECTROMAGNETIC RADIATION


1. MAXWELL'S EQUATIONS
Maxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave. Individually, the equations are known as Gauss's law, Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction. The set of equations is named after James Clerk Maxwell.
These four equations, together with the Lorentz force law are the complete set of laws of classical electromagnetism. The Lorentz force law itself was actually derived by Maxwell under the name of "Equation for Electromotive Force" and was one of an earlier set of eight equations by Maxwell.
 Conceptual description
This section will conceptually describe each of the four Maxwell's equations, and also how they link together to explain the origin of electromagnetic radiation such as light. The exact equations are set out in later sections of this entry.
• Gauss's law relates electric charge contained within a closed surface (Gaussian surface) to the surrounding electric field. It describes with mathematical clarity how the divergence of an electrical field is affected by charges (electric field lines diverge from positive charges and are drawn towards negative charges). It also states that the total electric flux through a Gaussian surface is unrelated to the shape and size of that surface.
• Gauss's law for magnetism states that the total magnetic flux through a Gaussian surface is zero. It is equivalent to saying that the magnetic field is a solenoidal vector field. This is due to real world magnetic charges coming in pairs (referred to as dipoles), with the two charges giving rise to opposite magnetic field divergences which cancel each other out. The theoretical single magnetic charge is referred to as a magnetic monopole. Magnetic monopoles have never been observed, but if they do exist, this law would need to be modified.
• The Maxwell equation that is known as 'Faraday's law' was so named by Oliver Heaviside. It describes how a changing magnetic field is related to the induced electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators. It should be noted however that this particular equation only caters for the time varying aspect of electromagnetic induction, and not for the motionally induced aspect, and that it takes on a different mathematical form than Michael Faraday's original law. In the original Faraday's law of induction, both aspects of electromagnetic induction are catered for.
• Ampère's law with Maxwell's correction states that magnetic fields can be generated in two ways: By electrical current (this was the original "Ampère's law") and by changing electric fields. The idea that a magnetic field can be induced by a changing electric field follows from the modern concept of displacement current which was introduced to maintain the solenoidal nature of Ampère's law in a vacuum capacitor circuit. This modern displacement current concept has the same mathematical form as Maxwell's original displacement current. Maxwell's current applies to the polarization current in a dielectric medium, and it sits adjacent to the modern displacement current in Ampère's law.
Maxwell's correction to Ampère's law was particularly important. In 1864, Maxwell derived the electromagnetic wave equation by linking the displacement current to the time-varying electric field that is associated with electromagnetic induction. This is described in his A Dynamical Theory of the Electromagnetic Field, where he wrote:
"The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws."
Formulation in terms of free charge and current
Name Differential form
Integral form

Gauss's law



Gauss's law for magnetism



Maxwell–Faraday equation
(Faraday's law of induction)



Ampère's circuital law
(with Maxwell's correction)  


Formulation in terms of total charge and current
Name Differential form Integral form
Gauss's law


Gauss's law for magnetism


Maxwell–Faraday equation
(Faraday's law of induction)


Ampère's circuital law
(with Maxwell's correction)



 Maxwell's equations in terms of E and B for linear materials
Substituting in the constitutive relations above, Maxwell's equations in linear, dispersionless, time-invariant materials (differential form only) are:




These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), and only free charges and currents are included (instead of all charges and currents). Unless that material is homogeneous in space, ε and μ cannot be factored out of the derivative expressions on the left-hand sides.
Electromagnetic radiation (sometimes abbreviated EMR) is a ubiquitous phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation. Electromagnetic radiation is classified into several types according to the frequency of its wave; these types include (in order of increasing frequency and decreasing wavelength): radio waves, microwaves, terahertz radiation, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. A small and somewhat variable window of frequencies is sensed by the eyes of various organisms; this is what we call the visible spectrum, or light.
Electromagnetic waves were first postulated by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz. Maxwell derived a wave form of the electric and magnetic equations, revealing the wave-like nature of electric and magnetic fields, and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave.
According to Maxwell's equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa. Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form an electromagnetic wave.
A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics.


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized wave propagating from right to left. The electric field is in a vertical plane, the magnetic field in a horizontal plane.


The physics of electromagnetic radiation is electrodynamics, a subfield of electromagnetism. Electric and magnetic fields obey the properties of superposition so that a field due to any particular particle or time-varying electric or magnetic field will contribute to the fields present in the same space due to other causes: as they are vector fields, all magnetic and electric field vectors add together according to vector addition. For instance, a travelling EM wave incident on an atomic structure induces oscillation in the atoms of that structure, thereby causing them to emit their own EM waves, emissions which alter the impinging wave through interference. These properties cause various phenomena including refraction and diffraction.
Since light is an oscillation it is not affected by travelling through static electric or magnetic fields in a linear medium such as a vacuum. However in nonlinear media, such as some crystals, interactions can occur between light and static electric and magnetic fields — these interactions include the Faraday effect and the Kerr effect.
In refraction, a wave crossing from one medium to another of different density alters its speed and direction upon entering the new medium. The ratio of the refractive indices of the media determines the degree of refraction, and is summarized by Snell's law. Light disperses into a visible spectrum as light is shone through a prism because of the wavelength dependent refractive index of the prism material (Dispersion).
EM radiation exhibits both wave properties and particle properties at the same time (see wave-particle duality). Both wave and particle characteristics have been confirmed in a large number of experiments. Wave characteristics are more apparent when EM radiation is measured over relatively large timescales and over large distances while particle characteristics are more evident when measuring small timescales and distances. For example, when electromagnetic radiation is absorbed by matter, particle-like properties will be more obvious when the average number of photons in the cube of the relevant wavelength is much smaller than 1. Upon absorption the quantum nature of the light leads to clearly non-uniform deposition of energy.
There are experiments in which the wave and particle natures of electromagnetic waves appear in the same experiment, such as the diffraction of a single photon. When a single photon is sent through two slits, it passes through both of them interfering with itself, as waves do, yet is detected by a photomultiplier or other sensitive detector only once. Similar self-interference is observed when a single photon is sent into a Michelson interferometer or other interferometers.
 Electromagnetic spectrum


Generally, EM radiation (the designation 'radiation' excludes static electric and magnetic and near fields) is classified by wavelength into radio, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays and gamma rays. Arbitrary electromagnetic waves can always be expressed by Fourier analysis in terms of sinusoidal monochromatic waves which can be classified into these regions of the spectrum.
The behavior of EM radiation depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. When EM radiation interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. Spectroscopy can detect a much wider region of the EM spectrum than the visible range of 400 nm to 700 nm. A common laboratory spectroscope can detect wavelengths from 2 nm to 2500 nm. Detailed information about the physical properties of objects, gases, or even stars can be obtained from this type of device. It is widely used in astrophysics. For example, hydrogen atoms emit radio waves of wavelength 21.12 cm.


2. WAVE EQUATION ANALYSIS
Wave equation models a pile as a series of masses connected by springs and a hammer blow as a compressive stress wave which travels trough the pile. General form of wave equation
analysis is:

where u is longitudinal displacement of a point, E is modulus of elasticity, ρ is density, t is time, x is longitudinal direction, and Rd is soil resistance.

Wave equation analysis programs are utilized more reliable pile driving result. In this paper, the MICROWAVE is conducted. The wave equation analysis will provide driving system of pile-hammer combination suitability, driving stresses, as well as pile drivability or adequacy of the driving system to achieve required bearing capacity.
The wave equation is usually used to investigate bearing capacity graph which is a plot of ultimate soil resistance versus set. Another is to give information about equipment compatibility-solutions for determining the type of hammer. A pile-hammer system is a set of discrete element which can be solved by using of springs and dampers. Smith (1960) proposed five basic equations for wave equation
analysis:

Dm,t = Dm,t-l + Vm,t-l.dt

where: Dm,t = displacement of element m when t = t
Dm,t-1 = displacement of element m when t = t-1
Cm,t = spring compression of element m when t = t
Km = pile’s spring constant include cap, capblock, and cushion
Fm,t = spring force of element m
R’ = soil resistance, include damping effect
Rm,t = final force resultant
Vm,t = velocity of element m when t = t
Wm = weight of element m
g = gravitation
Δt = time interval
Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. If you inspect Maxwell's equations without sources (charges or currents) then you will find that, along with the possibility of nothing happening, the theory will also admit nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell's equations for free space:




where
is a vector differential operator (see Del).
One solution,
,
is trivial.
To see the more interesting one, we utilize vector identities, which work for any vector, as follows:

To see how we can use this take the curl of equation (2):

Evaluating the left hand side:

where we simplified the above by using equation (1).
Evaluate the right hand side:

Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely


Applying a similar pattern results in similar differential equation for the magnetic field:
.

These differential equations are equivalent to the wave equation:

where
c0 is the speed of the wave in free space and
f describes a displacement
Or more simply:

where is d'Alembertian:

Notice that in the case of the electric and magnetic fields, the speed is:

Which, as it turns out, is the speed of light in free space. Maxwell's equations have unified the permittivity of free space ε0, the permeability of free space μ0, and the speed of light itself, c0. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism.
But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwell's equations. Let's consider a generic vector wave for the electric field.

Here is the constant amplitude, f is any second differentiable function, is a unit vector in the direction of propagation, and is a position vector. We observe that is a generic solution to the wave equation. In other words
,
for a generic wave traveling in the direction.
This form will satisfy the wave equation, but will it satisfy all of Maxwell's equations, and with what corresponding magnetic field?


The first of Maxwell's equations implies that electric field is orthogonal to the direction the wave propagates.


The second of Maxwell's equations yields the magnetic field. The remaining equations will be satisfied by this choice of .
Not only are the electric and magnetic field waves traveling at the speed of light, but they have a special restricted orientation and proportional magnitudes, E0 = c0B0, which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as .
From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left; but this picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization.