Download light as a form of electromagnetic wave in time harmonic fields

African Journal of Science and Research,2015,(4)3:13-15
ISSN: 2306-5877
Available Online: http://ajsr.rstpublishers.com/
LIGHT AS A FORM OF ELECTROMAGNETIC WAVE IN TIME HARMONIC
FIELDS (THEMF).
1Nwogu
1, 2
Uchenna, 2Emerole Kelechi, 3Akujuobi Azubuike
Department of Electrical Electronic Engineering Federal Polytechnic Nekede, Owerri, Imo State, Nigeria.
3 Department of Computer Science, University of Portharcourt, Portharcourt Rivers State, Nigeria.
Email: [email protected]
Received:13, Mar,2015
Accepted: 24,Apr,2015.
Abstract
Light can also be shown to exist in time harmonic fields, where complex phasor analysis, Maxwell’s equations and constituent
relations from vector calculus are deployed to unearth the plane velocity embedded in electromagnetic radiations. A sinusoidal
vector phasor radiation that contains this phenomenon in phase, direction and magnitude is x-rayed and using vector calculus
equations and constitutive relations to bring out the phenomenon of light (plane velocity) in relation to the speed of light in a
medium.
Keywords: light, electromagnetic, vector, domain, frequency, velocity, harmonic.
INTRODUCTION
Fields for which the time variation is sinusoidal are
called time harmonic fields. To further work in this domain,
phasor analysis (complex quantities) are employed to get a
simple frequency steady state response.
⃗ (x,y,ƶ) = Re[E(x,y,ƶ, t) ejɷt]………….(1)
From (1) above, ⃗ (x,y,ƶ) is a vector phasor that contain the
information in direction, magnitude and phase.
Also in the time harmonic field, the rate of change of E can
be written as;
ɷ
(x,y,ƶ,t) = Re[jɷE(x,y,ƶ, t) ej t]……….(2)
We can deduce that if the electric field vector E(x,y,ƶ)
is represented in phasor form as E(x,y,ƶ), then (x,y,ƶ,t)
can be represented by the phasor, jɷ E(x,y,ƶ).
Also the integral ∫
can be represented by the
phasor (x,y,ƶ,)
These time harmonic relationships can be applied to
both magnetic and electric field quantities in addition with
Maxwell equations and other constitutive relations to
expose the underlying entity called light.
LIGHT IN TIME HARMONIC FIELDS
Consider the field phasor or vector (E , H) and source
phasor (β, j) in a simple linear isotropic medium as shown
in Fig 1.
Fig. 1 Field phasor and Vector phasor diagrams
When the H and E fields vary with time sinusoidally, the
time derivations can be replaced by factors of jw and
Maxwell’s equation can be deduced to [4]:
V X H = E + jwϵE………(3)
V X E = - jwH…………….(4)
V. B = 0…………………..(5)
V.D= ……………………(6)
Where = jɷD,
- jɷB, D = Є Ē, B = μH and
from ohm’s law, J = Ē
Consider a magnetic field density B represented in the
harmonic field as B = Bcos(wt-BZ)y, it can be seen that the
signal is an oscilloscope in a ;positive B direction (direction
of wave propagation) and the magnetic field in the y
direction. But by right hand rule, the electric field
component will be in the x direction. Decomposing the
equation 7 and assuming free space.
Emerole Kelechi et.al.
14
B
:.
γ √
⁄ )… .………….(15)
The real and imaginary part of (15) gives the wave
attenuation , and the waves propagation constant of radio
signals in Air β
H
:.
= attenuation =
√ ⁄
⁄
)2 -1
………………………………..……(16)
E
......
A
A
....
....
S
.........
E
S
β = propagation constant = ɷ√ ⁄
D
⁄
)2 +1
………..…………………..(17)
.
D2
I.
Field vectors
Fig. 2 Field vectors
B = μH = B˳cos(ɷZ - βZ)y
H=
cos(ɷt - βZ)y
In free space,
β˳ =
⇒ H = H˳cos(ɷt –βƵ)y
H = H˳cos(ɷt –βƵ)y Amp/meters
From Max. equation, V X H = jɷЄ˳Ē
:.
E
=
=
electric
field
component............………………………(9)
From Equa. (9), V X H gives H˳cos (cot – βƵ) y
= ɷ√ ⁄
=H˳sin (ɷt – βƵ) (since, Ē XH fields has a 900 or
phase lead in time harmonic)
:. ɷt – βƵ = 0
:. ɷt – βƵ = = = phase velocity =
Where Ƶ = distance propagated
T = displacement
:.
=
= time harmonic phase velocity of light
…………………..……………..(10)
GENERAL WAVE
HARMONIC FIELDS
EQUATION
IN
Electromagnetic wave propagation and physical
interpretations.
In free space, the conductivity, δ 0. This shows
that the magnitude of both E and H alternates as it is being
propagated in any direction in a given medium. Applying
the complex wave propagation equation to decouple the
propagation components of E and H, we have:
Consider a case of perfect dielectric, where δ 0 (free
space)
Solving for
and β in equation (16) and (17) and
substituting δ 0
TIME
From Maxwell equations in the time harmonic fields,
Equ (1) = V X H = (δ + jɷϵ)
Equ (2) = V X E = -jɷμH
Taking the curl of (1) & (2)
X (
X H) = (δ + jɷϵ)(
X Ē)
………………………………………….(11)
X
(
X
E)
=
jɷμ
(
X
H)
…….………………………………..….(12)
Note that the X ( X A)
( .A) – 2 A2
ubstituting the values of X E and X H in (11) and (12)
and taking note of the Laplacian of a vector in the
Cartesian co-ordinate yields the vector wave equation
where γ complex wave propagation constant [4]:
V2.H = jɷμ (δ + jɷЄ)H γ2H ......…… (13)
V2.E = jɷμ (δ + jɷЄ) E = γ2E……….…(14)
:.
γ2 jɷμ (δ + jɷЄ)
= 0 (attenuation)
For β, substituting the value of δ in (17)
= ɷ√ ⁄
=ɷ
Β ɷ
....................………………(18)
Equation (18) shows why signal propagates in a dielectric
medium
:.
Β ɷ
In free space, μ μ˳, ϵ = ϵ˳
:. =
...........................…………(19)
Comparing equation (10) and (19), we have that.
phase velocity of light in an
electromagnetic field
Substituting the values of
for free space, we
have
3X108m/s = velocity of light (Q.E.D)
This shows that light is a form or an integral part of
both the electric E and magnetic field H. This also shows
that as a result of speed component of light, signal
(electromagnetic) is able to propagate in a medium even
when the attenuation is zero as in a perfect dielectric or
when you rapid alternation (
as in a perfect
conductor.
CONCLUSION
It has been shown that Maxwell’s equations leads
directly to Electrical and Magnetic fields satisfying the wave
equation for which the solution are linear combinations of
plane waves travelling with the speed of light C =
. The
African Journal of Science and Research,2015,(4)3:13-15
R.H.S (Right Hand Side) of the above equation is a
quantity relating to the equation governing electrical and
magnetic fields. It has units of velocity, but can be seen
that its derivatives components (E and B) forces had no
physical velocity components initially [5]. The fact is,
Maxwell’s equations explain how these waves can
physically propagate through space: the changing
magnetic field creates a changing electric field from
Faraday’s law. In turn, the Electric field creates a changing
Magnetic field through Maxwell’s correction to Ampere
laws. The perpetual cycle allows these waves, now known
as electromagnetic radiation to move through free space at
a velocity C, which is the speed of light.
From the decoupling procedures, it could deduce this
component speed of light (C), in the form of plane velocity
vsp embedded in the electromagnetic radiation is related to
the speed of light. (The ratio of the absolute
electromagnetic unit charge to the absolute electrostatic
unit charge, which is the ratio
).
References
1)Electronic Communication Systems by Kennedy and
Davis (5th Edition, 2000)
2)Communications Electronic by Johnson I. Ejimanya
(2005)
3)Understanding Fiber Optics by Jeff Hedit (2006)
4)Radiation Fields of Helical Antenna by S.A Adekola, Ike
Moete,A Ayorinde(2010)
5)Basic Electromagnetic Wave Properties of Light by
Matter J. Pary Hill (2009)
6)www.icbse.org/education: introduction to light waves
(2012).
7)www.jewave.com (2013)
8)www.nptel.acrom/course117.