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ISSN 2229-5518
Amol S Jagtap
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Any accelerating charge particle produces
M. Newstead et al[7] have concluded that the EM wave
has zero mass at speed c, and when collides with matter its
hν
electromagnetic wave. Electromagnetic (EM) wave consists of
mass is given by m =
c2
. But studies [8-11] have shown that
electric and magnetic field which travels perpendicular to each other and perpendicular to its direction of propagation. James
photon always carries mass equivalent to its energy. Since
photon is an EM wave, the EM wave has to carry mass
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Maxwell have derived a wave form of the electric field and
magnetic field equations [1,2]. The speed of electromagnetic wave calculated from Maxwell’s equation shows that these waves are travelling at the speed of light in vacuum i.e. at a speed (c) of 2.9979 X 108 m/s, from this Maxwell concluded that light itself is an EM wave [1,2,3]. A. Einstein Shows that a light is emitted as well as absorbed in discrete quanta of
energy ( hν ) called photon [4]. It has shown that the photon is
the particle of electromagnetic field [5]. According to wave- particle duality principle, electromagnetic wave exhibits both wave and particle properties at the same time [3,6].
Corresponding Auther: Amol S Jagtap
Department of Radiotherapy, Cama & Albless Hospital, Mahapalika Marg,
equivalent to its energy. In present work, I have shown that
EM wave is always associated with mass equivalent to its energy along with Electric and magnetic field and the amplitude of electromagnetic field is depends upon energy and mass of EM wave.
James Maxwell’s equations are the foundation of classical electrodynamics, classical optics and electrical circuit. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents [2].
Let us see the Maxwell’s equations [1-3] by considering electromagnetic wave as a monochromatic wave in empty space, with no currents or charges present.
Mumbai-400001, India.
Email: amol_jagtapm@yahoo.com
∇.E = 0
∇.B = 0
(1)
(2)
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 36
ISSN 2229-5518
mc
∇× E = − ∂B
∂t
(3)
E = Em sin
( x − ct )
B = B
mc
x − ct
∇× B = µ ε ∂E
(4)
m sin
( )
0 0 ∂t
Where, E
is the electric field, B is the magnetic field, µ0
From the Maxwell’s third equation (3) we can write the x
is the permittivity, and ε 0
is the permeability of the free
components of the equation as,
space.
According to Maxwell's equations, a spatially varying electric field is always associated with a magnetic field that changes over time and given by following equations [11]
∂E = − ∂B
∂x ∂t
By putting the values of E & B in above equation and by differentiating we get,
mc mc
mc2
mc
E = Em sin (κ x − ωt )
(5)
Em cos
( x − ct ) = − −
Bm cos
( x − ct )
B = Bm sin (κ x − ωt )
(6)
=mc
=mc
Where, κ = 2π / λ
is the wave number,
Em cos
( x − ct ) = cBm cos
( x − ct )
ω = 2πν
is the angular frequency,
Therefore,
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Em and Bm are the amplitudes of the electric and magnetic
c = Em
Bm
(11)
field.
The energy and momentum carried by photon are related to frequency (ν ) and wavelength ( λ ) i.e.
The above equation gives the relation between speed of light or photon and the amplitude of electric and magnetic
component of EM wave.
Energy ( E ) = hν = ω
(7)
ω 2πν ν
Also,
c = = =
Momentum ( p ) = h = κ λ
(8)
κ 2π / λ λ
ω Em
But, the energy and momentum of photon [8-11] in terms of
its mass is given by
Therefore,
c = =
κ Bm
E = mc2
p = mc
From equations (7) to (10), we can write,
mc2
ω =
κ = mc
(9) (10)
The speed of light or photon is also related to the
permittivity and permeability of the medium. Speed of photon when travelling in vacuum is given by following equation [1,2,11]
c = 1 (12)
µ0ε 0
By putting the values of speed of photon ( c ), from equations (11) and (12), into the equation (9) and (10), we
Therefore equation (5) and (6) of electromagnetic field in terms of mass can be written as follows,
can write the equation of energy and momentum carried by electromagnetic wave of mass ( m ) is,
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 37
ISSN 2229-5518
E = mc2 = m
p = mc = m
1
µ0ε 0
1
µ0ε 0
E 2
= m m
m
= m Em
Bm
(13)
(14)
2. Wikipedia: the free encyclopaedia [Internet]. [Modified 2013 April 19]. Available from: http://en.wikipedia.org/wiki/Electromagnetic_wa ve_equation.
3. Wikipedia: the free encyclopaedia [Internet].
Above equation gives the energy and momentum of
electromagnetic wave in terms of its mass and its electric and magnetic field amplitudes.
Therefore from energy and momentum relation, the mass of
EM wave can be written as,
B2
m = E µ ε = E m
0 0 2
m
[Modified 2013 June 3]. Available from: http://en.wikipedia.org/wiki/Electromagnetic_rad iation.
4. Jevremovic T. Nuclear principles in engineering.
Springer, USA; 2005. p. 26.
5. Baierlein R. Does nature converts mass into energy?. Am. J. Phys. 2007; 75(4) 320-325.
m = p
B
µ ε = p m
6. Wikipedia: the free encyclopaedia [Internet].
0 0
Em
From above equations it is shown that energy and momentum of EM wave are always associated with its mass
[Modified 2013 June 12]. Available from:
https://en.wikipedia.org/wiki/Light.
7. Mark AN, Stephen CN. Mass of an
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along with electric and magnetic component of the
electromagnetic wave. Also mass and energy of EM wave depends upon the medium through which EM wave travels.
Here I have shown that the electromagnetic wave always carry mass depending upon its energy along with its electric and magnetic field component. Because the speed of light is depends upon the medium, the mass and energy of light are also depends upon the medium through which is passes. This theory explains how EM wave exhibits wave as well as particle nature at same time. Electromagnetic field gives wave nature and mass gives particle nature of EM wave.
1. Wikipedia: the free encyclopaedia [Internet]. [Modified 2013 June 18]. Available from: https://en.wikipedia.org/wiki/Maxwell%27s_equ
ations.
Electromagnetic wave. [Submitted 2011 May
27]. Available from:
http://www.rxiv.org/pdf/1105.0041v2.pdf.
8. Jagtap AS. On the conservation of mass in pair production and annihilation process and energy of emitted photon in annihilation process, European Journal of Scientific Research, 2013;
109 (01): 101-106.
9. Tu LC, Luo J, Gillies G T. The mass of the photon. Rep. Prog. Phys. 2005; 68: 77–130.
10. Kidd R, Adrini J, Anton A. Evalution of the
Modern Photon. Am. J. Phys.1989; 57(1).
11. Halliday D, Resnik R. Fundamentals of Physics
(9th ed.), John Wiley & Sons, USA; 2011. p. 891-
896.
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