FIELDS AND RADIATION FROM A MOVING ELECTRIC CHARGE
Musa D. Abdullahi, U.M.Y. University
P.M.B. 2218, Katsina, Katsina State, Nigeria
E-mail: musadab@outlook.com, Tel: +2348034080399
Abstract
A particle of charge Q and mass m, moving at time t with velocity v, is associated with electrostatic
field Eo and potential ψ at a point in space. It has kinetic energy ½mv2, self-potential U and intrinsic
energy ½QU. A magnetic field Hφ and an electric field Ev are generated due to velocity and a reactive
electric field X due to acceleration. The sum of energies of Hφ and Ev is equal to the kinetic energy of
the particle. The reactive electric field X acts on the same charge to produce a force, QX, equal and
opposite to the accelerating force m(dv/dt), thereby causing inertia. An oscillating particle of charge Q
emits electromagnetic radiation with directivity of 1.5. It is shown that En = ½ Mc2 is the sum of
intrinsic energies of the electric charges constituting a neutral body of mass M. For a moving body the
total energy is E = ½ M(c2 + v2), in contrast to the relativistic mass-velocity formula E = Mc2.
Keywords: Acceleration, charge, energy, fields, mass, potential, radiation, spherical shell, velocity
1.
Introduction
1.1
Configuration of a Stationary Electric Charge
If a stationary electric charge is to assume any configuration, it is most likely to be a spherical
shell of charge Q and radius a as shown in Fig.1. Such a figure has electrostatic field of
intensity Eo and potential ψ at a distance r from the charge. The field at the outer surface of
the shell pushes the charges inward while the surface tension pushes them outward.
Equilibrium is reached at a certain formation and a stable particle is obtained. Such particles
occur in pairs of opposite charges as a doublet. If there is a God-particle, such a pair is one.
Eo. Eo
r
ψ
Eo,
Eo.
Q a
Eo
Eo
a
U
Eo.
Eo
Eo. Eo
Fig. 1. A stationary particle of positive charge of magnitude Q in the form of spherical shell of radius a,
with constant electrostatic potential U and zero electrostatic field inside the shell and electrostatic field
of intensity Eo and potential ψ at a distance r outside. A negative charge has opposite fields.
A particle of charge Q and mass m, moving at time t with velocity v, has electrostatic
field of intensity Eo and potential of magnitude ψ at a point in space. The moving charge
1
generates magnetic fields Hφ and electric Ev due to velocity and a reactive electric field X
due to acceleration. The moving charge exhibits a reactive force or inertia QX = –mdv/dt and
it emits electromagnetic radiation if it undergoes acceleration. The potential ψ and fields Hφ
and X are well known in electrodynamics but not Ev. Deducing expressions for the fields Hφ,
Ev and X and explaining the source of electromagnetic radiation and the cause of inertia are
the purposes of this paper.
The reactive electric field X acts on the same charge to bring about the inertial force or
reactive force QX, equal and opposite to the accelerating force m(dv/dt). This explains the
cause of inertia as a reaction to accelerating force. In the process, an equation en = ½mc2 is
derived as the intrinsic energy of a charged particle, in terms of mass m that is independent of
velocity. For a body of mass M, a composite of numerous charged particles, the electrostatic
energy becomes En = ½ Mc2, in contrast to the relativistic equation En = Mc2 [1, 2], where:
c=
1
(1)
µoε o
where µo is the permeability and εo the permittivity of a vacuum, as shown by Maxwell [3].
1.2
Kinetic Energy of a Moving Charged Particle
Newton’s second law of motion gives the impressed force f on a particle of mass m moving
with velocity v at time t, in terms of its acceleration dv/dt, as vector equation:
f =m
dv
dt
(2)
For motion through a distance s in a straight line, equation (2) becomes a scalar expression:
f =m
dv
ds dv
dv
=m
= mv
dt
dt ds
ds
Work done by an external agent in increasing the speed of the particle from 0 to v is the
kinetic energy k of the particle, given by the integral:
s
v
o
o
k = ∫ f (ds) = ∫ mv(dv) =
1 2
mv
2
(3)
In addition, the agent, such as an electric field, may do work against a frictional force, which
appears as heat or light radiation [4].
The energy of the latitudinal magnetic field Hφ of a moving electric charge is not
equal to the kinetic energy of the particle. The discrepancy is accounted for in the energy of a
radial electric field Ev. The sum of energies of the magnetic field Hφ and dynamic electric
field Ev is equal to the kinetic energy ½mv2 of the moving particle. While the well-known
latitudinal magnetic field Hφ takes part in the generation of reactive electric field X and
emission of electromagnetic radiation the less-known radial field electric field Ev is not
involved in radiation. The radial field Ev serve as a storage medium for the balance of kinetic
energy of a particle moving with velocity v relative to an observer.
1.3
Total Energy of a Moving Charged Particle
It is shown in this paper that the total energy e of a moving charged particle is the sum of the
intrinsic energy en and the kinetic energy k as given by:
2
1 2
1
mc + k = m ( c 2 + v 2 )
(4)
2
2
For a body of mass M composed of equal numbers or equal amounts of positive and negative
electric charges, moving at speed v with kinetic energy K = I/2 Mv2, the total energy E is:
1
1
E = Mc 2 + K = M ( c 2 + v 2 )
(5)
2
2
e = en + k =
This is in contrast to the relativistic mass-velocity formula, which makes the total energy
content E = Mc2 [1, 2].
2.
Electrostatic Field, Potential and Energy of a Spherical Charge
A stationary charge shown in Fig. 1 is a spherical shell of radius a. The figure has intrinsic
potential U and intrinsic energy en. It has electrostatic field Eo and potential ψ at a point
outside the sphere. For points outside the shell, it is as if the charge Q is located at the centre.
2.1
Electrostatic Field of a Spherical Charge
The electrostatic field intensity at a point distance r outside the shell, enclosing the charge Q,
is given by Gauss law [4, 5] as:
Eo =
Q
4πε o r
2
û=−
∂ψ
û = −∇ψ
∂r
(6)
where εo is the permittivity of vacuum, ψ is the electric potential at a point distance r from the
centre of the charge, û is a unit vector in the radial direction and ∇ψ denotes the ‘gradient’ of
a scalar ψ. A closed surface inside the shell does not enclose any charge. So Gauss law makes
the electric field inside the spherical shell zero and the potential there becomes a constant U.
2.2
Intrinsic Potential of a Spherical Charge
In Fig. 1 the space inside the spherical shell is a region of constant potential U. But the
potential ψ varies inversely as the distance outside the sphere, thus:
ψ = −∫
r
∞
Q
4πε o r
2
(dr ) =
Q
4πε o r
(7)
The electric potential is the work done in bringing a unit positive charge from infinity to a
point distance r from the charge. The intrinsic potential U at a point on the surface of a
spherical shell of charge Q and radius a, as well as inside the shell, is given by:
U=
Q
4πε o a
(8)
2.3
Intrinsic Energy of a Spherical Charge
The intrinsic energy en is work done, against the electric field, in composing the magnitude of
the electric charge from 0 to Q, at constant radius a, by infinitesimal amounts dq, thus:
en = ∫
Q
0
q
4πε o a
(dq ) =
Q2
1
= QU
8πε o a 2
(9)
Equation (9) may also be obtained by integrating the energy density (energy per unit volume)
throughout volume Y of space occupied by an electric field Eo, to give:
3
1
Q2
1
2
en = ∫ ε o Eo (dY ) =
= QU
Y
2
8πε o a 2
3.
(10)
Magnetic and Electric Fields of a Moving Electric Charge
Fig. 2 shows a particle of charge Q in the form of a spherical shell of radius a moving at time
t with velocity v and acceleration dv/dt. The charge is associated with a magnetic field Hφ
and a dynamic electric field Ev, as shown in Fig. 2 below. The orthogonal fields Hφ and Ev, in
spherical coordinates (r, θ, φ), with respective unit vectors (û, ȋ, ê), are given, according to
the law of electromagnetism [4, 5], by:
ẑ
v dv/dt
Eo.
X
X
Ev
.
Eo.
Hφ
Hφ
Hφ
Hφ
Hφ
X
Ev
X
Ev
û
ψ Eo
rP
….………… a………………..xHφ
……………… .θ…………….xHφ
.……………Q…………….....xHφ
.………………… …………..xHφ
...……………………………..xHφ
Q
Q
Ev
Ev
Ev
X
X
X
X
Hφ = εov×
×Eo = êεovEosinθ
Ev = û(v/c).Eo) = û(v/c)Eocosθ
Er = Eo + Ev = Eo{1 + (v/c)cosθ}
Eo
Fig. 2 An electric charge of magnitude Q, in the form of a spherical shell of radius a with centre O,
moving at time t with velocity v and acceleration dv/dt in the ẑ -direction, generating a magnetic field
Hφ and an electric field Ev due to velocity and a reactive electric field X due to acceleration. The
magnetic field Hφ (dotted horizontal arrows), in the latitudinal ê-direction, is out of the page on the lefthand side and into the page on the right-hand side and the electric field Ev (green radial arrows), in the
radial û direction, is away from the centre O at the top and toward the centre at the bottom. The
direction of the reactive electric field X (dashed vertical arrows) is opposite the acceleration dv/dt. The
charge moves with its radial electrostatic field of intensity Eo (red radial arrows) and has potential ψ at
a point distance r and at an angle θ to the velocity v. The total electric field at a point P due a charge
moving with v is Er = Eo{1 + (v/c)cosθ}, where θ is the angle between v and OP.
Hϕ = ε o ( v × Eo ) = êε o vEo sin θ
(11)
v
v
Ev = û .Eo = û Eo cos θ
c
c
(12)
Er = Eo + Ev = Eo {1 + (v / c) cos θ }
(13)
The latitudinal magnetic field Hφ is well known but not the dynamic (radial) electric field Ev.
The total electric field Er due to a charged particle moving with velocity v, relative to an
observer, is given by equation (13).
3.1
Energy of Magnetic Fields due to a Moving Electric Charge
Energy of magnetic field Hφ and dynamic electric field Ev due to an electric charge moving
with velocity v is obtained, from equations (11) and (12) as volume integral:
4
v 2 en
1
1
2
2
2
2 2 2
2
2
( µ Hϕ + ε Ev ) (dY ) = 2 ∫Y Eo ( µ ε v )( sin θ + cos θ ) (dY ) = c2
2 ∫Y
(14)
where Y is the volume of space occupied by the field and en is the intrinsic energy.
3.2
Reactive Electric Field due to an Accelerated Electric Charge
Equations (11) can also be expressed as:
Bϕ = µo Hϕ = µoε o ( v × Eo ) = − µoε o v × ∇ψ = µoε o∇ ×ψ v = ∇× A
(15)
where ∇ψ denotes the gradient of scalar potential ψ, ∇. denotes the divergence of a vector,
∇× is the curl of a vector, E = -∇ψ, ∇.v = 0 and ∇×v = 0. The magnetic vector potential, at a
point P in space, is:
A = µoε oψ v
(16)
If the charge is accelerated, a reactive field X is generated, given by equation (16) and
Faraday’s law of electromagnetic induction [5, 6], thus:
∂Bϕ
∂
∂A
∇ × A = −∇ ×
∂t
∂t
∂t
∂A
∂v
X=−
= − µoε oψ
∂t
∂t
∇×X = −
4.
=−
(17)
Radiation from an Oscillating Electric Charge
For a charged particle oscillating with angular frequency ω and velocity v = vmsinωτ of
amplitude vm, the magnetic vector potential A, measured by an observer at a point distance r
from the charge and at present time t and retarded time τ = t – r/c, is:
A = µoε oψ v m sin ωτ =
µoQv m
 r
sin ω  t − 
4π r
 c
(18)
where c is the speed of light in a vacuum. The retardation of vector potential in equation (18),
as a result of finite speed of light c, is responsible for electromagnetic radiation.
Equations (17) and (18) give Bφ and X as:
Bϕ = ∇ × A = û ×
µ Qv
∂A
= −û × o m
4π r
∂r
X=−
1
 r ω
 r 
 sin ω  t −  + cos ω  t −  
 c c
 c 
r
∂A −ωµo Qv m
 r
=
cos ω  t − 
∂t
4π r
 c
(19)
(20)
Taking the vector (cross) product with X and Hφ gives the Poynting vector:
2
 Q 
 r  1
 r ω
 r 
X × Hϕ = ωµo 
 v m × ( û × v m ) cos ω  t −   sin ω  t −  + cos ω  t −  
 4π r 
 c r
 c c
 c 
(21)
Equation (21) gives the power flow across unit area. This is made up of power Ps
stored in the magnetic field Hφ and power Pr dissipated as electromagnetic radiation into
space. The total power flow P, across surface area S surrounding the charge, is:
5
2
 Q 
 r  1
 r ω
 r 
P = ∫ ωµo 
cos ω  t −   sin ω  t −  + cos ω  t −   v m × (û × v m ).(dS)

S
 4π r 
 c r
 c c
 c 
2
2
 Qvm   ωµO
 r  µ oω
 r 
P=∫ 
sin 2ω  t −  +
cos 2 ω  t −   sin 2 (dS )

S 4π r 
c
 c
 c 

  2r
2
2
 Qv   ωµ
 r µω
P = ∫  m   o sin 2ω  t −  + o
S 4π r
2c
 c

  2r

 r   2
1 + cos 2ω  t −   sin θ (dS )
 c  

(22)
where:
( v m × û ) . ( v m × (d S) ) = vm2 sin 2 θ (dS )
4.1
Power in the Fields of an Oscillating Electric Charge
Equation (22) gives the power in the magnetic and electric fields as:
2
2
 Qv   ωµ
 r µ ω
 r 
Ps = ∫  m   o sin 2ω  t −  + o cos 2ω  t −   sin 2 θ (dS )
S 4π r
2c
 c
 c 

  2r
(23)
Power is exchanged between the magnetic field and the electric field. Over one cycle or
several cycles, the energy dissipated in the fields is zero.
4.2
Energy Radiated by an Oscillating Electric Charge
Equation (22) has a steady term, which indicates dissipation of power. The power radiated is:
2
 ωQvm 
Pr =
sin 2 θ (dS )
∫
S
2c  4π r 
µo
2
Pr =
µo  ωQvm 
2c  4π 
Pr =
µo
µ
2
2
(ωQvm ) ∫ sin 3 θ (dθ ) = o (ωQvm )
16π c
12π c
where dS = r2sinθ(dθ)(dφ) and
∫
π
0
1
∫∫ r
2
sin 3 θ (dθ ) =
(24)
sin 2 θ {r 2 sin θ (dθ )(dϕ )}
(25)
4
. The energy radiated, in time T, is TPr.
3
4.3
Directivity of Emission
Equation (24) gives the maximum intensity of radiation (radiant power per unit area)
Rm, with θ = π/2 radians, as:
µ  ωQvm 
Rm = o 
2c  4π r 
2
An isotropic radiator with power intensity R emitting the same power Pr (equation 25),
through spherical surface area 4πr2, makes:
µ  ωQvm 
R= o
3c  4π r 
2
The directivity DR is:
DR =
Rm
= 1.5
R
6
(26)
5.
Inertia of a Moving Electric Charge
An electric charge of magnitude Q in the form of spherical shell of radius a, under
acceleration, generates a reactive electric field X as given by equation (17). This field acts on
the self-same charge Q to produce a reactive force F equal and opposite to the accelerating
force in accordance with Newton’s second law of motion, thus:
F = QX = − µoε oψ Q
∂v
∂v
= −m
∂t
∂t
(27)
Equation (27) explains the cause of inertia.
At the location of the charge, the potential ψ = U and zero everywhere else, and,
therefore, equation (27) becomes:
µoε oUQ
∂v
∂v
=m
∂t
∂t
m = µoε oUQ = 2en µoε o
(28)
(29)
where en is the intrinsic energy of the charge as given by equation (10). Substituting for U
from equation (8) we obtain equation (29) as:
m=
µo Q 2
2e
= 2en µoε o = 2n
4π a
c
(30)
Equation (30), expressing mass m in terms of electrical quantities, gives:
en = ½ mc2
(31)
Substituting for en from equation (10) gives kinetic energy k as:
k=
1 2 v2
1
mv = 2 en = µoε o2 v 2 Eo2 (dY )
∫
Y
2
c
2
(32)
Equation (32) is the same as equation (14).
6.
Conclusion
Equations (12), (13), (14), (15), (16), (17), (25), (26) and (27) are what this paper set
out to derive. The dynamic electric field Ev is missing in electrodynamics. The kinetic energy
of a moving charged particle is contained in the magnetic field Hφ and electric field Ev.
Equation (25) shows that whenever an electrically charged particle is somehow
disturbed or shaken, it emits electromagnetic radiation of power proportional to the square of
product of amplitude of disturbance and frequency of oscillation. So, electric charges
oscillating in the electric fields of an atom also emit corresponding electromagnetic radiations
at the microwave frequencies of oscillation. This is in addition to emission of heat and light
radiation, as shown by the author [4], due to ‘frictional’ motion of charged particles, in the
electric field of the nucleus of an atom or in the electric fields between the charged particles.
This may explain the source of microwave background radiation [7].
An interesting result of this paper is the derivation of directivity of emission from an
oscillating charge. Equation (26) gives the directivity as 1.5, same as that of a dipole antenna.
The paper also succeeded in explaining the cause of inertia, expressed in equation
(27), as due to the reactive electric field X, generated by an accelerated charged particle,
acting on the same charge to produce the inertial force equal and opposite to the accelerating
7
force. For a body of mass M composed of electrically charged particles moving with velocity
v at time t, the reactive electric fields act at the location of the respective charges to produce
the sum of the inertial forces equal and opposite the accelerating force M(dv/dt).
In this paper, the mass of a charged particle is expressed in terms of electrical
quantities and the mass-energy equivalence law for a body of mass M moving with speed v is
derived as E = ½ M(c2 + v2), where En = ½ Mc2 is the electrostatic energy of the electric
charges composing the body. The mass-energy equivalence law, derived here from basic
electrical principles, differing from the relativistic equation by a factor of 1/2, is noteworthy.
References
[1] M. Jammer; Concept of Mass in Classical and Modern Physics, Cambridge, MA,
Harvard University Press (1961)
[2] M. Lange; “The most famous equation”, Journal of Philosophy (2001), 98, 219-238
[3] J. C. Maxwell; A Treatise in Electricity and Magnetism, Oxford, 3rd ed. (1892) Part 1V
[4] M. D. Abdullahi “An Alternative Electrodynamics to the Theory of Special Relativity”
Online at: www.musada.net/Papers/Paper1.pdf
[5] D. J. Griffith; Introduction to Electromagnetism, Prentice-Hall, Englewood Cliff, New
Jersey (1996)
[6] I. S. Grant & W. R. Phillips; Electromagnetism, John Wiley & Sons, N, York (2008)
[7] en.wikipedia.org/wiki/Cosmic_microwave_background
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