Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 361
Electromagnetic Forces on Charged Particles
Zi-Hua Weng
School of Physics and Mechanical & Electrical Engineering
Xiamen University, Xiamen 361005, China
Abstract— The paper presents a theoretical method with the quaternion algebra to define the
electromagnetic force from the linear momentum. It predicts that a new force of electromagnetic
field will cause a test charge to move along the magnetic force line. In a constant magnetic field,
the test charge will do uniform circular motion in the direction perpendicular to the magnetic
force line, and do uniform variable rectilinear motion along the magnetic force line. The resulting
track of the test charge is a helix with variable screw-pitch. The deduction can explain why the
solar wind varies velocity along the magnetic force line in the solar system.
1. INTRODUCTION
The force is one of important physical quantities for electromagnetic field. There is only Lorentz
force in the electromagnetic field described by the vectorial quantity. But this viewpoint can not
explain why the solar wind varies velocity along the magnetic force line in the interplanetary space
or the coronal hole on the sun [1].
One of J. C. Maxwell’s research characteristics is to demonstrate the electromagnetic property
with two methods simultaneously, the vectorial quantity and quaternion algebra [2]. The quaternion
was invented by W. R. Hamilton in 1843. Subsequently the quaternion was decomposed as the
scalar quantity and vectorial quantity. J. C. Maxwell mixed two methods to expatiate upon the
electromagnetic properties. Nowadays, it is found that there are a few discrepancies between two
descriptions with the vectorial quantity and quaternion algebra respectively [3, 4]. The discrepancy
has being become an attentive focus [5, 6].
The electromagnetic theory described by the quaternion asserts that there are a few new forces
besides Lorentz force [7]. One of them can cause a test charge to move along the magnetic force line.
This consequence does not violate any principle of physics, but it must be validated experimentally.
In the electromagnetic theory with the vector, it does not deal with the charge’s force along the
magnetic force line, and we were used to suppose the force to be zero. However, the deduction
has never been validated before. Currently, this question leaves the blank theoretically, and is
not validated experimentally. The prediction about new forces has become an emergent and key
question to be broke through in the researching of electromagnetic field.
The paper presents a theoretical method with quaternions to define the electromagnetic force
from the linear momentum. It attains a result that a new one of electromagnetic forces will cause
the charge to move along the magnetic force line.
2. ELECTROMAGNETIC FORCE
The gravitational field and electromagnetic field both can be illustrated by the quaternion, and their
quaternion spaces will be combined together to become the octonion space. In other words, the
characteristics of gravitational field and electromagnetic field can be described with the octonion
space at the same time.
In the quaternion space for the gravitational field, the basis vector Eg = (1, i1 , i2 , i3 ), and the
radius vector Rg = (r0 , r1 , r2 , r3 ), with the velocity Vg = (v0 , v1 , v2 , v3 ). For the electromagnetic
field, the basis vector Ee = (I0 , I1 , I2 , I3 ), the radius vector Re = (R0 , R1 , R2 , R3 ), and the velocity
Ve = (V0 , V1 , V2 , V3 ), with Ee = Eg ◦ I0 .
The Ee is independent of the Eg . Both of them can be combined together to become the basis
vector E of the octonion space.
E = (1, i1 , i2 , i3 , I0 , I1 , I2 , I3 )
(1)
The radius vector R(r0 , r1 , r2 , r3 , R0 , R1 , R2 , R3 ) in the octonion space is
R = r0 + i1 r1 + i2 r2 + i3 r3 + I0 R0 + I1 R1 + I2 R2 + I3 R3
(2)
and the velocity V(v0 , v1 , v2 , v3 , V0 , V1 , V2 , V3 ) is
V = v0 + i1 v1 + i2 v2 + i3 v3 + I0 V0 + I1 V1 + I2 V2 + I3 V3
(3)
PIERS Proceedings, Moscow, Russia, August 18–21, 2009
362
where, r0 = v0 t; v0 is the speed of light; t is the time; the ◦ denotes the octonion multiplication.
When the electric charge is combined with the mass to become the electron or the proton etc.,
we obtain the Ri Ii = ri ii ◦ I0 and Vi Ii = vi ii ◦ I0 , with i0 = 1. i = 0, 1, 2, 3; j = 1, 2, 3.
Table 1: The octonion multiplication table.
1
i1
i2
i3
I0
I1
I2
I3
1
1
i1
i2
i3
I0
I1
I2
I3
i1
i1
−1
−i3
i2
−I1
I0
I3
−I2
i2
i2
i3
−1
−i1
−I2
−I3
I0
I1
i3
i3
−i2
i1
−1
−I3
I2
−I1
I0
I0
I0
I1
I2
I3
−1
−i1
−i2
−i3
I1
I1
−I0
I3
−I2
i1
−1
i3
−i2
I2
I2
−I3
−I0
I1
i2
−i3
−1
i1
I3
I3
I2
−I1
−I0
i3
i2
−i1
−1
In the electromagnetic field theory, the Aharonov-Bohm effect states that the field potential is
more essential than the field strength. As a result, we can define the field strength and field source
successively from the electromagnetic field potential by means of the quaternion operator.
The potential of the gravitational and electromagnetic fields are Ag = (a0 , a1 , a2 , a3 ) and Ae =
(A0 , A1 , A2 , A3 ) respectively. They are combined together to become the potential A = Ag + keg Ae ,
with keg being the coefficient.
The strength B(b0 , b1 , b2 , b3 , B0 , B1 , B2 , B3 ) consists of the gravitational strength Bg and the
electromagnetic strength Be . The gauge equations satisfy b0 = 0 and B0 = 0.
B = ♦ ◦ A = Bg + keg Be
(4)
where, the quaternion operator ♦ = ∂0 + Σ(ij ∂j ); ∂i = ∂/∂ri ; ∇ = Σ(ij ∂j ).
The gravitational strength Bg includes two components, g/v0 = ∂0 a + ∇a0 and b = ∇ × a.
g/v0 = i1 (∂0 a1 + ∂1 a0 ) + i2 (∂0 a2 + ∂2 a0 ) + i3 (∂0 a3 + ∂3 a0 )
b = i1 (∂2 a3 − ∂3 a2 ) + i2 (∂3 a1 − ∂1 a3 ) + i3 (∂1 a2 − ∂2 a1 )
(5)
(6)
where, a = Σ(aj ij ). a = 0 and b = 0 in the Newtonian gravity.
At the same time, the electromagnetic strength Be involves two parts, E/v0 = (B01 , B02 , B03 )
and B = (B23 , B31 , B12 ).
E/v0 = I1 (∂0 A1 + ∂1 A0 ) + I2 (∂0 A2 + ∂2 A0 ) + I3 (∂0 A3 + ∂3 A0 )
B = I1 (∂3 A2 − ∂2 A3 ) + I2 (∂1 A3 − ∂3 A1 ) + I3 (∂2 A1 − ∂1 A2 )
(7)
(8)
In the electromagnetic field theory with the quaternion, some invariants are the same as them
with vectors respectively, including the definition of field strength, Maxwell’s equations, d’Alembert
equation, and continuity equation. But some are different, such as the definition of gauge equation,
direction of displacement current, and electromagnetic force.
The electric current density Se = qVg ◦ I0 is the source for the electromagnetic field, and the
linear momentum density Sg = mVg for the gravitational field. The source S satisfies,
µS = (B/v0 + ♦)∗ ◦ B = µgg Sg + keg µge Se + B∗ ◦ B/v0
(9)
where, m is the mass density; q is the electric charge density; µ, µgg , and µge are the constants; ∗
2 = µg /µg ; B∗ ◦ B/(2µg ) is the field energy density.
denotes the conjugate of octonion; keg
g
e
g
B∗ ◦ B/µgg = B∗g ◦ Bg /µgg + B∗e ◦ Be /µge
(10)
In the electromagnetic field, the linear momentum is defined as one function of field source.
The electromagnetic force is defined from the linear momentum by the quaternion operator, and
includes the Lorentz force with a few new forces. When the quaternion is decomposed to the scalar
quantity and vectorial quantity, a few differences will exist in two descriptions.
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 363
The force density F is defined from P = µS/µgg , which is the extension of the Sg .
F = v0 (B/v0 + ♦)∗ ◦ P
(11)
where, the force density includes the gravity density, inertial force density, Lorentz force density,
and interacting force density between electromagnetic strength with magnetic moment, etc.
In case of the gravitational field and the electric field both can be neglected, the above can be
decomposed in the basis vector (i1 , i2 , i3 ) further. Besides the inertial force, we can obtain the
electromagnetic forces as follows.
fe = qB × V − qB ◦ V0
(12)
where, B = Σ(Bj Ij ), V = Σ(Vj Ij ), V0 = V0 I0 .
In the above, the first term is Lorentz force, and the second term is a new force. This new
force of electromagnetic field will cause a test charge to move along the magnetic force line, but
the phenomena was used to be attributed to other factors. In a constant magnetic field, the charge
will do uniform circular motion in the direction perpendicular to the magnetic force line, and do
uniform variable rectilinear motion along the magnetic force line. And then the resulting track of
the test charge is a helix with variable screw-pitch.
3. CONCLUSION
In a constant magnetic field, the test charge will vary velocity along the magnetic force line, and
has a deviation from the uniform rectilinear motion. This states that it will violate the prediction
of Lorentz force along the magnetic force line. The deduction can explain why the solar wind varies
velocity along the magnetic force line in the solar system.
In theoretical analysis, Maxwell’s equations can not contain electromagnetic forces. Describing
the electromagnetic force with the quaternion algebra can deduce a few new forces. In experimental
investigation, the charge’s force along the magnetic force line has never been validated before. At
present, some related validating trials have been brought forward. This research improves our
knowledge about the electromagnetic force.
It should be noted that the investigation for the electromagnetic force has examined only one
simple case, of which one new force causes a test charge to move along the magnetic force line.
Despite its preliminary character, this study can clearly indicate that the test charge does uniform
variable motion along the magnetic force line. For the future studies, the research will concentrate
on only the predictions about new forces in the vast bulk and constant magnetic field.
ACKNOWLEDGMENT
This project was supported partially by the National Natural Science Foundation of China under
grant number 60677039.
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