Sec. 2.2 Origin of the Geomagnetic Field
2.2.1 Magnetic effect between moving charged particles
There is a magnetic coupling among moving charged particles. The charged
particles attract each other when those particle run paralleing in the
same direction. An example is "pinch effect of narrowing plasma in
nuclea-fusion experiments" where a large current flow in the flux
of plasma. Arc welding is another example, in which the arc of molten metal
at the tip of the rod is narrowed. These phenomena indicates a magnetic
coupling among parallel running charged particles. In traditional understanding
on the phenomenon that the moving charged particles are captured by magnetic
field lines is not correct. The magnetic action caused by moving charged
particles must not handle by static magnetic field lines. Becaue, the magnetic
field line is not real existence. The line indicates only direction of
magnetic force. The real existence of charge particle does not react with
the imagined line. The electric lines (E) of electric force is not real entities also. So, the reaction of electric
force lines caused by charged particle does not handled. In quantum mechanics,
this phenomenon is handled as energy state. Assuming that the vector potential
(A) caused by a moving charged particle acts on the other moving charged
particles. The magnetic coupling energy can be calculated. The energy of
moving charged particle with a velocity (v) in the A is given by Eq.(1).
Um= - (qv)・A (1).
The resulting reduced energy is evaluated as the effect by interaction among moving charged particles.
It is the electrostatic potential (φ) that gives electrostatic energy for a point charge. The electric field (E) is the gradient of locational potential φ, i.e. grad φ=E.
On the other hand, the kinetic energy of the motioning point charge (Um) is given by product of current by the charged particle and the vector potential (A).
The relationship between A and magnetic flux density (B) is defined by Eq.(2).
B = rot A (2)
Here, the negative sign is set to be low when v and A are parallel.
(rot) is an operation that the orbital direction of vector (A) at a minute area around a circle are added. It can be expressed in integral
form as follows. The magnetic flux (Φ) of the penetrating area is given by a line integration
of point field (A)
around the penetrating area.
The Aharonov–Bohm effect is a quantum mechanical phenomenon that electrons
undergo in a field with a vector potential A. This effect is shown in Fig. 2.2.1. The phenomenon is happened outside
of solenoid. The phase of the matter wave of electrons passing up and down
is different, and interference fringes appear. The upper and lower magnetic
field (B) on the solenoid are the same, but the direction of the current that generates the magnetic field is opposite up and down,
So, the vector potential (A) that the magnetic copling effect is opposite between up and down. Therefore,
a difference occurs in the phase of electrons passing above and below in
the outside of the solenoid. This effect was successfully demonstrated
in 1986 by Dr. Akira Tonomura on a solenoid covered with a superconductor.
Fig. 2.2.1 Interaction between electron flow outside the solenoid and current
flowing in the solenoid
2.2.2 Magnetic action between moving charged particles by vector potential
Fig.2.2.1 illustrates the relationship between the vector potential A and the magnetic field B.
Fig. 2.2.2 Relationship between vector potential A and magnetic field B
A torque is the rotational change of the energy on the motioning charged
particle with velocity v in the field of Vector Potential A.
FL = rot Um = - (qv)・rotA (3).
A charged particle (q) with velocity v receives the torque expressed by Eq.(4)
FL = - (qv) xB (4)
Tthe properties of A are clarified by using Maxwell's Eq.(5).
rot B =μ0(j -ε0∂E/∂t) (5).
From Eq (1) and (5),
rot (rot A) =μ0(j -ε0∂E/∂t) (6).
Maxwell's equation of ∂B/∂t=rot E and Eq.(1) yields ∂A/∂t=-B・E.
Therefore, Eq. (7) is obtained from Eq. (6).
rot (rot A) = μ0(j -ε0∂2A/∂t2) (7).
Eq.(10) is obtained using Eq.(8) of the vector analysis formula and Eq.（9)
of gauge transformation.
rot(rotA)=grad(divA) - △A (8)
div (A+ gradΛ) = 0 (9).
△A=(μ0ε0) ∂2A/∂t2 - μ0j (10).
Here, △≡∂2/∂x2+∂2/∂y2+∂2/∂z2.
In the case of (ε0)∂2A/∂t2≫- j , Eq. (10) becomes wave equaion (11).
△A=(μ0ε0) ∂2A/∂t2 (11).
In case of (ε0)∂2A/∂t2≪-j, Eq.(9 is corresponds to Poisson's equation (△V= -ρ/ε0) which shows the relationship between charge and potential. We get the
equivalent equation (12) for vector potential.
△A= - μ0j (12 ).
According to Eq. (12), A is proportional to the current (i) i.e., the product of the density and velocity.
This change changes the energy by rotation A. Therefore, ∂A/∂t=-E, Instantaneously electric field E acts on charged particles. The energy of rotation accompanies with partial
current (j) is expressed Eq.(1) also. A changes
with time, ∂A/∂t=-E, The electric field E acts on the charged particles.
Thus, there is an attractive effect of magnetic coupling at the moment when charged
particles having the same polarity and move side by side.When particles
with different polarities move side by side, the force in direction of
magnetic repulsion acts instantaneously.
2.2.3 Magnetism of Earth's moving charged particles and solar wind
Fi.g.2.2.3 shows the flow of hydrogen ions of solar wind those affect
the Earth's geomagnetic field. Near the equator in the daytime hemisphere,
the solar wind collides head-on with the atmosphere. Due to the angular
momentum in the counterclockwise direction due to the rotation of the Sun,
the atmosphere moves in the clockwise direction to drive the trade winds.
On the other hand, the solar wind accelerates the atmosphere by action
of parallel currents of H+ of the solar wind. That is, the atmosphere above the eastern side of the
Earth's mid-latitudes accelerates in the counterclockwise direction. As
a result, westerly winds blow in mid-latitudes of the Earth.
Fig. 2.2.3 Influence of solar wind on molecules and ions of the atmosphere
above the Earth
-Moving charged particles that forme the Van Allen belt and the Earth's geomagnetic field-
Since the Earth rotates counterclockwise, charged particles inside of
the Earth also rotate and charged particles in upper sky of the Earth orbit
counterclockwise direction. Since the velocity of those charged particles
are fastest near the equator, magnetic coupling acts strongly on charged
particles running parallel above the equator. The rotation speed of the
area disappears and only the Earth's gravity acts causing charged particles
to fall in the Arctic and Antarctic regions. So, charged particles orbiting
the Earth are distributed as doughnut shapes, as shown in Fig.2.2.4. References[1] Shinji Karasawa (2022) Effects of Solar Wind on
Earth's Climate. Geol Earth Mar Sci Volume 4(2): 1–5. DOI: 10.31038/GEMS.202242
The geomagnetism on the ground is affected by Earth's crustal structure and the distribution of charged particles in the sky, the magnetic action of charged particles inside the Earth and charged particles in the sky cancel each other out
on the ground of the Earth. The geomagnetism on the ground is changed by
the Earth's crustal structure and the distribution of charged particles
in the sky. Instead of assuming magnetic field lines of the geomagnetic field, consider
the interaction of the motion of charged particles.
In case of the influence of charged particles in the upper sky is strong,,
the magnetic field lines of the geomagnetic field at the surface of the
Earth directs from the south pole to the north pole of the Earth's as shown
in Fig.2.2.4. However, on Jupiter, the direction of the geomagnetic field
is opposite to that of the Earth. As for Jupiter, there are a large amount
of moving charged particles inside of Jupiter that are responsible for
the geomagnetism. It can be thought that the effect of charged particles
in the sky above exceeds the effects of charged particles in the sky. (last
modified April 1, 2023)
Fig. 2.2.4 Upper geomagnetic generated by H+ orbiting the Earth in the sky
to index -2.2-