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2 Basic Electromagnetism
the vector potential seems not to be simple. It is not difficult, however, to use the
vector potential. In particular, the vector potential gives a clearer physical concept,
as will be shown later. In addition, since the field caused by the magnetic potential is
nothing else than a gradient, the field with rotation cannot be given. That is, the use
of the magnetic potential is inconveniently limited only in regions where the current
does not flow, as in vacuum.
The essence of the magnetic flux density is that it is a rotational field that is caused
by current but has no divergence. This characteristic is asymmetrical with respect
to the electric field that describes the electrostatic phenomena. When a magnetizing
current of density i m coexists with current density i, i can be replaced by i + i m in
the above equations.
2.3 E-B Analogy
The electric field E is a general field produced by the electric charge and polarization
charge, and the magnetic flux density B is a general magnetic field produced by the
current and the magnetic moment. These fields exert the Lorentz force
F = q(E + v × B)
(2.39)
on a particle with electric charge q and velocity v. This phenomenon provides a
single connection between electromagnetism and dynamics. The first term is the
Coulomb force and the second term is the Lorentz force in the narrow sense. In
electromagnetism, the electric field E and the magnetic flux density B are treated as
important independent variables, and such a form is called the E-B analogy.
In the framework of the present electromagnetism, electric substances are roughly
classified into conductors and dielectric materials (insulators). Electric charges (electrons) can move freely inside a conductor and keep the interior electric field zero
regardless of any condition on the outside. On the other hand, electrons cannot move
freely due to bonding by nuclei in a dielectric material. Thus, electrons are slightly
displaced when an electric field is applied from the outside, resulting in a partial
shielding with a reduced interior electric field. This phenomenon is called electric
polarization. In this condition, the electrical neutrality condition is broken, and part
of the electric charge appears on the surface of the material. This electric charge is
called the polarization charge. The polarization charge cannot be transferred outside.
The polarization charge produces the electric field. The electric field produced in the
dielectric material is also called the electric polarization and is represented by P.
This is associated with the polarization charge as
∇ · P = −ρ p ,
(2.40)
where ρ p is the polarization charge density. It should be noted that the sign is different
from (2.7) for the relationship between the electric field and the electric charge
2 Basic Electromagnetism
the vector potential seems not to be simple. It is not difficult, however, to use the
vector potential. In particular, the vector potential gives a clearer physical concept,
as will be shown later. In addition, since the field caused by the magnetic potential is
nothing else than a gradient, the field with rotation cannot be given. That is, the use
of the magnetic potential is inconveniently limited only in regions where the current
does not flow, as in vacuum.
The essence of the magnetic flux density is that it is a rotational field that is caused
by current but has no divergence. This characteristic is asymmetrical with respect
to the electric field that describes the electrostatic phenomena. When a magnetizing
current of density i m coexists with current density i, i can be replaced by i + i m in
the above equations.
2.3 E-B Analogy
The electric field E is a general field produced by the electric charge and polarization
charge, and the magnetic flux density B is a general magnetic field produced by the
current and the magnetic moment. These fields exert the Lorentz force
F = q(E + v × B)
(2.39)
on a particle with electric charge q and velocity v. This phenomenon provides a
single connection between electromagnetism and dynamics. The first term is the
Coulomb force and the second term is the Lorentz force in the narrow sense. In
electromagnetism, the electric field E and the magnetic flux density B are treated as
important independent variables, and such a form is called the E-B analogy.
In the framework of the present electromagnetism, electric substances are roughly
classified into conductors and dielectric materials (insulators). Electric charges (electrons) can move freely inside a conductor and keep the interior electric field zero
regardless of any condition on the outside. On the other hand, electrons cannot move
freely due to bonding by nuclei in a dielectric material. Thus, electrons are slightly
displaced when an electric field is applied from the outside, resulting in a partial
shielding with a reduced interior electric field. This phenomenon is called electric
polarization. In this condition, the electrical neutrality condition is broken, and part
of the electric charge appears on the surface of the material. This electric charge is
called the polarization charge. The polarization charge cannot be transferred outside.
The polarization charge produces the electric field. The electric field produced in the
dielectric material is also called the electric polarization and is represented by P.
This is associated with the polarization charge as
∇ · P = −ρ p ,
(2.40)
where ρ p is the polarization charge density. It should be noted that the sign is different
from (2.7) for the relationship between the electric field and the electric charge