Chapter 7
Maxwell’s Equations
Maxwell’s equations consist of four first-order partial differential equations. First we
deal with basic properties of Maxwell’s equations. Next we show how equations of
electromagnetic wave motion are derived from Maxwell’s equations along vector
analysis. It is important to realize that the generation of the electromagnetic wave is a
direct consequence of the interplay between the electric field and magnetic field that
both change with time. We deal with behaviors of electromagnetic waves in dielectric media where no true charge exists. At a first glance, this restriction seems to
narrow a range of application of principles of electromagnetism. In practice, however, such a situation is universalistic; topics cover a wide range of electromagnetic
phenomena, e.g., light propagation in dielectrics including water, glass, polymers,
etc. Polarized properties characterize the electromagnetic waves. These include
linear, circular, and elliptic polarizations. The characteristics are important both
from a fundamental aspect and from the point of view of optical applications.
7.1 Maxwell’s Equations and Their Characteristics
In this chapter, we first represent Maxwell’s equations as vector forms. The equations are represented as a differential form that is consistent with a viewpoint based
on “action trough medium.” The equation of wave motion (or wave equation) is
naturally derived from these equations.
Maxwell’s equations of electromagnetism are expressed as follows:
div D ¼ ρ,
ð7:1Þ
div B ¼ 0,
ð7:2Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_7
269
Maxwell’s Equations
Maxwell’s equations consist of four first-order partial differential equations. First we
deal with basic properties of Maxwell’s equations. Next we show how equations of
electromagnetic wave motion are derived from Maxwell’s equations along vector
analysis. It is important to realize that the generation of the electromagnetic wave is a
direct consequence of the interplay between the electric field and magnetic field that
both change with time. We deal with behaviors of electromagnetic waves in dielectric media where no true charge exists. At a first glance, this restriction seems to
narrow a range of application of principles of electromagnetism. In practice, however, such a situation is universalistic; topics cover a wide range of electromagnetic
phenomena, e.g., light propagation in dielectrics including water, glass, polymers,
etc. Polarized properties characterize the electromagnetic waves. These include
linear, circular, and elliptic polarizations. The characteristics are important both
from a fundamental aspect and from the point of view of optical applications.
7.1 Maxwell’s Equations and Their Characteristics
In this chapter, we first represent Maxwell’s equations as vector forms. The equations are represented as a differential form that is consistent with a viewpoint based
on “action trough medium.” The equation of wave motion (or wave equation) is
naturally derived from these equations.
Maxwell’s equations of electromagnetism are expressed as follows:
div D ¼ ρ,
ð7:1Þ
div B ¼ 0,
ð7:2Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_7
269