Φ ¼ kρ and E ¼ E
j j ¼ E 0 e
ikρ
:
ð7:68Þ
In Fig. 7.5, a real part of E is shown with E ¼ E 0 at ρ ¼ 0. The sinusoidal curve
representing the electric field in Fig. 7.5 is shifted with time from left to right with its
form unchanged.
In this situation, the electric field is to be strengthened with a negative value at a
phase P 1 with time according to (7.68), whereas at another phase P 2 the electric field
is to be strengthened with a positive value with time. As a result, in a region tucked
between P 1 and P 2 a spiral magnetic field is generated toward the ε m direction, i.e.,
upper side of a plane of paper. The magnitude of the magnetic field is expected to be
maximized at a center point of P 1 P 2 (i.e., the origin of the coordinate system) where
the electric field is maximized as well. Thus, we conclude that E and H have the
same phase. 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. It is truly based upon the nature of the electric and
magnetic fields that are clearly represented in Maxwell’s equations.
Next, we consider the situation where two electromagnetic waves are propagated
in the same direction but with a different phase. Notice again that we are considering
the electromagnetic wave that is propagated in a uniform and infinite dielectric
media without BCs.
7.4 Superposition of Two Electromagnetic Waves
Let E 1 and E 2 be two electric waves described such that
E 1 = E 1 e 1 e
i kzÀωt
ð
Þ and E 2 = E 2 e 2 e
i kzÀωtþδ
ð
Þ ,
where E 1 (>0) and E 2 (>0) are amplitudes and e 1 and e 2 represent unit polarization
vectors in the direction of positive x-axis and y-axis; we assume that two waves are
P 1
P 2
Phase
Electric field
−
0
0
Fig. 7.5 Electric field of a
plane wave at some instant
as a function of phase. The
sinusoidal curve
representing the electric
field is shifted with time
from left to right with its
form unchanged
7.4 Superposition of Two Electromagnetic Waves
285
j j ¼ E 0 e
ikρ
:
ð7:68Þ
In Fig. 7.5, a real part of E is shown with E ¼ E 0 at ρ ¼ 0. The sinusoidal curve
representing the electric field in Fig. 7.5 is shifted with time from left to right with its
form unchanged.
In this situation, the electric field is to be strengthened with a negative value at a
phase P 1 with time according to (7.68), whereas at another phase P 2 the electric field
is to be strengthened with a positive value with time. As a result, in a region tucked
between P 1 and P 2 a spiral magnetic field is generated toward the ε m direction, i.e.,
upper side of a plane of paper. The magnitude of the magnetic field is expected to be
maximized at a center point of P 1 P 2 (i.e., the origin of the coordinate system) where
the electric field is maximized as well. Thus, we conclude that E and H have the
same phase. 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. It is truly based upon the nature of the electric and
magnetic fields that are clearly represented in Maxwell’s equations.
Next, we consider the situation where two electromagnetic waves are propagated
in the same direction but with a different phase. Notice again that we are considering
the electromagnetic wave that is propagated in a uniform and infinite dielectric
media without BCs.
7.4 Superposition of Two Electromagnetic Waves
Let E 1 and E 2 be two electric waves described such that
E 1 = E 1 e 1 e
i kzÀωt
ð
Þ and E 2 = E 2 e 2 e
i kzÀωtþδ
ð
Þ ,
where E 1 (>0) and E 2 (>0) are amplitudes and e 1 and e 2 represent unit polarization
vectors in the direction of positive x-axis and y-axis; we assume that two waves are
P 1
P 2
Phase
Electric field
−
0
0
Fig. 7.5 Electric field of a
plane wave at some instant
as a function of phase. The
sinusoidal curve
representing the electric
field is shifted with time
from left to right with its
form unchanged
7.4 Superposition of Two Electromagnetic Waves
285