(Retarded electric field / Moving electric field / Oscillatory electric field)
The three interesting points
discussed in this section are retarded electric field, moving electric field,
and oscillatory electric field.
1. Retarded electric field:
“The electric field E due to a
positive charge whose retarded acceleration is a′ (Feynman
et al., 1963, section 29–1 Electromagnetic waves).”
According to Feynman, if a charge is oscillating with a very small
amplitude, the electric field at an angle θ from the axis of the motion is in a
direction at right angles to the line of sight and in the plane containing the
line of motion and the line of sight. However, it is not straightforward for
students to visualize the retarded electric field based on Fig. 29–1. Specifically, Feynman suggests that the
understanding of the factor a(t−r/c) is to take the
simplest case, θ = 90°, and plot the field
graphically. In a sense, the simplest case should be θ = 0° because the formula E(t)
= −qa(t−r/c) sin θ/4πϵ0c2r
clearly becomes zero. In short, the retarded electric field at any point on the
vertical axis (or line of motion) through the charge is 0.
In Fig. 29-1, Feynman shows that the retarded electric field E due to a positive charge is a′. One may recall the equation (28.6) Ex(t) = (−q4πϵ0c2/r)ax(t−r/c) and this leads to the equation (29.1) E(t) = −qa(t−r/c)sin θ/4πϵ0c2r by simply writing ax(t−r/c) as a(t−r/c)sin θ. (Note: a(t−r/c) means at′=t−r/c) It is worthwhile to elaborate the retarded electric field at θ = 90° refers to any point on the horizontal plane as shown in Fig. 29-1 that is perpendicular to the line of motion. That is, the retarded electric field at any point on any horizontal axis is −qa(t−r/c)/4πϵ0c2r (maximum value) and it is in the opposite direction to the line of motion. Simply put, the direction of the retarded electric field at any point on the horizontal plane in Fig 29-1 is anti-parallel to the acceleration vector.
2. Moving electric field:
“That is, as time goes on the field moves as a
wave outward from the source. That is the reason why we sometimes say
light is propagated as waves (Feynman et al., 1963, section 29–1 Electromagnetic waves).”
According to Feynman, light is sometimes described
as a propagation of waves because the retarded electric field moves as a
wave outward from the source. It is equivalent to saying its electric field
is spreading outward as time goes on. In Volume II, Feynman adds that “[i]f
this electric field tries to go away, the changing electric field would create
a magnetic field back again. So by a perpetual interplay—by the swishing back
and forth from one field to the other—they must go on forever... (Feynman et
al., 1964, Chapter 18).” However, we may clarify that the electric field is moving
outward in the sense that there is energy in the electric field and its field
energy has momentum (See Feynman et al., 1964, Chapter 27).
Feynman explains that the retarded electric field
at farther points is determined by a charge’s acceleration at earlier times. It
can be illustrated by the curve of electric field in Fig. 29–3 that is a “reversed” plot of the charge’s acceleration
as a function of time. To be more specific, one may elaborate that the curve of
electric field is a “reverse and inverse” plot of the charge’s acceleration
graph. That is, the acceleration graph is “reversed” or flipped from left to
right because the acceleration of the charge at earlier times affects the
electric field at farther points. In addition, the acceleration graph is also
“inversed” or flipped from up to down because of the negative sign in E(t) = −qa(t−r/c)sin θ/4πϵ0c2r.
3. Oscillatory electric
field:
“An interesting special case is that where the charge q is moving
up and down in an oscillatory manner (Feynman et al.,
1963, section 29–1 Electromagnetic waves).”
Feynman describes a simple example of oscillatory
electric field as a result of a charge accelerating up and down along a line
having a very small amplitude. In the previous chapter, he has simplified the Heaviside-Feynman expression of electric field by assuming “a still simpler
circumstance in which the charges are moving only a small distance at a
relatively slow rate. Since they are moving slowly, they do not move an
appreciable distance from where they start, so that the delay time is
practically constant (Feynman et al., 1963, Section 28-2).” In essence, the
simplified electric field formula (29.1) is applicable to any charge that is moving at a relatively slow speed (without
relativistic effects) and it is relatively far away from the observer.
Feynman expresses the oscillatory electric field of
a charge as E = −qsin θa0cos ω(t−r/c)/4πϵ0rc2 by substituting the charge’s acceleration a
=−ω2x0cos ωt = a0cos ωt
into E(t) = −qa(t−r/c)sin
θ/4πϵ0c2r.
However, some may ask whether the expression could
be in terms of sine instead of cosine. As an alternative, we may use a =−ω2x0sin (ωt + a) in which the phase angle a is dependent on the time we choose to start the
experiment. Similarly, the expression sin θ in the expression of
electric field may be cos (θ + b) instead of sin θ because it depends on the
reference line where the angle θ + b is measured from. In summary, sin θ is
dependent on the reference line in space, whereas cos ω(t−r/c)
is dependent on the clock measured in time.
Review Questions:
1. How would you visualize the direction of the retarded electric
field using Fig. 29–1?
2. How would you
explain the moving electric field as the “reversed”
plot of the charge’s acceleration as a function of time?
3. What are the assumptions of the oscillatory
electric field?
The moral of the lesson: light may be described as a propagation of waves because its electric
field is spreading outward from the charge.
References:
1. Feynman, R. P., Leighton, R.
B., & Sands, M. (1963). The
Feynman Lectures on Physics, Vol
I: Mainly mechanics,
radiation, and heat. Reading, MA: Addison-Wesley.
