(Unit vector / Inverse distance dependence / x-component
of acceleration)
The three
interesting concepts discussed in this section are the unit vector in Feynman’s
formula for the electric field, its property of inversely proportional to
distance, and the x-component of acceleration in the simplified formula.
1. Unit vector:
“What this term says is: look at the charge and note
the direction of the unit vector (we can project the end of it onto the surface
of a unit sphere) (Feynman et al., 1963, section 28–2 Radiation).”
Feynman defines the unit vector er′ as pointed toward the
apparent position of the charge. In addition, the end of er′ may move like a curve so
that its acceleration has two components. In spherical coordinates, a vector r can be represented as
follows: r = r × rˆ(q, f) in which r is the magnitude of the vector r, rˆ is the unit vector, q is the polar angle, and f is the azimuthal angle.
Specifically, the vector r may be described as the line of sight vector if it is from the
perspective of an observer. Perhaps it is confusing to describe the transverse component
as going up and down because it is a sidewise component that could be going
east-west, north-south, or the resultant of these two directions.
Feynman claims that it is easy to see the wigglings
of er′ of a given source would vary inversely as the distance when it is
moving farther away. Although Feynman considers the unit vector to be easy,
but the rules of unit vector can be complicated. For example, it is not simple
for some students to deduce drˆ/dt = (drˆ/dq)(dq/dt) + (drˆ/df)(df/dt) = (icos q cos f + jcos q sin f − ksin q)(dq/dt) + (−i
sin q sin f + j sin q cos f)(df/dt) = qˆ(dq/dt) + fˆ(sin q)(df/dt). Furthermore, a = (d2r/dt2 – r[dq/dt]2 – r[df/dt]2sin2 q)rˆ + (r[d2q/dt2] + 2[dr/dt][dq/dt] – r[df/dt]2sin q cos q)qˆ + (r[d2f/dt2] sin q + 2[dr/dt][df/dt]sin q + 2r[dq/dt][df/dt]cos q)fˆ. Worst still, this unit vector is dependent on the retarded time r′/c and retarded
distance r′.
2. Inverse distance dependence:
“To discuss these phenomena, we must select from Eq.
(28.3) only that piece which varies inversely as the
distance and not as the square of the distance (Feynman
et al., 1963, section 28–2
Radiation).”
Feynman mentions that the electric field at a point is
inversely proportional to the distance from the charge, but the expression of
the electric field is given as a law and it will be learnt in detail next year.
In vol II, he says: “[i]t turns out that we won’t quite
make it—that the mathematical details get too complicated for us to carry
through in all their gory details (Feynman et al., 1964, chapter 21).” He
simplifies the situation by suggesting some charges are moving only a small
distance at a slow rate. This tiny motion is effectively at a constant distance
implies that the unit vector can be represented using Cartesian coordinates, rˆ = (x/r, y/r, z/r), in which r
= Ö(x2 + y2 + z2). In essence, the charged object must be moving at
a slow speed radially such that r can be considered as a constant (thus d2rˆ/dt2 = d2(x/r)/dt2 = ax/r).
In vol II, Feynman uses a “bullet” analogy to
explain how an electric field of a point charge is inversely proportional to
the square of the distance. In his words, “[i]f the gun is enclosed in a
surface, whatever size and shape it is, the number of bullets passing through
is the same—it is given by the rate at which bullets are generated at the gun…
(Feynman et al., 1964, section 4-5).” Similarly, Zangwill (2012) explains the
inverse distance dependence using an expanding annular ring. In this case, the total
electric flux captured by the ring is Ea2πRdR and the constant value of the
total electric flux leads to Ea(R) µ 1/R. One should be cognizant that many forces in physics such as the
nuclear force and molecular force do not obey the inverse square law.
3. x-component
of acceleration:
“…r is practically constant, the x-component
of d2er′/dt2 is simply the acceleration of x itself at an earlier time
divided by r, and so finally we get the law we want, which is Ex(t) = (−q4πϵ0c2/r)ax(t−r/c). (Feynman
et al., 1963, section 28–2 Radiation).”
According to Feynman, if a charged object is moving in a very small motion and
it is laterally displaced by the distance x, then the unit vector er′ is displaced by x/r. The x-component
of d2er′/dt2 is simply the acceleration
of x at an earlier time divided by r provided r is constant. This also
implies the acceleration of r(t) or possibly z(t)
is zero. On the other hand, one may consider the distance x(t) to
be close to zero because the charged object is moving at a relatively slow
speed and it is far away from the observer. In other words, the distance x
that is laterally displaced, is possibly very short in comparison to r such
that we can use the small-angle approximation formula q @ sin q = x/r.
Feynman elaborates that Eq. (28.5) is the complete and correct formula for radiation,
and even relativity effects are all contained in this formula. This statement is
not completely correct because Feynman’s formula for the electric field is a
special case of Jefimenko’s
equations. One may elaborate that Lorentz’s factor and
the constant speed of light in all inertial frames are hidden in Maxwell’s equations.
In an article titled Why is Maxwell’s Theory
so hard to understand, Dyson (1999) writes, “[w]e may hope that a deep understanding of Maxwell’s theory
will result in dispersal of the fog of misunderstanding that still surrounds
the interpretation of quantum mechanics.” In short, Maxwell’s field theory does not only contain
relativity effects, but the prototype of quantum field theory.
Review Questions:
1. Would you consider the unit vector to be simple
and describe its transverse
component as only going up and down?
2.
How would you explain the electric field
is inversely proportional to the distance?
3. Is Eq. (28.5) the complete and correct formula for radiation
and only relativity effects are all contained in this formula?
The
moral of the lesson: if a charge is moving at a slow
speed and it is laterally displaced by the distance x(t), then
the angle that the unit vector er′ is displaced is x/r, and since r is
practically constant, we have Ex(t)= (−q4πϵ0c2/r)ax(t−r/c).
References:
1. Dyson, F. (1999). Why is Maxwell’s theory so hard to
understand? In James Clerk Maxwell Commemorative Booklet, Fourth
International Congress Industrial and Applied Mathematics, Edinburgh, Scotland.
2. 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.
3. Zangwill, A. (2012). Modern Electrodynamics. New York: Cambridge University Press.
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