Question: What is the refractive index of a
medium?
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A student in Singapore complained the
following examination question:
What is the refractive index of a medium?
A. the ratio of the speed of light in air to the speed of light in the medium.
B.
the ratio of the speed of light in the medium to the speed of light in air.
C.
the ratio of the speed of light in the medium to the speed of light in vacuum.
D.
the ratio of the speed of light in vacuum to the speed of light in the medium.
The student chose the answer “A,” but his physics teacher told him that the
answer should be “D.” However, this does not seem reasonable to him because the
recommended textbook states that “the refractive index is a ratio between the speed of light in air or
vacuum and the speed of light in a medium (Chew & Chow, 2007, p. 231).” Subsequently, this led to complaints that the textbook has errors and
the ministry of education has recommended the wrong physics textbook. Alternatively, we can
define absolute index of refraction as a ratio of the speed of light in a vacuum to the speed of light
in a medium. Thus, this is also a problem of terminology and thus, the
examination question should be phrased more carefully.
How would Feynman answer?
An ability to state the refractive index as a ratio of the speed of
light in vacuum and the speed of light in a medium does not imply a genuine
understanding of the concept. In short, Feynman would explain that
the speed of light waves in a medium remains constant, but it is the effective
speed of the light waves that is decreased. We should have some knowledge of
Feynman’s answer from the perspective of wave mechanics, electromagnetism, quantum
mechanics, and problems of defining index of refraction as shown below.
1. Effective
speed of light waves: Strictly speaking, it is imprecise and incomplete to
state that the speed of light is reduced in a transparent material or medium. In
general, there are different kinds of speed (or velocity) of light waves such
as phase velocity, group velocity, and effective speed. In Feynman’s words,
“[i]t is approximately true that light or any electrical wave does
appear to travel at the speed c/n through a material whose index of
refraction is n, but the fields are
still produced by the motions of all the charges — including the charges
moving in the material — and with these basic contributions of the field
travelling at the ultimate velocity c.
Our problem is to understand how the apparently slower velocity comes
about (Feynman et al., 1963, section 31–1 The
index of refraction).” Simply phrased, light or light waves still travel at the
same ultimate speed of light, c,
instead of c/n. However, the constant speed of light waves appears to slow down
as a result of moving charges in the material.
Importantly, there should be a deeper understanding of the
meaning of effective speed (or apparent speed) of light. We should
distinguish the term absolute speed
and effective speed for the speed of
light in the material. Feynman explains that “[b]efore we proceed with our
study of how the index of refraction comes about, we should understand that all
that is required to understand refraction is to understand why the apparent
wave velocity is different in different materials. The bending of
light rays comes about just because the effective speed of the waves is
different in the materials (Feynman et al., 1963, section 31–1 The index of refraction).” Although the
bending of light is sometimes explained by the principle of least time, it is
also important to understand the concept of absolute speed of light in
different materials. Better still, we should emphasize that the refractive
index is related to the effective speed of light in the material or medium.
Furthermore, the effective or apparent speed of light can be explained
by the phase shift of light waves. Feynman elaborates that “[i]n spite of the
fact that it is said that you cannot send signals any faster than the speed of
light, it is nevertheless true that the index of refraction of materials at a
particular frequency can be either greater or less than 1. This just means that
the phase shift which is produced by the scattered light can be either
positive or negative (Feynman et al., 1963, section 31–3 Dispersion).” Note that the refractive index of a material is
related to the phase velocity or speed of nodes of the waves. Moreover, there
is a phase difference between an incident light wave and the emitted light wave
generated by an atom. If the phase of the emitted light wave is delayed, the
effective speed of light is slowed down.
2. Theories of
refractive index:
The concept of refractive index can be explained by wave
mechanics, electromagnetism, and quantum mechanics. Generally speaking, the
oscillation of atoms in a medium can be modeled by using wave mechanics.
Feynman mentions that “[y]ou may think that this is a funny
model of an atom if you have heard about electrons whirling around in orbits.
But that is just an oversimplified picture. The correct picture of an atom,
which is given by the theory of wave mechanics, says that, so far as
problems involving light are concerned, the electrons behave as though they
were held by springs (Feynman et al., 1963, section 31–2 The field due to the material).” In a sense, we can idealize the
electrons as tiny oscillators with a resonant frequency and having a linear restoring force. Therefore, the driven
motion of the electrons can re-emit light waves through the material.
In addition, the physical principles behind the oscillation of molecules
are based on electromagnetism. Feynman clarifies that “[t]he electric
field of the light wave polarizes the molecules of the gas, producing
oscillating dipole moments. The acceleration of the oscillating charges
radiates new waves of the field. This new field, interfering with the old
field, produces a changed field which is equivalent to a phase shift of the
original wave. Because this phase shift is proportional to the thickness of the
material, the effect is equivalent to having a different phase velocity in the
material (Feynman et al., 1964, section 32–1 Polarization of matter).” Essentially, Feynman has simplified
the discussion by excluding complications that arise from the effects of light waves
changing the electric fields at the oscillating charges. He has assumed the
forces on the charges in the atoms came only from the incoming wave, and
did not delve deeper in the re-emitted waves from all other atoms.
Fundamentally speaking, we can have a quantum interpretation of
the equation derived by wave mechanics on the refractive index. In a footnote
of his lecture, Feynman states that “[i]n quantum mechanics even an atom with
one electron, like hydrogen, has several resonant frequencies. Therefore
Nk is not
really the number of electrons having the frequency ωk, but is replaced instead by Nfk, where N is the number of atoms per unit
volume and fk (called the oscillator strength) is a factor that tells how
strongly the atom exhibits each of its resonant frequencies ωk (Feynman et al., 1963,
p. 31-8).” In short, the interactions of light with a material can be
visualized as the absorptions and re-emissions of photons. As a result, the absorptions and emissions of light waves
in the material cause the phase shift and the reduction of (effective) speed of
light.
3. Problems
of defining refractive index:
The refractive index is not a simply a constant
and it is dependent on the frequency (or wavelength) of light waves. According
to Feynman, “we have also learned how the index of refraction should vary with the
frequency ω of the light. This is something we would never understand from the
simple statement that ‘light travels slower in a transparent material’ (Feynman
et al., 1963, section 31–3
Dispersion).” In essence, the speed of light is dependent on the color and the
refractive index is not exactly defined by the ratio of the
speed of light in vacuum to the speed of light in the medium. Mathematically, the index of refraction can
be modeled by the equation n = 1 + Nqe2/2ϵ0m(ω02 − ω2)
in which N is the number of atoms per
unit volume in a plate. We can use this equation to explain the phenomenon of
dispersion.
Interestingly, the refractive index can be represented by a complex
number. Feynman explains that “the index of refraction is now a complex
number! What does that mean? By working out what the real and
imaginary parts of n are we could
write n = n′ − in′′, where n′ and n″ are real numbers (Feynman et al., 1963, section 31–4 Absorption).” The imaginary part of
refractive index means that some energy of light waves can be absorbed by the material
or medium. This implies a decrease in the amplitude of the light
waves that is proportional to the thickness of the medium such as a piece of
glass plate. In other words, the light waves that come out from the other side of
the glass plate have lesser energy. Therefore, n″ is sometimes known as the “absorption index.”
On the other hand, there is still a problem of
incomplete knowledge in the refractive index. In Feynman’s words, “We still have the problem, of course, of knowing how many atoms per unit
volume there are, and what is their natural frequency ω0. We do not
know this just yet, because it is different for every different material, and
we cannot get a general theory of that now. Formulation of a general theory of
the properties of different substances — their natural frequencies, and so on —
is possible only with quantum atomic mechanics (Feynman et al., 1963, section 31–3 Dispersion).” Therefore, it is challenging
to have a general mathematical equation for all refractive indices that is
applicable to all materials. Currently, there are meta-materials in which the
refractive index can even be negative.
To conclude, the knowledge of
speed of light is reduced in a transparent material does not mean
a good understanding of the concept of refractive index. Nevertheless, Feynman
would explain that it is the effective speed of light that is reduced and this
is related to the phase shift of light waves in the material. Importantly, the concept of refractive index can be modeled by using wave
mechanics, electromagnetism, and quantum mechanics. Furthermore, we should be
cognizant of problems of defining refractive index such as it is depending on
the wavelengths of light and it is possible to have an imaginary part in the
refractive index.
Note: Feynman has a short and
interesting explanation on the bending of light in a medium, “[f]inding the
path of least time for light is like finding the path of least time for a
lifeguard running and then swimming to rescue a drowning victim: the path of least distance has too much
water in it; the path of least water has too much land in it; the path of least
time is a compromise between the two (Feynman, 1985, p. 51).”
References:
1. Chew, C. & Chow, S. F. (2007). GCE ‘O’ Level Physics Matters. Singapore: Marshall Cavendish.
2. Feynman,
R. P. (1985). QED: The strange theory of light and matter.
Princeton: Princeton University Press.
3. 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.
4. Feynman, R. P., Leighton, R. B., & Sands,
M. (1964). The Feynman Lectures on
Physics, Vol II:
Mainly
electromagnetism and matter. Reading, MA: Addison-Wesley.
