Video Transcript
Which of the following definitions
best describes the critical angle for total internal reflection? A) The critical angle is the angle
at which all incident light at a boundary is reflected. B) The critical angle is the angle
of incidence minus the angle of refraction. C) The critical angle is the angle
of incidence plus the angle of refraction. D) The critical angle is the angle
at which the refracted ray travels along the boundary from which the incident ray is
reflected. And then, last but not least,
option E) the critical angle is the angle of refraction minus the angle of
incidence.
Now in these answer options, we’ve
seen a lot of descriptions of angles: angles of refraction, angles of incidence,
angles of reflection. To get some clarity on what these
answer options mean, let’s sketch out a scenario that involves the critical
angle. Say we have an interface here
between two materials. The index of refraction of the
material on top we’ll call 𝑛 sub f, and that on bottom we’ll call 𝑛 sub i. And let’s say further that we have
a ray of light coming up through this first material and reaching the interface
between the two.
Now we don’t need to specify
exactly what 𝑛 sub f and 𝑛 sub i are. But it is important that 𝑛 sub i
be greater than 𝑛 sub f. The index of refraction of the
material the ray is trying to get into must be less than the index of refraction
that the material is already in. This is a necessary condition for
total internal reflection. In addition to all this, let’s say
that our incoming ray has an angle of incidence that is the critical angle — we’ll
call it 𝜃 sub c — that has a very specific meaning in terms of what the refracted
ray looks like.
But before we get to that ray,
let’s look at the reflected ray, the one that bounces off this interface between the
two materials. For this reflected ray, if we call
the angle of reflection 𝜃 sub r, then by the law of reflection, we know that that
reflection angle, 𝜃 sub r, is equal to the angle of incidence, which in this case
is the critical angle. So that’s the reflected ray in this
scenario.
But we also know there’s a
refracted ray. And because our angle of incidence
is the critical angle, we know that the angle of refraction — we can call it 𝜃 sub
f — is equal to 90 degrees. This is what it means for our
incoming ray to be at the critical angle. So we now have these three angles
marked out: the incident angle, the reflected angle, and the refracted angle. We’re now ready to evaluate these
five answer options to see which one best describes the critical angle.
We already have answer option E) on
screen. But let’s recall answer options A),
B), C), and D). Answer option A) said that 𝜃 sub
c, the critical angle, is the angle at which all incident light is reflected. But looking at our diagram, we see
this isn’t so. At the critical angle, we still
have a refracted ray. It’s at 90 degrees, but it still
exists. So answer option A) isn’t our
choice.
Option B) says that the critical
angle is equal to the angle of incidence minus the angle of refraction. But remember, in the case of the
critical angle, that angle 𝜃 sub c is equal to 𝜃 sub i. And the angle 𝜃 sub f, the angle
of refraction, is 90 degrees. So this equation is essentially
saying that one quantity, 𝜃 sub c, is equal to another quantity, which we’ve said
is identical to 𝜃 sub c, minus 90 degrees. But mathematically, that can’t be
the case. So B) isn’t our choice either.
And for a similar reason, we won’t
choose option C). This option says that 𝜃 sub c, the
critical angle, is equal to the angle of incidence plus the angle of refraction. But again, at 𝜃 sub c, the
critical angle, the angle of incidence is the critical angle. So this option is saying that a
particular value is equal to itself, in this case, plus 90 degrees. So option C) isn’t our definition
for critical angle either.
Option D) is interesting. It says that the critical angle is
the angle at which the refracted ray travels along the boundary from which the
incident ray is reflected. Now as we look at our sketch, we
see that indeed this is happening. Our refracted ray drawn here is
moving along the boundary at which our incident ray is reflected. So option D) looks like a great
choice.
Let’s just consider option E) to
make sure we’re covering all the bases. This option says that the critical
angle is the angle of refraction minus the angle of incidence. But for the same reason that
options B) and C) were incorrect, option E) isn’t correct either. At the critical angle, the angle of
incidence is equal to the critical angle. So if we subtract this angle from
90 degrees, the angle of refraction, we won’t get the original angle back. This confirms that option D) is our
best answer.
The critical angle is the angle at
which the refracted ray travels along the boundary from which the incident ray is
reflected.
