### Video Transcript

The graph contains four lines that
may or may not correctly represent a plot of how 𝛼, the angle of deviation of the
triangular prism shown in the diagram, varies with Φ one, the angle of incidence of
light rays on it. Which of the lines represents the
relationship between Φ one and 𝛼 correctly? (A) Black, (B) blue, (C) green, (D)
yellow, (E) red.

In order to answer this question,
let’s first recall the equation that relates 𝛼, Φ one, 𝜃 two, and 𝐴. Remember that by using geometric
properties of triangles, we can relate the angles of incidence and refraction for a
prism. Careful examination of the diagram
of rays entering and exiting the prism leads us to this expression: 𝛼 equals Φ one
plus 𝜃 two minus 𝐴. We are asked to identify the line
on the graph that correctly represents the relationship between Φ one and 𝛼, so
let’s look at what kind of graph this expression will create.

We know that 𝐴, the apex angle, is
a constant and will not affect the way that Φ one changes with respect to 𝛼. So we can choose to ignore that
term. We also know that 𝜃 two is
dependent on Φ one. So the only term in the expression
that will directly affect 𝛼, the angle of deviation, is Φ one, the angle of
incidence. Notice that the graph in our
diagram only shows the general shape of a set of curves. To find our answer then, we’ll want
to understand roughly how the angle of deviation increases or decreases with respect
to the angle of incidence.

The minimum possible angle of
deviation happens when the incoming ray passes through the prism exactly parallel to
its base. When this happens, Φ one equals 𝜃
two, which means we can write that 𝛼 sub min equals two times Φ one minus 𝐴. Therefore, the minimum angle of
deviation occurs at roughly the middle of the range of possible angles of
incidence. At other angles of incidence,
either lower or higher than this one, the angle of deviation increases. This means we can expect the line
to curve upward on both sides of the point we’ve marked out. Overall, the curve will look a bit
like a smile.

We can look at the lines given as
possible answers and find the one that matches our graph the closest. We can quickly eliminate the yellow
and green lines since they are upside down compared to our curve. We can also eliminate the black
line since it is a straight line with no curve. We can see further that blue is not
the correct answer since it only curves down and does not curve back upward like our
line. That leaves us with the correct
option: (E) red. The shape of the red line most
closely matches the curve that we drew. This shows the correct relationship
between Φ one and 𝛼.