Video Transcript
Below is a series of ray diagrams
for convex mirrors. All four mirrors have identical
incident light reflecting off their surfaces. This light is shown by the yellow
lines. Which of the four mirrors has the
smallest radius of curvature?
To answer this question, we need to
understand the definition of the radius of curvature of a spherical mirror. All of the mirrors shown in the
question are spherical mirrors. This is because the mirrors are
shaped in such a way that they can be thought of as being part of the surface of a
sphere. Every sphere has a radius. It is the distance from the center
to the edge. The distances shown here are all
equal. They are all the radius of the
sphere. In other words, the distance
between the center of a sphere and its surface is the same in every direction.
For a spherical mirror, the
sphere’s radius is called the radius of curvature. The smaller the sphere, the smaller
this radius is. Therefore, we can answer our
question by figuring out which of our four mirrors is part of the smallest
sphere. That mirror will have the smallest
radius of curvature.
If we look at the curves that
represent the four mirrors, we know that the more a line curves, the smaller the
sphere that line is a part of. We want to know which line curves
the most. We can see it’s not mirror
three. That line curves the least of all
four mirrors. It’s also not mirror four, which
curves less than mirrors one and two. Considering mirrors one and two, it
looks like the sphere made partly by mirror two has a slightly smaller radius. We can confirm this by considering
the rays of light that reflect off of mirrors one and two.
Clearing some space, let’s consider
this spherical mirror with a ray of light incident on it. When this ray reflects off the
mirror, it will do so in a direction perpendicular to the mirror surface at that
point. In other words, if we draw a line
normal to the mirror at the point where the ray is incident, the reflected ray will
follow this line. And if we send in another ray
parallel to the first one, the same thing will happen. The ray will reflect along the line
normal to the mirror surface where the ray was incident.
Now, let’s trace our two reflected
rays backwards, behind the mirror. These virtual rays cross at a
point. And notice this point is some
distance from the center of the mirror. Now, if we were working with a
larger spherical mirror, the same incident rays would reflect in different
directions than before, because now the lines normal to the mirror surface point
differently. When traced backward, they cross at
a point much farther from the center of the mirror.
All this means we can compare the
curvatures of mirrors one and two by looking at where the virtual rays cross in each
one. We can see the virtual rays in
mirror two meet at a smaller distance from the center of the mirror than those in
mirror one. Therefore, mirror two has a greater
curve to its surface, which means it has a smaller radius of curvature. For our answer, we choose mirror
two.
