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PHYSICS
CONTEST
[Ulil'rorlull ol ulatop
"The empty mirror. lf you could really understand that,
there would be nothing left to look fs1'."-ysn de Wetering
by Arthur Eisenkraft and Larry D. Kirkpatrick
N I L'. ,?ff,il1#'#,fJl;
I -asic
shows and see the ma-
I gicirtt push sharp swords
through a box containing the
"lovely assistant" or ride the
"Haunted Mansion" and see the
ghost flying through the room at
Disneyland/ we are often surprised
and pleased by clever manipulations
of images.
In this contest problem, we'll
look at the image producedby a concave
mirror filled with water. Because
our con{idence in a physics solution
increases if different
approaches to the problem yield the
same result and there are many
ways of obtaining the position of the
image, we will want to discover as
many of them as possible. Perhaps
you will come up with a solution
that is fundamentally different from
the ones we expect.
Texts on geometrical optics often
begin by showing that the reflection
of light from plane mirrors follows
the principle that the angle of incidence
is equal to the angle of reflection.
If the mirror is curved, this behavior
still holds, but the geometry
of the parabolic mirror is such that
all parallel rays come to a focus for
a concave mirror, or appear to diverge
from the focus in the case of a
convex mirror. For a spherical mirror,
the spherical surface approximates
the parabolic curve and parallel
rays near the axis also come together
at (or diverge from) the focus.
The relationship between the
image and object is given by the
mirror formula
111
s s' f'
where s and s'are the distances of
the object and image from the surface
of the mirror and I is the focal
length of the mirror. The focal
length is often stamped on the mirror
and is equal to one half of the
radius of the spherical surface from
which the mirror is made. The focal
length can be measured by shining
a beam of parallel light onto a concave
mirror and measuring the distance
from the surface of the mirror
to the point where the beam is
brought to a focus. For a convex
mirror the light appears to diverge
from a focal point located behind the
mirror. Both of these points can be
determined by drawing several rays
parallel to the axis of the mirror, using
the law of reflection at the surface,
and locating where the rays
CIOSS.
To make effective use of the mirror
formula we must remind ourselves
of a number of conventions.
The fistance s is positive if the object
is located in front of the mirror.
This will always be the case for real
objects, but the "obiect" couldbe an
image produced by another optical
device. I:r this case, the object could
be located behind the mirror and s
would be negative. If the image is
located in front of the mirror, the
image distance s'is positive; if the
image is behind the mirror, s' is
negative. Finaily, f is positive for a
concave mirror and negative for a
convex mirror.
As an example, consider an object
located a distance 3/ in front of a
concave mirror:
1 r 1 I 1.2^ -=-:
-=---=--_ s'fsf33
Therefore, the image is located a
distance 3f 12 in front of the mirror.
This can also be shown with a diagram
that traces the rays. Convince
yourself that the image would be
3f l4behind the mirror if we use a
convex mirror instead of the concave
mirror.
The mirror formula also works
for lenses if we adopt the following
conventions: s is positive if the object
is located in front of the lens,
negative if the obfect is located behind
the lens; s' is positive if the
image is located behind the lens,
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