Topic
8. Shadows and geometrical optics
Validity of Geometrical Optics
Geometrical optics: The domain in which optical
waves can be treated as simple straight-line rays and the effect of optical elements
can be modeled using simple geometrical constructions. There are a number of
requirements for these approximations to be an accurate description of an
optical system. The most important is that l/D << 1, where l is the wavelength of the light signal and D
is the size of the object, lens, aperture, etc. that is being studied. This
requirement is almost always satisfied for visible light interacting with
everyday-sized objects (which is why Newton did not find a contradiction from
everyday experience when he stated that light was probably a stream of
particles).
There are a number of other requirements that must
be fulfilled in some cases. These requirements will become important in the
studies of lenses and curved mirrors.
Shadow: produced on a screen behind the object when
light from a source is blocked (either completely or only partially) by the
object. In the case where geometrical optics is appropriate, the shadow can be
constructed using simple straight-line rays.
A clear unambiguous shadow requires a “point” source
of light – something that either is very small compared to the other dimensions
of the experiment or else is very far away. The geometrical construction of the
shadow is particularly simple in this case, since all of the rays that strike
the object come from a single point, so that a given point of the image either
receives light or doesn’t.
If the source of light is too large to be regarded
as a “point” source, then the shadow is constructed using the principle of
superposition – the extended source is modeled as a large number of point
sources, and the shadow produced by each of these point sources is constructed
independently. This is an example of the “principle of superposition” – that is
that the effect of a number of elementary sources is the sum of the effects of
each one of them taken individually.
The shadow produced by two or more point sources is
also handled by the principle of superposition in the same way – the light
striking a screen behind the object is the sum of the contributions from all of
the light sources taken individually.
The umbra:
the portion of the shadow which is totally dark because light from all
sources is absent.
The penumbra: the portion of the shadow which
is illuminated by some of the sources and is therefore not completely dark.
In some situations (such as a transparency projector
or an x-ray picture), the information content of the image is really in the
shadows, and the unobstructed parts of the image are not of interest.
False shadows are often used to make a flat image
appear “3-dimensional.” This works because our eyes are accustomed to the idea
that a shadow is produced when an object is in front of a screen.
Solar and Lunar eclipses
Since the Moon orbits the earth with a period of
about 29.5 days, there are occasions when the Moon comes between the Earth and
the Sun or when the Earth comes between the Moon and the Sun. These are called
eclipses, and they can be understood using the principles of shadows outlined
above. (The orbit of the Moon is not
exactly in the same plane as the orbit of the Earth around the Sun, so that an
eclipse does not happen every 29.5 days.)
Although the Moon is much smaller than the Sun, it
is much closer to the Earth, so that its shadow can totally block the light of
the Sun if the alignment is exactly correct. This is a “total” eclipse; it is
quite rare, since the alignment is usually not exact. The more usual situation
is a “partial” eclipse in which only part of the Sun is blocked. Even during a
total eclipse, the umbra of the Moon does not completely cover the Earth, so
that only some places on the Earth see the eclipse as total. Other locations
see only a partial eclipse, and yet other locations see nothing at all, since
they are neither in the umbra nor in the penumbra.
A lunar eclipse occurs when the Earth comes between
the Sun and the Moon. As above, the eclipse may be total or partial, depending
on the exact alignment. Since the Moon shines by reflecting light from the Sun,
the moon almost disappears during an eclipse. However, even in a total eclipse,
it is often possible to see a very faint image of the Moon because some light
from the Sun is deflected by the atmosphere and still reaches the Moon.
Return to Physics 1230 main page