The reflection and refraction of light
7-27-99
Rays and wave fronts Light is a very complex phenomenon, but in many situations its behavior
can be understood with a simple model based on rays and wave fronts. A
ray is a thin beam of light that travels in a straight line. A wave
front is the line (not necessarily straight) or surface connecting all
the light that left a source at the same time. For a source like the
Sun, rays radiate out in all directions; the wave fronts are spheres
centered on the Sun. If the source is a long way away, the wave fronts
can be treated as parallel lines.
Rays and wave fronts can generally be used to represent light when
the light is interacting with objects that are much larger than the
wavelength of light, which is about 500 nm. In particular, we'll use
rays and wave fronts to analyze how light interacts with mirrors and
lenses.
The law of reflection
Objects can be seen by the light they emit, or, more often, by the
light they reflect. Reflected light obeys the law of reflection, that
the angle of reflection equals the angle of incidence.
For objects such as mirrors, with surfaces so smooth that any hills or
valleys on the surface are smaller than the wavelength of light, the
law of reflection applies on a large scale. All the light travelling in
one direction and reflecting from the mirror is reflected in one
direction; reflection from such objects is known as specular
reflection.
Most objects exhibit diffuse reflection, with light being reflected
in all directions. All objects obey the law of reflection on a
microscopic level, but if the irregularities on the surface of an
object are larger than the wavelength of light, which is usually the
case, the light reflects off in all directions.
Plane mirrors A plane mirror is simply a mirror with a flat surface; all of us use
plane mirrors every day, so we've got plenty of experience with them.
Images produced by plane mirrors have a number of properties,
including:
1. the image produced is upright
2. the image is the same size as the object (i.e., the magnification is m = 1)
3. the image is the same distance from the mirror as the object appears
to be (i.e., the image distance = the object distance)
because the light rays do not actually pass through the image. This
also implies that an image could not be focused on a screen placed at
the location where the image is. A little geometry Dealing with light in terms of rays is known as geometrical optics, for
good reason: there is a lot of geometry involved. It's relatively
straight-forward geometry, all based on similar triangles, but we
should review that for a plane mirror.
Consider an object placed a certain distance in front of a mirror,
as shown in the diagram. To figure out where the image of this object
is located, a ray diagram can be used. In a ray diagram, rays of light
are drawn from the object to the mirror, along with the rays that
reflect off the mirror. The image will be found where the reflected
rays intersect. Note that the reflected rays obey the law of
reflection. What you notice is that the reflected rays diverge from the
mirror; they must be extended back to find the place where they
intersect, and that's where the image is.
Analyzing this a little further, it's easy to see that the height of
the image is the same as the height of the object. Using the similar
triangles ABC and EDC, it can also be seen that the distance from the
object to the mirror is the same as the distance from the image to the
mirror.
Spherical mirrors Light reflecting off a flat mirror is one thing, but what happens
when light reflects off a curved surface? We'll take a look at what
happens when light reflects from a spherical mirror, because it turns
out that, using reasonable approximations, this analysis is fairly
straight-forward. The image you see is located either where the
reflected light converges, or where the reflected light appears to
diverge from. A spherical mirror is simply a piece cut out of a reflective
sphere. It has a center of curvature, C, which corresponds to the
center of the sphere it was cut from; a radius of curvature, R, which
corresponds to the radius of the sphere; and a focal point (the point
where parallel light rays are focused to) which is located half the
distance from the mirror to the center of curvature. The focal length,
f, is therefore: focal length of a spherical mirror : f = R / 2
This is actually an approximation. Parabolic mirrors are really the
only mirrors that focus parallel rays to a single focal point, but as
long as the rays don't get too far from the principal axis then the
equation above applies for spherical mirrors. The diagram shows the
principal axis, focal point (F), and center of curvature for both a
concave and convex spherical mirror.
Spherical mirrors are either concave (converging) mirrors or convex
(diverging) mirrors, depending on which side of the spherical surface
is reflective. If the inside surface is reflective, the mirror is
concave; if the outside is reflective, it's a convex mirror. Concave
mirrors can form either real or virtual images, depending on where the
object is relative to the focal point. A convex mirror can only form
virtual images. A real image is an image that the light rays from the
object actually pass through; a virtual image is formed because the
light rays can be extended back to meet at the image position, but they
don't actually go through the image position.
Ray diagrams To determine where the image is, it is very helpful to draw a ray
diagram. The image will be located at the place where the rays
intersect. You could just draw random rays from the object to the
mirror and follow the reflected rays, but there are three rays in
particular that are very easy to draw. Only two rays are necessary to locate the image on a ray diagram,
but it's useful to add the third as a check. The first is the parallel
ray; it is drawn from the tip of the object parallel to the principal
axis. It then reflects off the mirror and either passes through the
focal point, or can be extended back to pass through the focal point. The second ray is the chief ray. This is drawn from the tip of the
object to the mirror through the center of curvature. This ray will hit
the mirror at a 90° angle, reflecting back the way it came. The chief
and parallel rays meet at the tip of the image.
The third ray, the focal ray, is a mirror image of the parallel
ray. The focal ray is drawn from the tip of the object through (or
towards) the focal point, reflecting off the mirror parallel to the
principal axis. All three rays should meet at the same point.
A ray diagram for a concave mirror is shown above. This shows a few
different things. For this object, located beyond the center of
curvature from the mirror, the image lies between the focal point (F)
and the center of curvature. The image is inverted compared to the
object, and it is also a real image, because the light rays actually
pass through the point where the image is located.
With a concave mirror, any object beyond C will always have an
image that is real, inverted compared to the object, and between F and
C. You can always trade the object and image places (that just reverses
all the arrows on the ray diagram), so any object placed between F and
C will have an image that is real, inverted, and beyond C. What happens
when the object is between F and the mirror? You should draw the ray
diagram to convince yourself that the image will be behind the mirror,
making it a virtual image, and it will be upright compared to the
object.
A ray diagram for a convex mirror
What happens with a convex mirror? In this case the ray diagram looks like this:
As the ray diagram shows, the image for a convex mirror is virtual, and
upright compared to the object. A convex mirror is the kind of mirror
used for security in stores, and is also the kind of mirror used on the
passenger side of many cars ("Objects in mirror are closer than they
appear."). A convex mirror will reflect a set of parallel rays in all
directions; conversely, it will also take light from all directions and
reflect it in one direction, which is exactly how it's used in stores
and cars.
Drawing a ray diagram is a great way to get a rough idea of how big
the image of an object is, and where the image is located. We can also
calculate these things precisely, using something known as the mirror
equation. The textbook does a nice job of deriving this equation in
section 25.6, using the geometry of similar triangles.
Magnification
In most cases the height of the image differs from the height of the
object, meaning that the mirror has done some magnifying (or reducing).
The magnification, m, is defined as the ratio of the image height to
the object height, which is closely related to the ratio of the image
distance to the object distance:
A magnification of 1 (plus or minus) means that the image is the same
size as the object. If m has a magnitude greater than 1 the image is
larger than the object, and an m with a magnitude less than 1 means the
image is smaller than the object. If the magnification is positive, the
image is upright compared to the object; if m is negative, the image is
inverted compared to the object.
Sign conventions What does a positive or negative image height or image distance
mean? To figure out what the signs mean, take the side of the mirror
where the object is to be the positive side. Any distances measured on
that side are positive. Distances measured on the other side are
negative. f, the focal length, is positive for a concave mirror, and negative for a convex mirror. When the image distance is positive, the image is on the same side
of the mirror as the object, and it is real and inverted. When the
image distance is negative, the image is behind the mirror, so the
image is virtual and upright.
A negative m means that the image is inverted. Positive means an upright image.
Steps for analyzing mirror problems There are basically three steps to follow to analyze any mirror
problem, which generally means determining where the image of an object
is located, and determining what kind of image it is (real or virtual,
upright or inverted).
* Step 1 - Draw a ray diagram. The more careful you are in
constructing this, the better idea you'll have of where the image is.
* Step 2 - Apply the mirror equation to determine the image distance.
(Or to find the object distance, or the focal length, depending on what
is given.)
concave mirror with a focal length of 10.0 cm. Where is the image? How
tall is the image? What are the characteristics of the image?
The first step is to draw the ray diagram, which should tell you
that the image is real, inverted, smaller than the object, and between
the focal point and the center of curvature. The location of the image
can be found from the mirror equation:
which can be rearranged to:
The image distance is positive, meaning that it is on the same side
of the mirror as the object. This agrees with the ray diagram. Note
that we don't need to worry about converting distances to meters; just
make sure everything has the same units, and whatever unit goes into
the equation is what comes out.
Calculating the magnification gives:
Solving for the image height gives:
The negative sign for the magnification, and the image height, tells us that the image is inverted compared to the object.
To summarize, the image is real, inverted, 6.2 cm high, and 17.7 cm in front of the mirror.
Example 2 - a convex mirror The same Star Wars action figure, 8.0 cm tall, is placed 6.0 cm in
front of a convex mirror with a focal length of -12.0 cm. Where is the
image in this case, and what are the image characteristics? Again, the first step is to draw a ray diagram. This should tell
you that the image is located behind the mirror; that it is an upright,
virtual image; that it is a little smaller than the object; and that
the image is between the mirror and the focal point.
The second step is to confirm all those observations. The mirror equation, rearranged as in the first example, gives:
Solving for the magnification gives:
This gives an image height of 0.667 x 8 = 5.3 cm.
All of these results are consistent with the conclusions drawn from
the ray diagram. The image is 5.3 cm high, virtual, upright compared to
the object, and 4.0 cm behind the mirror.
Refraction When we talk about the speed of light, we're usually talking about the
speed of light in a vacuum, which is 3.00 x 108 m/s. When light travels
through something else, such as glass, diamond, or plastic, it travels
at a different speed. The speed of light in a given material is related
to a quantity called the index of refraction, n, which is defined as
the ratio of the speed of light in vacuum to the speed of light in the
medium: index of refraction : n = c / v
When light travels from one medium to another, the speed changes,
as does the wavelength. The index of refraction can also be stated in
terms of wavelength:
Although the speed changes and wavelength changes, the frequency of the
light will be constant. The frequency, wavelength, and speed are
related by:
The change in speed that occurs when light passes from one medium to
another is responsible for the bending of light, or refraction, that
takes place at an interface. If light is travelling from medium 1 into
medium 2, and angles are measured from the normal to the interface, the
angle of transmission of the light into the second medium is related to
the angle of incidence by Snell's law :
