Refraction of Light by GlassConverging and Diverging Lenses©98

Experiment 10

Objective:To determine the law of refraction of light by glass using geometric ray-tracing; to become familiar with the characteristics of converging and diverging lenses.

DISCUSSION:

When a ray of light in the air strikes the surface of a transparent medium, such as glass, at some angle θ1 to the normal of a surface, that part of the ray which passes into the glass is bent or refracted so that it leaves the surface at some other angle θ2. The situation is shown in Fig. 1.

The ancients were aware of these phenomena and found that it was roughly true that

(1)

where k is a constant unique to the medium the light entered from air. However, a better law (discovered in the early 17th century by Willebord Snell) is that

(2)

where n1 is a constant unique to medium 1 and is known as the index of refraction of medium 1 relative to medium 2. Similarly, n2, is the index of refraction of medium 2 relative to medium 1. When a ray of refracted light enters the eye, the observer interprets the ray as having traveled along a straight line from its source. The direction of this line is the direction at which light enters the eye. Consider Fig. 2. Medium 2 is a rectangular glass plate, and medium 1 is air. Pins are placed at P1and P2. The observer views these pins through the glass, and then places a third pin at P3so that, when seen through the glass, the 3 pins are aligned. Actually the line of light connecting P1, P2and P3is bent at the first and second surfaces of the glass. These positions P1, P2and P3, along with the normal to the first surface, enable one to measure the angles θ1and θ2.


If n1n2, as in going from air to glass, the beam of light is bent toward the normal at the first surface as in Fig. 2. However if n1n2, as in going from glass to air, the beam of light is bent away from the normal, as shown at the second surface of Fig. 3. Sometimes, when a light ray hits the boundary, it does so at such an angle that it cannot be refracted. The boundary also cannot absorb the light, and so it must either be carried along the boundary or reflected back into the medium as seen in Fig. 4. The criticalangleθc is the angle at which the light will follow the boundary of the medium, making θ2 of Snell's Law equal to 90º. Since sin 90º = 1.0, Snell's Law then becomes n2sin θ2 = n2 for this particular case. But, we know that the sine function has a maximum value of 1.0, and so the term n1sin θ1must be equal or greater than n2 in order to make the statement true. Angles larger than the critical angle will cause total internal reflection while angles less than the critical angle will cause the light ray to be refracted as usual.

Note: The index of refraction of a medium with respect to air is commonly called n. Since nairis about 1.0003, the common measure of n will indicate the index of refraction of a given material with respect to air, instead of a vacuum.

A converging lens is a piece of transparent optics, usually glass or plastic, that causes parallel rays of light to converge toward the central axis of the lens. For example, a simple converging lens can concentrate enough sunlight (striking the Earth as parallel rays) to start a fire on a dry sheet of paper. In contrast, a diverging lens causes parallel rays of light to diverge or spread away from the central axis of the lens. For example, a simple diverging lens would spread out sunlight that went through it as though the sunlight were coming from a point along the central axis in front of the lens.

The opticalaxis of a lens is a line through the center of the lens and perpendicular to the plane of the lens. The focalpoint of the converging lens is that point on the optical axis at which rays of light from an infinitely remote source on the optical axis (like the sun) converge. The focal point of the diverging lens is that point on the optical axis from which the rays of light passing through the lens, from an infinitely remote source on the optical axis, seem to diverge. It is also the point from which the rays of light that enter the lens parallel to the optical axis seem to diverge after passing through the lens. Figs.1 and 2 show how rays of light travel through a converging and a diverging lens, respectively.


The rules for constructing the image of an object, or a "ray diagram" are as follows:

  1. From a point on the object lying off the optical axis, draw a straight line to the lens, parallel to the optical axis, as shown in Figs. 1 and 2. Change the direction of this line on the other side of the lens so that it:
  2. passes through the focal point, if the lens is converging
  3. seems to originate from the focal point, if the lens is diverging.
  4. From the same point on the object, draw a second straight line through the center of the lens, continuing it on the other side of the lens with its direction unchanged.
  1. If the extensions of the two lines constructed above converge after passing through the lens, draw the real image of the object- point at the intersection of the two lines. Or if the extensions of the two lines diverge, draw the virtual image of the object – point at the point from which the extended lines seem to diverge.

The images formed by diverging lenses are always virtual because the transmitted rays of light always diverge. But the images formed by converging lenses may be real or virtual. They are real if the object is farther from the lens than is the focal point, and they are virtual if the object is nearer the lens than is the focal point.

It can be shown from the geometry of the rays in Figs. 1 and 2 that in general

(1)

where so isthe distance of the object from the lens, siis the distance of the image from the lens, and f is the distance of the focal point from the lens. The following sign conventions are followed by optical scientists:

  1. The focal length f is positive for converging lenses and negative for diverging lenses.
  2. The object distance so is positive when the object is on the incoming side of the lens, negative otherwise.
  3. The image distance si is positive when the image is on the outgoing side of the lens, negative otherwise.

If two lenses are used in combination and are placed in contact with one another, the image that would be produced by the first lens acting alone becomes the object for the second lens. In this case, the effective focal length f of the combination is given by;

(2)

where f1 and f2 are the focal lengths of the separate lenses.

EXERCISES FOR PRISMS:

  1. Refraction in a square rectangular piece of glass:
  2. Using the pins as objects, trace several independent rays of light through the rectangular prism. Suggested angles of θ1 are 5˚, 10˚, 15˚, 20˚, 30˚, and 70˚.
  3. For each ray of light, determine the index of refraction of the glass using Snell's Law. Is this index of refraction a constant? Determine the percent deviation of each n2 value from the average n2 value.
  4. Calculate the value of k in Eq. (1) for each ray of light. Determine the percent deviation of each k value from the average k value.
  1. Reflection and Refraction in a prism:
  2. Trace several independent rays of light through the glass triangle.
  3. Using the appropriate angles, compute the index of refraction of the glass.
  4. Calculate the "critical angle" and verify that your observations of total internal reflections are valid.

EXERCISES FOR LENSES:

  1. Measure the focal lengths of the lenses marked red and yellow using a distant object outside the window.
  2. Place a screen on the end of the optical bench opposite the window.
  3. Place one lens holder with a lens on the optical bench between the screen and the window.
  4. Move the lens back and forth on the optical bench until you have the best image of a distant object.
  5. Lock the lens in place, then measure and record the distance between the lens and the screen. This is the focal length of the lens.
  6. Repeat for the other lens.
  1. Determine the focal lengths of the same lenses by imaging nearby objects.
  2. Add the light source to the optical bench on the opposite end from the screen.
  3. Calculate the focal lengths of the same lenses by using Eq. (1) and measuring the image and object distances due to an object placed in the neighborhood of the lens.
  4. Repeat this exercise for different object distances and calculate an average focal length for each lens.
  5. How does this calculated average focal length compare to the measured focal length from Exercise 1?
  6. Repeat for remaining lens.
  7. Using so and si found above and the calculated f, draw a to-scale ray diagram for each lens (in cm!).
  8. Determine the focal length of the diverging lens by combining it with a suitable converging lens and using Eq. (2). (note: you may have to hold the lenses together as they will not both fit in the lens holder).
  9. Observe the interchangabilty of object and image distances.
  10. Set the object and the screen at least four focal lengths apart for a particular lens.
  11. Find the two lens positions that will give a clear image on the screen.
  12. How are the two images similar/different?
  13. Explain this phenomenon based on Eq. (1).
  1. Build a simple telescope.
  2. Remove the screen and light source from the optical bench.
  3. Add a second lens holder.
  4. Put the longer focal length lens in the lens holder closer to the window. This is the objective of the telescope.
  5. Put the shorter focal length lens in the other holder. This lens is the eyepiece.
  6. While keeping the objective lens stationary, move the eyepiece until you can see a clear image through the two lenses. (note: the eyepiece should not be right against your eye.)
  7. Measure and record the distance between the two lenses.
  8. Calculate the magnification of your telescope using the following equation:

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