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Illustration 33.1: Mirrors and the Small-Angle Approximation
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Shown is an optics bench that allows you to add various optical elements (lens, mirror, and aperture) and light sources (beam, object, point source) and see their effect. Elements and sources can be added to the optics bench by clicking on the appropriate button and then clicking inside the applet at the desired location. Moving the mouse around in the applet gives you the position of the mouse, while a click-drag will allow you to measure angle (position is given in centimeters and angle is given in degrees).
Add a mirror to the optics bench by clicking on the mirror button and then clicking inside the animation to place the mirror. Adjust the focal point of the mirror by dragging on the round hotspots. Notice that you can make the mirror either concave or convex. Make the mirror concave with a focal length of 0.5 cm and place it near the right-hand side of the applet. Now add a source of light by clicking on the object button and then clicking inside the animation. You can later add other sources of light.
Notice the rays emanating from the object, their reflection from the mirror, and the resulting image. Click the head of the arrow (the object) and move it around. First note that there are three rays that emanate from the head of the arrow. One ray comes off parallel to the principal axis (the yellow centerline) and is reflected through the focal point, one ray comes off at an angle to hit the mirror on the principal axis and is reflected, and one ray passes through the principal axis at the focal point of the mirror and is reflected parallel to the principal axis.
Do the rays always behave as you expect them to? Probably not. As you drag the head of the source and change its height and position, what do you notice about the rays when they reflect from the mirror? The rays reflect from the vertical line tangent to the mirror's surface. If you click on the mirror, you will see this line in green. This applet uses what is called the small-angle approximation. This approximation assumes that the object is of a much smaller dimension than the mirror. For larger focal length mirrors, you may barely notice the approximation, but for smaller focal length mirrors it becomes increasingly noticeable. In this Illustration a focal length of f < 1 cm will yield a noticeable difference between the rays you expect and the result of the small-angle approximation. Click on the mirror and drag the round hotspots to change the mirror's focal length to see the effect.
The optics bench allows you to try many different configurations to see how light will interact with a mirror. Take some time to play with the applet. You may also find it helpful to refer back to this Illustration as you develop your understanding of optics. A brief description of the three sources is given below.
- The "Beam" button adds a beam of parallel light rays. The angle of the light rays can be changed by dragging the hotspot after clicking on the beam.
- The "Object" button adds an arrow as an object. A ray diagram is drawn for the object if an optical element is present.
- The "Source" button adds a point source of light. The spread of the light rays can be adjusted by dragging the hotspot after clicking on the source.
Illustration authored by Mario Belloni and Melissa Dancy.