Reflection can occur from either rough or smooth surfaces. In either case the law of reflection applies. Let's start by looking at a single light ray striking a highly polished plane mirror surface.

1. Find the angle of reflection for a ray of light that strikes a mirror at an incident angle of 28° with respect to the normal.

In this case, the angle of incidence is given to be 28°, and because this angle is measured with respect to the normal line (perpendicular to the surface of the mirror), the angle of reflection is also 28°.

This may seem like a trivial calculation, but the concept is critical to our ability to draw rays and thus determine the position of images that are formed by plane mirrors. Simple straight-line drawings can show us a great deal about how mirrors and lenses work, so make sure that you carefully study the figures and diagrams in the textbook that illustrate this process.

Figure 7.3 in the textbook is a good example of how geometric optics (ray tracing) can determine the location and properties of an image formed by a simple optical device; in this case, a plane mirror. Notice that you must have not only a ruler or other straight-edge to do this drawing, but you also must have a protractor to measure the angles and make sure that the angle of reflection is really exactly equal to the angle of incidence. Also note that for a plane mirror, the image is the same size as the object, it is upright, and it is located the same distance behind the mirror that the object is in front of the mirror.


2. If a girl is 1.42 meters tall, what is the height of the smallest plane mirror that she could look into and still see her hair and her shoes at the same time while she is standing upright?

Because her image in a plane mirror must be the same size as she is, a small mirror will only show a portion of her entire figure at one time. When the angles of incidence and reflection are equal, she will be able to see her entire height in any mirror that is at least one half as tall as she is. In this case the mirror must be at least 1.42 m / 2, or 0.71 meters in height. See Fig. 7.4 in the textbook, where ray tracing methods have been used to determine the solution to this same type of question.

So far we have done almost no mathematical calculations in solving these optics problems. Ray tracing is a physical method of problem solving that is often much easier than the mathematical methods that we would have to use. There are, however, some problems in this chapter that require a more direct mathematical approach to their solution, as demonstrated in the following exercises.


3. The index of refraction of diamond is the highest of any known transparent substance. This means that light travels slowest through this material. At what percentage of the speed of light does a light ray travel as it passes through diamond?

Here we must use the index-of-refraction equation, but we must first solve it for the speed of light in the particular medium, diamond, where n_diamond = 2.42. (See Table 7.1 in the textbook for the index of refraction of other transparent materials.) The speed of light, c, is equal to 3.00 x 10⁸ m/s, but we do not need to use this value to find the percentage.

n = c / c_m

c_m = c / n = 1.00 c / 2.42 = 0.413 c

or expressed as a percentage, c_m = 41.3% of c

Notice that because we are given the index of refraction of diamond to three significant figures, we can calculate the percentage of the speed of light in diamond to three significant figures. Exercise 5 in the textbook solves the above problem as an actual speed instead of as a percentage.


4. The speed of light in a particular type of glass is measured to be 1.60 x 10⁸ m/s. What is the index of refraction of this glass?

Here we know the speed of light, but we need to calculate the index of refraction.

n = c / c_m = 3.00 x 10⁸ m/s / 1.60 x 10⁸ m/s = 1.90

Note that the units (m/s) cancel, so that no unit is expressed for an index of refraction.



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