MAX 12/30/03MAX 12/28/03, PAUL 12/28/03 kierana11/26/03   aaaaaxxyy 11/17/03 email to rhr: rhromer@amherst.edu

hi, bob

Of course the basic reason that things look different in the passenger-side mirror, not the way they do when you look at them directly, is that the surface of the mirror is curved. If you feel it with your hand, you can almost feel that it is a convex surface. A sure-fire way to tell is to take something straight, e.g., a 3" x 5" index card held edgewise or a 6" ruler, and hold it against the mirror; you can easily see from this that the mirror is by no means flat. If you have someone hold a little flat mirror next to that mirror, then by comparing what you see in the two mirrors, you can see that indeed things do look smaller (and thus as if they are farther away than they really are) in the curved mirror. And a handy convex mirror to experiment with is a spoon: hold it with the bottom of the spoon facing toward you and look at your reflection in the spoon. Almost everyone has looked at reflections in curved mirrors, especially concave mirrors, for instance the type that are found in many bathrooms for giving you a closeup look at your face. x 3/21/03 aaa

To be sure, the light that reaches the driver's eyes from the passenger-side mirror does have to go a little bit farther to get to her eyes than does light from the driver's-side mirror. And so things seen in the passenger-side mirror would look a tiny bit smaller, even if that mirror were flat. But the extra distance is only a foot or so, and if you're looking at a truck 70 feet or so behind you, a couple of feet more or less would only make an extremely small change in the apparent size of that truck. By itself, that slight extra distance can't be the explanation. Besides, all you have to do to test that idea is to stop the car, undo your seat belt, and move over into the passenger seat - things seen in that mirror still look farther away than they really are.

We'll see later, though, that there is just a germ of truth in the silly remark that things look smaller than they really are because the light has to go farther to get to your eye; the amount of reduction in size does turn out to depend on how far away from a curved mirror your eye is.

How to work it out? The important factor determining how big an object looks is the angle it "subtends" at your eye. In the case of a truck behind you on the highway, let's take this to be the angle between two light rays coming to your eye, one from the bottom of the truck, the other from the top. If the truck is 8 feet high and is 4 car lengths (about 70 feet) behind you, the angle (if you look at it directly - or in a flat mirror, for that matter) is about 8/70 = 0.1 (in radians, which is about 7 degrees). Almost all the angles we deal with in this problem are small, and we use "small angle approximations" because it simplifies the math. We will usually express angles in radians, and 10% accuracy is fine for this problem. So we'll write 8/70 = 0.1, instead of 8/70 = 0.11429....

Here's a sketch (not to scale!) of a truck behind you, a convex mirror, and your eye. You see light from all of the truck, but the sketch simply shows two rays of light, one from the bottom of the truck and one from the top, both of which are reflected from the mirror and arrive at your eye.

There are plenty of other rays, but the others from the top and bottom of the truck miss your eye. (We're treating the eye as if it were very small - which it is, in this context.) And we don't show any of the light coming from intermediate points, such as the radiatior cap, etc. Light reflects from a curved mirror just as it would from a small flat mirror that is tangent to the curved mirror at the point where the light reflects. "Locally", the curved mirror acts no differently from a small flat mirror. Both of the rays shown are drawn so that they follow the usual rule of reflection: angle of incidence = angle of reflection. That's the only basic physics involved here - though being careful about applying this physics and thinking about what angles are important, etc. is also an important part of the practice of physics.

From the sketch, it certainly looks as though the angle between the two rays coming to your eye from the top and the bottom of the truck is indeed quite small, smaller than the angle you would get by looking directly at the truck. To get quantitative about this, it is useful to introduce some of the formalism of geometrical optics. A convex mirror like this has a "focal length" (f) , which is equal to R/2, where R is the radius of curvature of the mirror. (When we say that the focal length is f, we mean that a very distant object will form a "virtual image" at about a distance f behind the surface of the mirror.)  You can measure the radius of curvature of such a mirror by putting a short straight ruler across it, measuring the distances shown in the following sketch, and doing a little geometry. (We are only after approximate results, so there is no need to be fussy. Approximate! For instance, don't worry about whether you measure the distance x the way it's shown (perpendicular to the ruler) or instead perpendicular to the mirror surface.)

For the passenger-side mirror on a 2000 Chevy Prizm, with y = 3" (we used a 6” ruler), we found x to be about 2 millimeters (sorry about the mixed units), and then some geometry and conversion of units leads to the conclusion that R is about 56” (and so f is about 28” = 2.3’).

If we're looking at a truck behind us, then the distance to the truck (call that distance s) is surely >> f. And the size of the image (say its height, h) is less than the actual height of the truck (H) approximately by the ratio of the distance s to the distance f:

h/H = f/s.

(It takes some drawing of light rays and work with similar triangles to work out that result.)

If f = 2.3' and s is, say 70’, then h is much smaller than H. But the image of the truck is much closer than the real truck, and what really matters is how large an angle there is between the light coming to us from the top of the truck and the light from the bottom of the truck. Call that angle A for the case where we're looking directly at the truck. A is surely small enough so that small angle approximations are OK, so

A = H/s

With the numbers used earlier,  A = 8/70 = 0.1 radians.

Call A' the angle when we're looking at the image, which has a height h. The image is a distance f behind the mirror, so its distance from our eye is (E + f), where E is the distance from our eye to the surface of the mirror, and so:

A' = h/(E + f) = Hf/s(E + f)

and what we really care about is the "magnification" (M), the ratio of those two angles:

M = A'/A = f/(E + f).

M is obviously less than 1 (things do look smaller), but we still call it the "magnification". So M depends on two things: the focal length (and thus the amount of curvature of the mirror) and the distance from the driver's eyes to the mirror. For the driver of our Prizm, E is about 4.2’, and so for her (we’ll leave out a bit of arithmetic – you can fill it in), M is about 0.56, which seems reasonable. That is, this is roughly what one's subjective sense is of how much smaller and farther away things appear in that mirror.

This is our main result. For the driver, M is around 0.5 or 0.6, definitely less than 1, but not enormously smaller, not down around 0.1 or anything like that.

The passenger, though, is closer to the mirror. For him, E is only about 1.8’, which makes M about 0.35. That is, our final results do show that how much smaller objects appear when looked at in a convex mirror does depend on how close your eye is to the mirror. Look at our final equation for M. Make E smaller (as it is for the pasenger) and M gets larger. In fact, if you make E very small ( << f ), our theory says that M approaches 1, i.e., no size reduction at all. (Put your eye very, very close to the surface of the mirror.) So in a sense the silly observation that one of our characters made ("That’s because you’re farther away from the mirror, so the light has to go farther to get to you.") does have just a little bit of truth to it.