Spherical mirror question

[mdunbar]

I thought about this one for a few hours and still couldn't come to any conclusion. Someone please help me.

I am very confused about focal points, specifically of a concave mirror. Based upon the equation f = r/2 it seems logical that any line drawn from the center of curvature to the surface of the mirror contains a focal point at its midpoint. Since an infinite number of lines can be drawn I predict that there would be an infinite number of focal points. However, all of the material I have read describes a spherical mirror as possessing 1 unique focal point, which makes since to me if the object's distance approaches infinity. In that case, all of the rays arrive more or less parallel to one another and parallel to a central axis (can be any one of the infinite number of possible axises described above) with its corresponding focal point, making all other focal points irrelevant. Where I get confused is when the object distance is just a little bit great greater than the focal focal length. If everything I have said so far holds true then wouldn't you end up with what I have drawn below, where for every focal point a different real image is created. And since there is an infinite number of focal points, does that mean that there is an infinite number of real images?

The black semi-circle is the surface of the concave mirror. The red semi-circle is a circle corresponding to every possible focal point if an infinite number of lines were drawn from the center of curvature, C, to the surface of the mirror. O is the object. F1 is focal point one (which actually corresponds to Image 2, I2, instead of image one (because I labeled it wrong)). F2 is focal point 2 and I1 is image 1.


Please someone tell me where I went wrong!

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[EkramVahsedi86]

After careful review (optics is my weak area) I think your confusion is due to your idea of the focal point.

The central axis is a very big deal. The focal point is by definition, must be on the central axis. What helps me is to remember that a "tilted mirror" does not exist, because if it did, then you could arbitrarily redefine which axis to pick your central axis, and as you stated there would be an infinite number of focal points.

So think of it like this: for every mirror, there is only one central axis and it is the de facto central axis and mirrors will never be drawn in such a way to appear tilted. Thus you never have to worry about the central axis nor the "tilting" of a concave mirror. It will always be setup in a clear, nonconfusing way.

Upon further review, I see what you are pointing out! There should be some sort of relationship if any ray hits any part of the mirror and if you know the angle of incidence and the angle of reflection and the value of theta for the normal line extended to the central axis.

If theta is the angle between the central axis and the normal line extended, then yes you should be able to solve EXTREMELY challenging optics problems, which would never be on the mcat.