Introductory physics: focusing mirrors

Focusing mirrors are mirrors cut from a parabola of reflecting material; the parabola is fabricated in such a way that distant rays will be bent through a single point i.e. the focus of the mirror. In fact, there are two types of curved mirrors; converging mirrors made from parabolas that are concave in shape , and diverging mirrors that are made from parabolas that are convex. In either case, the focal length of the mirror is half the radius of the sphere from which it is cut.

From the diagram below, you can see that in the case of a converging (concave) mirror, parallel rays are focused down to an image at the focal point (this is the point of such a mirror). In this type of mirror the rays reflected by the mirror actually pass through F and it is therefore a real focus.

Converging (concave) mirror

By contrast, parallel rays appear to come from a focal point behind the mirror in the case of a diverging mirror. i.e. the focus is virtual.

Diverging (convex) mirror

There is a simple set of rules to follow when finding the position of an image in curved mirrors:

1. Rays parallel to principal axis are reflected through the principal focus

2. Rays through the principal focus are reflected parallel to the principal axis

3. Rays passing through the centre of curvature are reflected back along their own path

These rules are not mysterious but smply a result of how the mirrors are fabricated.

In the diagram above, note that the object is close to the converging mirror, but outside of the focal length. Using the first 2 rules above, the intersection of the reflected rays gives the position of the image. You can see the image is inverted and diminished.

Image tracing in a diverging mirror

More quantiatively, for any object a distance u from the mirror of focal length f, the location v of the image can be found from the ‘mirror’ equation

1/u +   1/v =   1/f

Note that there are only 2 variables in this equation since f is fixed for a given mirror. Typically, one uses the formula to find the location of the image of an object a given distance from the lens. One can also calculate the height of the image; this is because the magnification m of the mirror is given by the equation

m =   -v/u

Note: in using both the above formulae, we use the convention that any distance that is real object is taken as a positive.

Correction

Actually, focusing mirrors are cut from parabolic surfaces, not spherical ones – I forgot this. See comment below by Norman.

Problems

1. An astronomer is observing a distant star with a reflecting telescope: use the mirror formula above to calculate where the photographic plate should be positioned. What kind of magnification can one expect?

2. If an object 5 cm high is placed 40 cm in front of a converging mirror of focal length 20 cm, calculate the position and height of the image.  Is the image real or virtual?

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2 Comments

Filed under Introductory physics

2 responses to “Introductory physics: focusing mirrors

  1. When you say that “Focusing mirrors are mirrors cut from a sphere”, this is not quite correct. Focusing mirrors are cut from a parabolic form. I offer you the following picture of a coffee cup to illustrate the problem

    The picture shows weak light reflected onto the coffee surface and the drawback of using spherical mirrors (such as a cup is). One clearly sees that only the furthest points of the cup from the light source – forms the effective weak focal point. Without going into the mathematics of a parabolic form, the coffee cup reflection gives a visualisation of the corrections needed to intensify the focus i.e. gradually decrease the curve as you move away from the centre of the mirror ensuring all light rays converge on one point.
    All the best from Frankfurt.

  2. cormac

    Thanks Norman! I remember now – even in school we used to call them parabolic mirrors!