Concave and Convex Mirrors

 

Carlyle Moore

 

            Spherical mirrors can be divided into two classes: concave and convex.  A spherical mirror is said to be concave if the reflecting surface and the center of curvature (the center of the sphere of which the mirror forms part) are on the same side of the mirror; convex if the reflecting surface and the center of curvature are on opposite sides of the mirror.  The center of the mirror P is called the pole and the radius through the pole is called the axis of the mirror.

 

Concave Mirrors

 

            Let us first consider the image formed by a concave mirror of a point O on the axis.  It was assumed earlier that an optical device forms a sharp point image of a point object.  In the case of a concave mirror, this is true only if the rays of light make small angles with the axis, i.e. strike the mirror close to the pole.  Such rays are said to be paraxial.  Rays outside the paraxial region do not converge to a sharp point, with the result that the image is blurred considerably, a defect known as spherical abberation.  If the image is sharply defined, its location can be found by tracing out the paths of any two rays emanating from O.  One of these may be conveniently chosen to be normal to the mirror, since it is just reflected back along the original path.  Let the other ray strike the mirror at Q, making an angle q with the radius CQ , C being the centre of curvature of the mirror. The image I is located at the intersection of the two reflected rays. We shall assume, for the moment, that OP is greater than the radius of curvature .  Denote by U and V, respectively, the distances of the object O and the image I from the pole P.  It follows from the geometry that

 

 

 

(1.1)

 

whence

(1.2)

 

Now because of the paraxial condition (  ), we have

 

(1.3)

and Equation (1.2) becomes

 

(1.4)

 

This formula gives the image distance V in terms of the object distance U and the radius of curvature R, assuming that .  We note the important special case in which ,  when Equation (1.4) reduces to .  This describes a situation in which a ray of light moving parallel to the axis strikes the mirror and is reflected so as to pass through a point G midway between the centre of curvature and the pole.  Indeed, all parallel rays (within the paraxial region) are brought to a focus at the point G, called the principal focus of the mirror.  A concave mirror is therefore a converging mirror.

 

 

            We are now in a position to use the ray-tracing method to determine the location and nature of the image of an extended object produced by a concave mirror.  Because of the paraxial

 

 

 

 

condition, only a very small portion of the mirror is illuminated, and this can be represented, to a good approximation, by a flat surface, the concavity or convexity of the mirror being indicated by means of the shaded triangles at the ends, as shown.  In the interest of clarity it is, of course, necessary to exaggerate considerably the dimensions of the mirror perpendicular to the axis.  Let  be an erect object of height  located a distance U from the pole.  The image I of the point O clearly lies somewhere along the axis of the mirror.  The image of  may be found by tracing out the paths of two rays, one passing through the center of curvature C  (i.e. normal to the mirror) which is reflected back along itself, the other making an angle q with the axis, as shown.  This image   lies at the intersection of the two reflected rays.  We note for future reference that the image II' is inverted.  The magnification m of the image may be defined as the ratio of the height  of the image to the height  of the object:

 

(1.5)

 

From the triangles POO' and PII', we have

 

(1.6)

 

whence

(1.7)

 

            It is instructive to repeat the above calculations in the case where the distance of the object from the pole of the mirror is less than the radius of curvature.  Using the same notation as before, we get

 

 

 

(1.8)

 

(1.9)

 

or

(1.10)

 

            Note the difference between Equation (1.10) which locates the image on the right of the mirror when the object is closer to the pole than the center of curvature, and Equation (1.4), which locates the image on the left of the mirror when the object is further from the pole than the center of curvature.  The formula for obtaining the position of the image would appear to depend on the position of the object.  To see how this undesirable situation might be remedied, let us use the ray-tracing method to find the magnification of an extended object .  The image is found to be upright and to be located on the right of the mirror.  From the triangles POO' and PII', we note that

 

 

 

(1.11)

 

and the magnification is given by

(1.12)

 

This is identical to Equation (1.7) for the magnification produced when the object is further from the pole than the center of curvature, even though in that case the image was inverted.

 

            In order to resolve this confusing situation, a suitable sign convention must be introduced.  In  each of the two cases considered above, both the object (which takes the form of an arrow oriented at right angles to the axis) and the center of curvature are on the left of the mirror, so the difference between the mirror equations (1.4) and (1.10) must be a reflection of the fact that the image is on the left in the former case and on the right in the latter.  Further, Equation (1.12) must be amended so as to indicate whether the image has been inverted, with respect to the object, or not.  These considerations suggest that the sign convention might be based on a Cartesian coordinate system in which the pole of the mirror is chosen as the origin and the axis of the mirror as the x-axis.  Let us denote by u, v, and r the x-coordinates of the the object, image and center of curvature, respectively, and by  the y-coordinates of the tips of the object and image arrows, respectively, the tails being located on the axis.  The (lateral) magnification  may then more usefully be defined as the ratio . In the first example (on p. 77), the center of curvature is on the left of the mirror (  ) and an erect object on the left (  ) produces an inverted image on the left (  ).  Equation (1.4) may then be written as

(1.13)

 

where  is the x-coordinate of the principal focus.  The quantity  is called the focal length of the mirror.  It is clearly a distance, hence inherently positive, while the coordinate f may be either positive or negative.  The lateral magnification becomes

 

(1.14)

 

In the second example, just discussed, an upright object on the left (  ) produces an upright image on the right of  mirror (  ).  Thus Equation (1.10) again becomes

 

(1.15)

 

and the (lateral) magnification is

 

(1.16)

 

(1.17)

 

as before.

 

            We have thus obtained a single mirror equation (1.15) and a single formula (1.17) for the (lateral) magnification which hold in both of the situations we have examined.  These results do, in fact, hold in all cases and will be used exclusively in what follows.  The sign convention used is called the Cartesian convention.  It has the virtue of extreme simplicity, though of course care must be taken to observe the correct signs of the coordinates.  It does not require a priori knowledge of any aspect of the image (e.g. whether it is real or virtual), since the information gleaned from an application of Equations (1.15) and  (1.17) is sufficient to describe the image completely.

 

            If v turns out to be positive (negative), the image is located to the right (left) of the mirror.  If m is positive, the object and the image are on the same side of the mirror axis. Thus an erect object produces an erect image, and an inverted object produces an inverted image.  On the other hand, if m is negative, the object and image are on opposite sides of the axis, i.e. an erect object produces an inverted image, and vice versa.  To simplify matters, however, it may be convenient to assume that the object is always erect.  If m is numerically greater (less) than unity, the image is magnified (diminished).  A real (virtual) image is produced if the reflected rays converge (diverge).  Thus a real (virtual) image produced by a concave mirror must be on the same (opposite) side of the mirror as the object.