1.3 Basic Geometrical Optics (E-II-3)

C. Mirror formulas for image location

In place of the graphical ray-tracing methods described above, we can use formulas to calculate the image location. We shall derive below a “mirror formula” and then use the formula to determine image location. The derivation is typical of those found in geometrical optics, and is instructive in its combined use of algebra, geometry, and trigonometry. (If the derivation is not of interest to you, you may skip to the next section, where the derived formula is used in typical calculations. Be sure, though, that you learn about the sign convention discussed below.)

1. Derivation of the mirror formula. The drawing we need to carry out the derivation is shown in Figure 3-18. The important quantities are the object distance p, the image distance q, and the radius of curvature r. Both p and q are measured relative to the mirror vertex, as shown, and the sign on r will indicate whether the mirror is concave or convex. All other quantities in Figure 3-18 are used in the derivation but will not show up in the final “mirror formula.”

Figure 3-18 Basic drawing for deriving the mirror formula

The mirror shown in Figure 3-18 is convex with center of curvature C on the right. Two rays of light originating at object point O are drawn, one normal to the convex surface at its vertex V and the other an arbitrary ray incident at P. The first ray reflects back along itself; the second reflects at P as if incident on a plane tangent at P, according to the law of reflection. Relative to each other, the two reflected rays diverge as they leave the mirror. The intersection of the two rays (extended backward) determines the image point I corresponding to object point O. The image is virtual and located behind the mirror surface.

Object and image distances measured from the vertex V are shown as p and q, respectively. A perpendicular of height h is drawn from P to the axis at Q. We seek a relationship between p and q that depends on only the radius of curvature r of the mirror. As we shall see, such a relation is possible only to a first-order approximation of the sines and cosines of angles such as a and j made by the object and image rays at various points on the spherical surface. This means that, in place of expansions of sin j and cos j in series as shown here,

we consider the first terms only and write

sin j @ j and cos j @ 1, so that

These relations are accurate to 1% or less if the angle j is 10° or smaller. This approximation leads to first-order (or Gaussian) optics, after Karl Friedrich Gauss, who in 1841 developed the foundations of this subject. Returning now to the problem at hand—that of relating p, q, and r—notice that two angular relationships may be obtained from Figure 3-18, because the exterior angle of a triangle equals the sum of its interior angles. Thus,

q = a + j in DOPC and 2q = a + a¢ in DOPI

which combine to give

a – a¢ = 2j

Using the small-angle approximation, the angles a, a¢, and j above can be replaced by their tangents, yielding

Note that we have neglected the axial distance VQ, small when j is small. Cancellation of h produces the desired relationship,

(3-5)

If the spherical surface is chosen to be concave instead, the center of curvature will be to the left. For certain positions of the object point O, it is then possible to find a real image point, also to the left of the mirror. In these cases, the resulting geometric relationship analogous to Equation 3-5 consists of the same terms, but with different algebraic signs, depending on the sign convention employed. We can choose a sign convention that leads to a single equation, the mirror equation, valid for both types of mirrors. It is Equation 3-6.

(3-6)

2. Sign convention. The sign convention to be used in conjunction with Equation 3-6 and Figure 3-18 is as follows.

Object and image distances p and q are both positive when located to the left of the vertex and both negative when located to the right.
The radius of curvature r is positive when the center of curvature C is to the left of the vertex (concave mirror surface) and negative when C is to the right (convex mirror surface).
Vertical dimensions are positive above the optical axis and negative below.

In the application of these rules, light is assumed to be directed initially, as we mentioned earlier, from left to right. According to this sign convention, positive object and image distances correspond to real objects and images, and negative object and image distances correspond to virtual objects and images. Virtual objects occur only with a sequence of two or more reflecting or refracting elements.

3. Magnification of a mirror image. Figure 3-19 shows a drawing from which the magnification—ratio of image height h_i to object height h_o—can be determined. Since angles q_i, q_r, and a are equal, it follows that triangles VOP and VIP¢ are similar. Thus, the sides of the two triangles are proportional and one can write

This gives at once the magnification m to be

When the sign convention is taken into account, one has, for the general case, a single equation, Equation 3-7, valid for both convex and concave mirrors.

(3-7)

If, after calculation, the value of m is positive, the image is erect. If the value is negative, the image is inverted.

Figure 3-19 Construction for derivation of mirror magnification formula

Let us now use the mirror formulas in Equations 3-6 and 3-7, and the sign convention, to locate an image and determine its size.

Example 7

A meterstick lies along the optical axis of a convex mirror of focal length 40 cm, with its near end 60 cm from the mirror surface. Five-centimeter toy figures stand erect on both the near and far ends of the meterstick. (a) How long is the virtual image of the meterstick? (b) How tall are the toy figures in the image, and are they erect or inverted?

Solution: Use the mirror equation

twice, once for the near end and once for the far end of the meterstick. Use the magnification equation

for each toy figure.

Near end:
Far end:
Near-end toy figure:
Far-end toy figure: