phy1020.DVI

If a ray diagram is drawn with great care and correctly to scale, it may used to measure (with a ruler)
the image distancediand image heighthi. You can tell whether the image is real or virtual, or whether it is
upright or inverted, simply by inspecting the diagram.

45.2 Algebraic Method

An alternative to the ray diagram method is thealgebraic method. This is simpler, faster, and more accurate
than the ray diagram method, but it does not give a good intuitive picture of what’s going on. Also, it isvery
easy to make a sign error with the algebraic method and get the wrong answer.
Solving a mirror optics problem algebraically involves three equations:

1. Focal length equation.If we aren’t given the focal length, we can find it from the radius of curvature
   using the equation given earlier:

fD

R

2

(45.2)

2. Mirror equation.This equation relates the image and object distances to the focal length:
   1
   di

C

1

do

D

1

f

(45.3)

Typically one is given the object distance and focal length, and solves this for the image distancedi.

3. Magnification equation.This equation lets us find the image heighthiand magnificationm:

mD

hi

ho

D

di

do

(45.4)

Typically, you’re given the image object distancedoand object heightho, and have found the image distance
difrom the mirror equation. You can then use this equation to find the image heighthiand magnificationm.
When using these equations, it isveryimportant that you give each quantity the correctsign. The sign
convention for mirrors in shown in Table 45-1.

Table 45-1. Sign conventions for mirrors.

Variable C

do real object virtual object

di real image virtual image

ho always —

hi,m upright image inverted image

f converging mirror diverging mirror

By inspecting the sign ofdi(which you find from the mirror equation), you can determine whether the
image is real or virtual. Also, when you computehi, its sign will tell you whether the image is upright or
inverted. So the equations above give you not only the image distancediand image heighthi, but theirsigns
give you additional information about the image (real/virtual, upright/inverted).

45.3 Segmented Mirrors

For astronomical telescopes, the bigger the mirror, the more light is collected and the better the resolution—
so generally bigger is better. But there is a limit to how large one may big a mirror in a reflecting telescope:
at some point very large mirrors become too costly impractical to manufacture.