phy1020.DVI

OBJECT

IMAGE

2

1

33

F

F

Figure 47.2: Ray diagram for a converging (bi-convex) lens.

47.2 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 lens optics problem algebraically involves three equations:

1. Lens maker’s equation.If we aren’t given the focal length, we can find it from the radii of curvature of
   the two lens surfaces and the index of refraction of the lens material, using thelens maker’s equation:

1

f

D



nlens

nair

 1



1

R 1

C

1

R 2



(47.1)

whereR 1 andR 2 are the radii of curvature of the two surfaces,nlensis the index of refraction of the lens
material, andnairD 1 is the index of refraction of the air.

2. Thin lens equation.This equation relates the image and object distances to the focal length, and is
   identical in form to the mirror equation:

1

di

C

1

do

D

1

f

(47.2)

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

3. Magnification equation.This equation (which is the same as it is for mirrors) lets us find the image
   heighthiand magnificationm: Magnification equation:

mD

hi

ho

D

di

do

(47.3)

Typically, you’re given the image object distancedoand object heightho, and have found the image distance
difrom the thin lens equation. You can then use this equation to find the image heighthiand magnification
m.
When using these equations, it isveryimportant that you give each quantity the correctsign. The sign
convention for lenses in shown in Table 47-1, and is essentially the same as the sign convention for mirrors.
Sign convention: