W9_parallel_resonance.eps

404 Week 12: Lenses and Mirrors

or

n 1 α+n 2 γ= (n 2 −n 1 )β (991)

We substitute in the small angle formulas and cancellto get:

n 1

s

+

n 2

s′

= (n 2 −n 1 )

1

r

(992)

In most cases of interest to us, the lenses in question will be made out of glass, plastic,

or collagen (in the case of the eye) surrounded or faced by air, in which case this will

simplify to:

1

s+

n

s′= (n−1)

1

r (993)

If there are two lensing surfaces separated by a very small distance, we have a so-called

thin lens. The relevant geometry of a thin lens surrounded by air is shown in (166).

The first surface struck by light from an object (presumed comingin from the left) has

r 2

r 1

Figure 166: Geometry of a thin lens surrounded by air.

positive radius of curvaturer 1. The second surface has a negative radius of curvature

r 2. The index of refraction of the lens isn.

Suppose we have an object on the left hand side of this lens at distances. From the

formula above, we have:

1

s

+

n

s′

= (n−1)

1

r 1

(994)

The image of the first lensing surface is avirtual objectfor the second lensing surface.

Because it is virtual (located to therightof the second surface, on the side light is going

to) and because we are going from the material with index of refractionninto air, the

formula for the second lensing surface is:

−n

s′ +

1

s′′= (1−n)

1

r 2 (995)

If we add these two formulae, thes′term cancels and, we get:

1

s+

1

s′′= (n−1)

(

1

r 1 −

1

r 2

)

=

1

f (996)

This is thethin lens formulawheres′′is the final location of the image of the entire lens.

Note that this isidenticalto the formula for the mirror. The focal length is given by the

lensmaker’s formula:

1

f= (n−1)

(

1

r 1 −

1

r 2

)

(997)

With the thin lens formula in hand, we can easily adaptexactlythe same rules for drawing

ray diagrams for locating images. Let’s draw a simple ray diagram for aconverging and

a diverging lens that are similar to the ray diagrams above for mirrors. We do the usual

algebra and arithmetic:s^1 ′= 101 − 301 = 302 sos′= 15.0 cm,m=−^12. The final image is

inverted, real, and smaller than the object.