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s = 14 cm

R = –14 cm

s’=–14 cm

Or:

Fish appears in the center but 33% larger!

In-class example: Fish bowlA spherical fish bowl has a 28.0 cm diameter and a fish at its center. What is the apparent position and magnification of the fish to an observer outside of the bowl?

A. s’ = –7 cm, m = 2.0

B. s’ = +7 cm, m = –2.0

C. s’ = –14 cm, m = 1.3

D. s’ = +14 cm, m = –1.3

E. s’ = –14 cm, m = 1.0

Lenses: through two spherical surfaces

nout

nout

Do the calculation twice, once for each surface.

nin

R1

R2

Overall effect

(combination of nin, nout, R1and R2)

Thin lens model

- In this model
- thickness of material <<distances to objects, images
- angles tend to be small
- consider doubly effective refraction at center of lens

Everything can then be described in terms of two focal points:

f: focal distance

|f |

|f |

F2

F1

Lensmaker’s equation

If we analyze a thin lens in terms of the two spherical surfaces it is made of (in the paraxial approximation), we obtain:

Proof: see book

Remember:

R > 0 if center of curvature is on the same side of surface as outgoing rays

R1

R2

diverging

Example: Diverging lensA double-concave lens is made of glass with n = 1.5 and radii 20 cm and 25 cm. Find the focal length.

R1 = –20 cm

R2 = +25 cm

Note: If we reverse the lens (R1 = –25 cm and R2 = +20 cm), the result is the same.

Principal rays

Of the many possible rays you could draw, 3 are very useful

- parallel to axis refracts through focus

2) through center (no net refraction due to symmetry)

3) through focus refracts parallel to axis

Typical shapes:

Converging lensObject at infinity forms a real image at F2

(observer sees rays appearing to originate from point F2)

Focal length f > 0

Small |f| more converging

Typical shapes:

Diverging lensObject at infinity forms a virtual image at F2

(observer sees rays as if coming from point F2)

Focal length f < 0

Small |f| more diverging

Eye intercepts reflected rays that come from a point of the image on screen

If we place a screen

at image location

Diffuse reflection

from screen

F

F

Converging or diverging image on screen“power” (in diopters = m-1 ):

The formulae…y

F

F

y’

f

f

s

s’

Valid for both convergent and diverging thin lenses (and mirrors) in the paraxial limit

Example: Diverging lens image on screen

Where is the image if this is a lens of -10 diopters?

30 cm

Virtual, smaller, upright image

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