Lecture 36
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Lecture 36. Spherical and thin lenses. Spherical lens sculpture. Thin lenses. Refraction in a spherical surface. n 1. n 2. h. s > 0. s ’ > 0. R > 0. Paraxial approximation. Magnification (spherical refracting surface). n 1. n 2. y. y ’. s > 0. s ’ > 0. s = 14 cm R = –14 cm

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Lecture 36

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Lecture 36

Lecture 36

Spherical and thin lenses

Spherical lens sculpture

Thin lenses


Refraction in a spherical surface

Refraction in a spherical surface

n1

n2

h

s > 0

s’> 0

R > 0


Lecture 36

Paraxial approximation


Magnification spherical refracting surface

Magnification (spherical refracting surface)

n1

n2

y

y’

s > 0

s’> 0


In class example fish bowl

s = 14 cm

R = –14 cm

s’=–14 cm

Or:

Fish appears in the center but 33% larger!

In-class example: Fish bowl

A 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

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

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

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


Example diverging lens

R1

R2

diverging

Example: Diverging lens

A 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

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


Converging lens

DEMO: Converging lens

Typical shapes:

Converging lens

Object 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


Diverging lens

DEMO: Diverging lens

Typical shapes:

Diverging lens

Object 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


Example converging lens

DEMO: Smiley on bulb.

Example: Converging lens

Where does the image of the arrow form?

F

F


Lecture 36

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


The formulae

Converging or diverging “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 lens1

Example: Diverging lens

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

30 cm

Virtual, smaller, upright image


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