Physics 1402: Lecture 31Today’s Agenda

• Announcements:

– Midterm 2: Monday Nov. 16 …

– Homework 08: due Wednesday (after midterm 2)Homework 08: due Wednesday (after midterm 2)

• Optics – Lenses

– Eye

oi

fh’

h

Rh

h’o-R

R-i

o

i

&

Summary• We have derived, in the paraxial (and thin lens) approximation, the

same equations for mirrors and lenses:

when the following sign conventions are used:

Variable

f > 0f < 0

o > 0o < 0

i > 0i < 0

Mirror

concaveconvex

real (front)virtual (back)

real (front) virtual (back)

Lens

convergingdiverging

real (front)virtual (back)

real (back) virtual (front)

This could be used as a projector. Small slide on big screen

This is a magnifying glass

This could be used in a camera. Big object on small film

Upright

Enlarged

Virtual

Inverted

Enlarged

Real

Inverted

Reduced

Real

Image Object

Inside F

Object

Image

Past 2F

Image

Object

BetweenF & 2F

3 Cases for Converging Lenses

1) Rays parallel to principal axis pass through focal point.

2) Rays through center of lens are not refracted.

3) Rays toward F emerge parallel to principal axis.

F

F

Object

P.A.

Image is virtual, upright and reduced.

Image

Diverging Lens Principal Rays

Multiple Lenses • We determine the effect of a system of lenses by considering the

image of one lens to be the object for the next lens.

For the first lens: o1 = +1.5, f1 = +1

For the second lens: o2 = +1, f2 = -4

f = +1 f = -4

-1 +3+10 +2 +6+5+4

Multiple Lenses • Objects of the second lens can be virtual. Let’s move the second lens

closer to the first lens (in fact, to its focus):

For the first lens: o1 = +1.5, f1 = +1

For the second lens: o2 = -2, f2 = -4

Note the negative object distance for the 2nd lens.

f = +1 f = -4

-1 +3+10 +2 +6+5+4

Multiple Lenses • If the two lenses are thin, they can be touching – i.e.

in the same position. We can treat as one lens.

ftotal = ???

Adding,

For the first lens: o=o1, i1 and f1

For the second lens: o2 = -i1, i2=i, f2

As long as,

The Lens Equation

– Convergent Lens:

i

fh’

o

h

The Lensmaker’s Formula• So far, we have treated lenses in terms of their focal lengths.

• How do you make a lens with focal length f ?

• Start with Snell’s Law. Consider a plano-convex lens:

Snell’s Law at the curved surface:

The bend-angle is just given by:

The bend-angle also defines the focal length f:

The angle can be written in terms of R, the radius of curvature of the lens :

Putting these last equations together,

RNair air

h

light ray

Assuming small angles,

More generally…Lensmaker’s Formula

Two curved surfaces…

Two arbitrary indices of refraction

R > 0 if convex when light hits it

R < 0 if concave when light hits it

The complete generalized case…

Note: for one surface Planar,

~fe

I1

eyepiece

I2

~fo

objectiveL

The

EYE

Retina

To brain

The Eye• What does the eye consist of?

– Sphere (balloon) of water.

- An aperture that controls how much light gets through – the Iris/pupil

- Bulge at the front – the cornea

- A variable focus lens behind the retina – the lens- A screen that is hooked up to your brain – the retina

Cornea

IrisLens

The Eye• The “Normal Eye”

– Far Point distance that relaxed eye can focus onto retina

= – Near Point closest distance that can be focused on to the retina = 25 cmTherefore the normal eye acts as a lens with a focal length which

can vary from 2.5 cm (the eye diameter) to 2.3 cm which allows

objects from 25 cm to be focused on the retina!

2.5cm

25cm

this is called “accommodation”

Diopter: 1/f Eye = 40 diopters, accommodates by about 10%, or 4 diopters

Lecture 31, ACT 1

When your eye adjusts to read versus see far objects, its muscles adjust so that the lens bulges and elongates. To read a book do we want a bulged lens or an elongated lens ?

Cornea LensF < DFN < FF

Near Case

We have f1 = fcornea, f2 = flens

For F to get smaller, so must flens

Smaller f means more curvature (see lensmakers formula)

Cornea LensD = FF

Far Away CaseD

Bonus: Calculate how much the radius of curvature of the lens changes as the eye adjusts from the far to the near point.

Now since,

Getting Old• As you age, the lens loses its ability to change its shape.

• It gets stuck in its relaxed position, the far point.

• Thus the eye is now just an unadjustable lens. Objects at different distances will focus at different places.

• Only objects at infinity will focus on the retina.

2.5cm

25cm

This is called presbyopia, it is not necessarily “farsightedness”.

An intuitive way to view eye correctionsNear-sighted eye is elongated, image forms in front of retina

Add diverging lens, image forms on retina

Far-sighted eye is short, image forms behind retina

Add converging lens, image forms on retinaNote: for old age (presbyopia), this sort of correction can only make one point in focus. If your relaxed eye naturally focuses either at infinity (for driving) or the near point (reading) then you only need one lens. Otherwise bifocals are needed. Could you design multifocals ??

Magnification• Our sense of the size of an object is determined by the size of

image on the retina.

– Consequently, the relevant magnification factor of a lens is just the ratio of the angular size with the lens to the angular size without the lens.

Lnp

h

Object at Near Point

~f

h

Object just inside Focal Pointof simple magnifier

Define Angular Magnification:

Compound Microscope

o1

h

O

I2h2

feye

h1

I1

i1

Objective(fob< 1cm)

fob

L

Eyepiece(feye~5cm)

Magnification:

Refracting Telescope

Star

feye

I2h2

fob

Objective(fob~ 250cm)

Eyepiece(feye~5cm)

i1

I1h1

AngularMagnification: