Physics 1502: Lecture 30Today’s Agenda

• Announcements:

– Midterm 2: Monday Nov. 16 …

– Homework 08: due FridayHomework 08: due Friday

• Optics – Mirrors

– Lenses

– Eye

oi

fh’

h

Rh

h’o-R

R-i

o

i

&

The Mirror Equation• We will now transform the geometric drawings into algebraic

equations:

R

object

h

image

oi

from triangles,

eliminating ,

Now we employ the small angle approximations:

Plugging these back into the above equation relating the angles, we get:

Defining the focal length f = R/2,

This eqn is known as the mirror eqn. Note that there is no mention of in this equation. Therefore, this eqn works for all , ie we have an image!

Magnification• We have derived the mirror eqn which determines the image

distance in terms of the object distance and the focal length:

• What about the size of the image? • How is h’ related to h??• From similar triangles:

Now, we can introduce a sign convention. We can indicate that this image is inverted if we define its magnification M as the negative number given by:

Rh

o

h’i

More Sign Conventions• Consider an object distance s which is less than the focal

length:

h’

i

Ray Trace:• Ray through the center of the sphere (light blue) is reflected straight back. R

h o

f

• We call this a virtual image, meaning that no light from the object passes through the image point.• Proof left to student: This situation is described by the same mirror equations as long as we take the convention that images behind the mirror have negative image distances s’. ie:

In this case, i < 0, which leads to M > 0, indicating that the image is virtual (i<0) and not inverted (M>0).

• Ray parallel to axis (red) passes through focal point f.

• These rays diverge! ie these rays look they are coming from a point behind the mirror.

Concave-Planar-Convex• What happens as we change the curvature of the mirror?

– Plane mirror:

» R = IMAGE:

virtualupright (non-inverted)

h’

h

o if

IMAGE: virtual

upright (non-inverted)

– Convex mirror:

» R < 0

Lecture 30, ACT 1• In order for a real object to create a real, inverted enlarged image, a) we must use a concave mirror.

b) we must use a convex mirror.

c) neither a concave nor a convex mirror can produce this image.

Mirror – Lens Definitions• Some important terminology we introduced last class,

– o = distance from object to mirror (or lens)

– i = distance from mirror to image

o positive, i positive if on same side of mirror as o.

– R = radius of curvature of spherical mirror

– f = focal length, = R/2 for spherical mirrors.

– Concave, Convex, and Spherical mirrors.

– M = magnification, (size of image) / (size of object)

negative means inverted image

R

object

h

image

oi

Lenses• A lens is a piece of transparent material shaped such that

parallel light rays are refracted towards a point, a focus:– Convergent Lens

»light moving from air into glass

will move toward the normal

»light moving from glass back into

air will move away from the normal

»real focus

– Divergent Lens

»light moving from air into glass

will move toward the normal

»light moving from glass back into

air will move away from the normal

»virtual focus

1) Rays parallel to principal axis pass through focal point.2) Rays through center of lens are not refracted.

3) Rays through F emerge parallel to principal axis.

Assumptions: • monochromatic light incident on a thin lens.• rays are all “near” the principal axis.

F

F

Object

P.A.

Image is: real, inverted and enlarged (in this case).

Image

Converging Lens Principal Rays

ACT 2: Converging Lens

Demo

F

F

Object

P.A.

Which way should you move object so image is real and diminished?

(1) Closer to lens

(2) Further from lens

(3) Converging lens can’t create real diminished image.

The Lens Equation• We now derive the lens equation which determines the image distance in terms of the

object distance and the focal length.– Convergent Lens:

i

fh’

o

h

Ray Trace:• Ray through the center of the lens (light blue) passes through undeflected.

two sets of similar triangles:

eliminating h’/h:same as mirror eqn

if we definei > 0 f > 0

magnification: also same as mirror eqn!! M < 0 for inverted image.

• Ray parallel to axis (white) passes through focal point f.

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

Which way should you move object so image is real?

1) Closer to lens

2) Further from lens

3) Diverging lens can’t create real image.

ACT 3: Diverging Lenses

DemoF

F

Object

P.A.

Lecture 30, ACT 4• A lens is used to image an object on a

screen. The right half of the lens is covered.

– What is the nature of the image on the screen?

(a) left half of image disappears(b) right half of image disappears(c) entire image reduced in intensity

object

lens

screen

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,

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:

~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