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36.3 Images Formed by Refraction

Images Formed by Refraction

- Consider two transparent media having indices of

refraction n1 and n2 - The boundary between the two media is a spherical

surface of radius R - Rays originate from the object at point O in the

medium with n n1

Images by Refraction, 2

- We will consider the paraxial rays leaving O
- All such rays are refracted at the spherical

surface and focus at the image point, I - The relationship between object and image

distances can be given by - (36.8)

Images by Refraction, 3

- The side of the surface in which the light rays

originate is defined as the front side - The other side is called the back side
- Real images are formed by refraction in the back

of the surface - Because of this, the refraction sign conventions

for q and R are opposite the reflection sign

conventions

Sign Conventions for Refracting Surfaces

Flat Refracting Surfaces

- If a refracting surface is flat, then R ??,

therefore - q ?(n2 / n1)p (36.9)
- The image formed by a flat refracting surface is

on the same side of the surface as the object - For n1 n2 a virtual image is formed between the

object and the surface - For n1 of the object

Active Figure 36.20

(SLIDESHOW MODE ONLY)

Example 36.5 Gaze Into the Crystal Ball

- A set of coins is embedded in a spherical plastic

paper weight of radius 3.0 cm, with n 1.50. One

coin is located 2.0 cm from the edge of the

sphere. Find the position of the image of the

coin. - Here n1 n2 so a virtual image is formed inside

the paperweight. - R is negative

36.4 Thin Lenses

- Lenses are commonly used to form images by

refraction - Lenses are used in optical instruments
- Cameras, telescopes, microscopes
- Images from lenses
- Light passing through a lens experiences

refraction at two surfaces - The image formed by one refracting surface serves

as the object for the second surface

Image Formed by a Lens

- The lens has an index of refraction n and two

spherical surfaces with radii of R1 and R2 - R1 is the radius of curvature of the lens surface

that the light of the object reaches first - R2 is the radius of curvature of the other

surface - The object is placed at point O at a distance of

p1 in front of the first surface

Image From Surface 1

- There is an image formed by surface 1
- Since the lens is surrounded by the air, n1 1

and n2 n ? - Equation (36.8) becomes
- (36.10)
- If the image due to surface 1 is virtual, q1 is

negative, and it is positive if the image is real

Image From Surface 2

- For surface 2, n1 n and n2 1
- The light rays approaching surface 2 are in the

lens and are refracted into air - Use p2 for the object distance for surface 2 and

q2 for the image distance, so equation (36.8)

becomes - (36.11)
- From the virtual image at surface 1 p2 q1

t - q1 is negative and t is the thickness of the lens
- From the real image at surface 1 p2 q1

t - q1 is positive

Image Formed by a Thin Lens

- A thin lens is one whose thickness t is small

compared to the radii of curvature - For a thin lens, the thickness, t, of the lens

can be neglected - In this case, p2 q1 for either type of image
- Hence equation (36.11) becomes
- (36.12)

Image Formed by a Thin Lens, 2

- Adding equations (36.10) and (36.12) we obtain
- (36.13)
- Then the subscripts on p1 and q2 can be omitted

as in the figure and rewrite equation (36.13) as - (36.14)

Lens Makers Equation

- The focal length f of a thin lens is the image

distance q that corresponds to an infinite object

distance - This is the same as for a mirror
- Making p ? ? and q ? f, so equation (36.14) will

become the lens makers equation - (36.15)
- Given n and f the lens maker can determine the

values of R1 and R2 - Given R1, R2 and n lens maker can calculate the

value of f

Thin Lens Equation

- Using equation (36.15) we can write equation

(36.14) in a for identical to equation (36.6) for

mirrors. - The relationship among the focal length, the

object distance and the image distance is the

same as for a mirror - (36.16)

Notes on Focal Length and Focal Point of a Thin

Lens

- Because light can travel in either direction

through a lens, each lens has two focal points - One focal point is for light passing in one

direction through the lens - The other is for light traveling in the opposite

direction - However, there is only one focal length
- Each focal point is located at the same distance

from the lens

Focal Length of a Converging Lens

- The parallel rays pass through the lens and

converge at the focal point - The parallel rays can come from the left or right

of the lens

Focal Length of a Diverging Lens

- The parallel rays diverge after passing through

the diverging lens - The focal point is the point where the rays

appear to have originated

Determining Signs for Thin Lenses

- The front side of the thin lens is the side of

the incident light - The back side of the lens is where the light is

refracted into - This is also valid for a refracting surface

Sign Conventions for Thin Lenses

Magnification of Images Through a Thin Lens

- The lateral magnification of the image is
- When M is positive, the image is upright and on

the same side of the lens as the object - When M is negative, the image is inverted and on

the side of the lens opposite the object

Thin Lens Shapes

- These are examples of converging lenses
- They have positive focal lengths
- They are thickest in the middle

More Thin Lens Shapes

- These are examples of diverging lenses
- They have negative focal lengths
- They are thickest at the edges

Ray Diagrams for Thin Lenses Converging

- Ray diagrams are convenient for locating the

images formed by thin lenses or systems of lenses - For a converging lens, the following three rays

are drawn - Ray 1 is drawn parallel to the principal axis and

then passes through the focal point on the back

side of the lens - Ray 2 is drawn through the center of the lens and

continues in a straight line - Ray 3 is drawn through the focal point on the

front of the lens (or as if coming from the focal

point if p

parallel to the principal axis

Ray Diagram for Converging Lens, p f

- The image is real
- The image is inverted
- The image is on the back side of the lens

Ray Diagram for Converging Lens, p

Ray Diagrams for Thin Lenses Diverging

- For a diverging lens, the following three rays

are drawn - Ray 1 is drawn parallel to the principal axis and

emerges directed away from the focal point on the

front side of the lens - Ray 2 is drawn through the center of the lens and

continues in a straight line - Ray 3 is drawn in the direction toward the focal

point on the back side of the lens and emerges

from the lens parallel to the principal axis

Ray Diagram for Diverging Lens

- The image is virtual
- The image is upright
- The image is smaller
- The image is on the front side of the lens

Active Figure 36.28

(SLIDESHOW MODE ONLY)

Image Summary

- For a converging lens, when the object distance

is greater than the focal length - (p )
- The image is real and inverted
- For a converging lens, when the object is between

the focal point and the lens, (p - The image is virtual and upright
- For a diverging lens, the image is always virtual

and upright - This is regardless of where the object is placed

Fresnal Lens

- Refraction occurs only at the surfaces of the

lens - A Fresnal lens is designed to take advantage of

this fact - It produces a powerful lens without great

thickness

Fresnal Lens, cont.

- Only the surface curvature is important in the

refracting qualities of the lens - The material in the middle of the Fresnal lens is

removed - Because the edges of the curved segments cause

some distortion, Fresnal lenses are usually used

only in situations where image quality is less

important than reduction of weight

Two Thin Lenses (Combination)

- If two thin lenses are used to form an image
- The image formed by the first lens is located as

if the second lens were not present - Then a ray diagram is drawn for the second lens
- The image of the first lens is treated as the

object of the second lens - The image formed by the second lens is the final

image of the system

Two Thin Lenses, 2

- If the image formed by the first lens lies on the

back side of the second lens, then the image is

treated as a virtual object for the second lens - p will be negative
- The same procedure can be extended to a system of

three or more lenses - The overall magnification is the product of the

magnification of the separate lenses

Two Thin Lenses, 3

- Consider a case of two lenses in contact with

each other - The lenses have focal lengths of 1 and 2
- If p1 p is the object distance for the

combination, equation (36.16) becomes - Since the lenses are in contact, p2 q1

Two Thin Lenses, final

- For the second lens q2 q ,
- Adding the two previous equations for the

combination of the two lenses - (36.17)
- Two thin lenses in contact with each other are

equivalent to a single thin lens having a focal

length given by the above equation

Example 36.6 Where is the Final Image?

Example 36.6 Where is the Final Image? , 2

- The location of the image formed by lens 1
- The image of lens 1 becomes object for lens 2,

with p2 20cm 15cm 5cm ? - Then, the total magnification will be

Material for the Midterm

- Examples to Read!!!
- Example 36.11 (page 1149)
- Examples in Class!!!
- Example 36.9 (page 1146)
- Example 36.10 (page 1147)
- Homework to be solved in Class!!!
- Question 9
- Problems 21, 28