Notation for Mirrors and Lenses

• The

object distance

is the distance from the object to the mirror or lens – Denoted by

p

• The

image distance

to the mirror or lens – Denoted by

q

is the distance from the image • The

lateral magnification

of the mirror or lens is the ratio of the image height to the object height – Denoted by

M

Images

• Images are always located by extending diverging rays back to a point at which they intersect • Images are located either at a point from which the rays of light

actually

diverge or at a point from which they

appear

to diverge

Types of Images

• A

real image

is formed when light rays pass through and diverge from the image point – Real images can be displayed on screens • A

virtual image

is formed when light rays do not pass through the image point but only appear to diverge from that point – Virtual images cannot be displayed on screens

Images Formed by Flat Mirrors

• Simplest possible mirror • Light rays leave the source and are reflected from the mirror • Point

I

image

is called the of the object at point

O

• The image is virtual

Images Formed by Flat Mirrors

• One ray starts at point

P

, travels to

Q

and reflects back on itself • Another ray follows the path

PR

and reflects according to the law of reflection • The triangles

PQR P’QR

are congruent and

Lateral Magnification

• Lateral magnification,

M

, is defined as

M

 Image height Object height 

h ' h

– This is the general magnification for any type of mirror – It is also valid for images formed by lenses – Magnification does not always mean bigger, the size can either increase or decrease • •

M M

less than 1 -> image size decreased greater than 1 -> image size increased

Reversals in a Flat Mirror

• A flat mirror produces an image that has an

apparent

left-right reversal – For example, if you raise your right hand the image you see raises its left hand

Spherical Mirrors

• A

spherical mirror

sphere has the shape of a section of a • The mirror focuses incoming parallel rays to a point • A

concave spherical mirror

has the silvered surface of the mirror on the inner, or concave, side of the curve • A

convex spherical mirror

has the silvered surface of the mirror on the outer, or convex, side of the curve

Convex vs Concave

Spherical aberation

No clear focusing of the incident rays

Spherical mirrors

R R f  R 2

Spherical Mirrors

2

i r

 

i

 S   C  

i

P  

r

  F

q

d

R p

2

R

2

d R

 

p d q p

1

q

1

f

  sin   tan  

d p

  sin   tan  

d R

  sin   tan  

d q

Ray Tracing

 C  F • Parallel ray hit the mirror and heads to the focus • Ray through focus hits the mirror and comes off parallel • Ray through center of curvature returns to center of curvature

Ray Tracing

F   C • Parallel ray hits the mirror and reflects as if it started at the focus • Ray heading to the focus hits the mirror and comes off parallel • Ray heading to the center of curvature returns along the same path

Magnification

h

 

h q p

Or: 

h

 

h q p M



h h o i



h



h

Gain is a better description since M might be < 1

M



h



h

 

q p

Sign Conventions

Example: Find the image

C F f = 20 cm p = 32 cm h = 5 cm

Images Formed by Refraction n sin 1   1 n sin 2 2 n 1 1 n 2 2    1      2    sin   tan  

d p

  sin   tan  

d R

  sin   tan  

d q

n 1 n 2  n 2  n 1   n 1 p  n 2 q   n 2  n 1  R

Sign Conventions for Refracting Surfaces

Flat Refracting Surfaces

• If a refracting surface is flat, then

R

is infinite • Then

q

= -(

n

2 /

n

1 )

p

– The image formed by a flat refracting surface is on the same side of the surface as the object • A virtual image is formed

Locating the Image Formed by a Lens • The lens has an index of refraction

n

and two spherical surfaces with radii of

R

1 and

R

2 –

R

1 is the radius of curvature of the lens surface that the light of the object reaches first –

R

2 is the radius of curvature of the other surface • The object is placed at point

O

distance of surface

p

1 at a in front of the first

n

1 

p n

2

q



n

2 

R n

1  1

p

1 

n q

1 

n

 1

R

1

n

1 

p n

2

q



n

2 

n

1

R



n p

2  1

q

2  1 

n R

2 p 2    1 t Small!!

1

p

1  1

q

2  

n

 1   1

R

1  1

R

2  

Lens Makers’ Equation

• The focal length of a thin lens is the image distance that corresponds to an infinite object distance – This is the same as for a mirror • The

lens makers’ equation

is 1

p q

(

n

 1)   1

R

1  1

R

2    1

ƒ

Thin lenses - converging

Thin lenses - diverging

Ray Tracing Rules

• A ray parallel to the axis exits through the focal point • A ray through the focal point leaves parallel to the axis • A ray through the center is undeviated

Ray Diagram for Converging Lens,

p

<

f

• The image is virtual • The image is upright • The image is larger than the object • The image is on the front side of the lens

Ray tracing for a diverging lens

Sign Conventions

Example

A 2 cm tall object is placed 30 cm in front of a diverging lens with a focal length of –20 cm. Find the location of the image, its classification, the size of the image, and the magnification. Construct a ray diagram.

Lateral (Transverse) magnification

h m  h  h

F

1

F

2 d o d i But: h  h  q p M  h  h   q p h’ Or:  h  h  q p

Multiple Thin Lenses

• Find the image distance of the first lens.

• Use the image of the first lens as the object of the second lens.

• Find the new object distance by correcting with the separation of the lenses.

• Find the image of the second lens.

• Transverse Magnification is the product of the magnifications of the individual lenses

f 1  20.0cm

Example

f 2  25.0cm

p  60cm Find the location and magnification of the final image formed

9.

A spherical convex mirror has a radius of curvature with a magnitude of 40.0 cm. Determine the position of the virtual image and the magnification for object distances of (a) 30.0 cm and (b) 60.0 cm. (c) Are the images upright or inverted?

11.

A concave mirror has a radius of curvature of 60.0 cm. Calculate the image position and magnification of an object placed in front of the mirror at distances of (a) 90.0 cm and (b) 20.0 cm. (c) Draw ray diagrams to obtain the image characteristics in each case.

12.

A concave mirror has a focal length of 40.0 cm. Determine the object position for which the resulting image is upright and four times the size of the object.

29.

The left face of a biconvex lens has a radius of curvature of magnitude 12.0 cm, and the right face has a radius of curvature of magnitude 18.0 cm. The index of refraction of the glass is 1.44. (a) Calculate the focal length of the lens. (b)

What If?

Calculate the focal length the lens has after is turned around to interchange the radii of curvature of the two faces.

31.

A thin lens has a focal length of 25.0 cm. Locate and describe the image when the object is placed (a) 26.0 cm and (b) 24.0 cm in front of the lens.

34.

A person looks at a gem with a jeweler’s loupe—a converging lens that has a focal length of 12.5 cm. The loupe forms a virtual image 30.0 cm from the lens. (a) Determine the magnification. Is the image upright or inverted? (b) Construct a ray diagram for this arrangement.