1

CHAPTER 23 Light: Geometric Optics

http://www.physicsclassroom.com/Class/refln/reflntoc.html

Units

• The Ray Model of Light • Reflection; Image Formed by a Plane Mirror • Formation of Images by Spherical Mirrors • Index of Refraction • Refraction: Snell’s Law • Total Internal Reflection; Fiber Optics • Thin Lenses; Ray Tracing • The Thin Lens Equation; Magnification • Combinations of Lenses • Lensmaker’s Equation

The Ray Model of Light

Light very often travels in straight lines. We represent light using rays, which are straight lines emanating from an object. This is an idealization, but is very useful for geometric optics.

The Reflection of Light

If a stone is dropped into a pond, circular waves emanate from the point where it landed. Rays, perpendicular to the wave fronts, give the direction in which the waves propagate.

As one moves farther from a point wave source, the wave fronts become more nearly flat.

2

Reflection; Image Formation by a Plane Mirror

Law of reflection: the angle of reflection (that the ray makes with the normal to a surface) equals the angle of incidence.

The Reflection of Light

Reflection from a smooth surface is called specular reflection; if the surface is rough, it is diffuse reflection.

Reflection; Image Formation by a Plane Mirror

With diffuse reflection, your eye sees reflected light at all angles. With specular reflection (from a mirror), your eye must be in the correct position.

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 is the distance from the image to the mirror or lens – Images are formed at the point where rays actually intersect or appear to originate – Denoted by q

3

• The lateral magnification of the mirror or lens is the ratio of the image height to the object height

– Denoted by M

Types of Images for Mirrors and Lenses

• A real image is one in which light actually passes through the image point – Real images can be displayed on screens

• A virtual image is one in which the light does not pass through the image point – The light appears to diverge from that point – Virtual images cannot be displayed on screens

More about Images

• To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror

Forming Images with a Plane Mirror

Light reflected from the flower and vase hits the mirror. Obeying the law of reflection, it enters the eye. The eye interprets the ray as having had a straight-line path, and sees the image behind the mirror.

Properties of Mirror Images Produced by Plane Mirrors: • A mirror image is upright, but appears reversed right to left. • A mirror image appears to be the same distance behind the mirror that the object is in front of the mirror. • A mirror image is the same size as the object.

Reflection; Image Formation by a Plane Mirror

What you see when you look into a plane (flat) mirror is an image, which appears to be behind the mirror. This is called a virtual image, as the light does not go through it. The distance of the image from the mirror is equal to the distance of the object from the mirror.

4

Flat Mirror

• Simplest possible mirror • Properties of the image can be determined by

geometry • One ray starts at P, follows path PQ and reflects

back on itself • A second ray follows path PR and reflects

according to the Law of Reflection

Properties of the Image Formed by a Flat Mirror

• The image is as far behind the mirror as the object is in front – q = p

• The image is unmagnified – The image height is the same as the object height

• h’ = h and M = 1 • The image is virtual • The image is upright

– It has the same orientation as the object • There is an apparent left-right reversal in the image

Application – Day and Night Settings on Auto Mirrors

• With the daytime setting, the bright beam of reflected light is directed into the driver’s eyes

• With the nighttime setting, the dim beam of reflected light is directed into the driver’s eyes, while the bright beam goes elsewhere

Spherical Mirrors

A spherical mirror has the shape of a section of a sphere. If the outside is mirrored, it is convex; if the inside is mirrored, it is concave.

5

Formation of Images by Spherical Mirrors

Spherical mirrors are shaped like sections of a sphere, and may be reflective on either the inside (concave) or outside (convex).

Spherical Mirrors

Spherical mirrors have a central axis (a radius of the sphere) and a center of curvature (the center of the sphere).

Formation of Images by Spherical Mirrors

Rays coming from a faraway object are effectively parallel.

Parallel rays striking a spherical mirror do not all converge at exactly the same place if the curvature of the mirror is large; this is called spherical aberration.

Concave Mirror, Notation

• The mirror has a radius of curvature of R • Its center of curvature is the point C • Point V is the center of the spherical segment • A line drawn from C to V is called the principle axis of the

mirror

6

Spherical Aberration

• Rays are generally assumed to make small angles with the mirror

• When the rays make large angles, they may converge to points other than the image point

• This results in a blurred image • This effect is called spherical aberration

When the Hubble Space Telescope was first launched, its optics were marred by spherical aberration. This was fixed with corrective optics.

Formation of Images by Spherical Mirrors

If the curvature is small, the focus is much more precise; the focal point is where the rays converge.

Ray Tracing and the Mirror Equation

We use three principal rays in finding the image produced by a concave mirror.

• The parallel ray (P ray) reflects through the focal point.

• The focal ray (F ray) reflects parallel to the axis.

• The center-of-curvature ray (C ray) reflects back along its incoming path.

These three rays are illustrated here.

7

Example 1: Image in a concave mirror. A 1.50-cm-high diamond ring is placed 20.0 cm from a concave mirror with radius of curvature 30.0 cm. Determine a) the position of the image

1d is positive, the image is 60.0 cm in front of the mirror.

b) The magnification is

The image height is 3.0 times the object height, and is

The minus sign states the image is inverted.

Ray Tracing and the Mirror Equation

This image shows how these three rays are used to find the image formed by a convex mirror. The image is located where the projections of the three rays cross. The size of the image can also be determined.

The process is similar for a concave mirror, although there are different results depending on where the object is placed.

11 1 1 1 10.0167

15.0 20.0i o

cmd f d cm cm

11/ 0.0167 60.0id cm cm

60.03.00

20.0

i

o

d cmm

d cm

( 3.00)(1.5 ) 4.5i oh mh cm cm

8

We derive the mirror equation using the ray diagrams:

Using the similar triangles and the fact that f = ½ R, we get the mirror equation:

Here, do is the distance from the mirror to the object, di is the distance from the mirror to the

image, and f is the focal length.

Formation of Images by Spherical Mirrors

Geometrically, we can derive an equation that relates the object distance, image distance, and focal length of the mirror:

We can also find the magnification (ratio of image height to object height).

The negative sign indicates that the image is inverted. This object is between the center of curvature and the focal point, and its image is larger, inverted, and real.

Example 2: Object closer to concave mirror. A 1.00-cm-high object is placed 10.0cm from a concave mirror whose radius of curvature is 30.0cm. a) Draw a ray diagram to locate (approximately) the position of the image.

9

b) Determine the position of the image and the magnification analytically. Since / 2 15.0f r cm the object is between the mirror and the focal point.

Therefore, 30.0id cm The minus sign means the image is behind the mirror.

The + sign means that the image is upright.

Example 3: Convex rearview mirror. An external rearview car mirror is convex with a radius of curvature of 16.0 m. Determine the location of the image and its magnification for an object 10.0 m from the mirror. The center of curvature of a convex mirror is behind the mirror, as is its focal point, so set r = -16m. So the focal length is / 2 8.0f r m . The object is in front of the mirror, / 2 8.0f r m .

Thus

Sign conventions: The image distance is negative, -44m, so the image is behind the mirror. The magnification is m = +0.44, so the image is upright and less than half as tall as the object.

Ray Tracing and the Mirror Equation

1 1 1 1 1 2 3 1

15.0 10 30.0 30.0i od f d cm cm cm cm

/ ( 30.0 ) /(10.0 ) 3.00i om d d cm cm

1 1 1 1 1 10.0 8.0 18

8.0 10.0 80.0 80.0i od f d m m m m

80.0 /18 4.4id m m

( 4.4 )0.44

(10.0 )

i

o

d mm

d m

10

Here are the sign conventions for concave and convex mirrors:

Sign Conventions for Refracting Surfaces

Ray Diagrams

• A ray diagram can be used to determine the position and size of an image • They are graphical constructions which tell the overall nature of the image • They can also be used to check the parameters calculated from the mirror and

magnification equations

Drawing a Ray Diagram

• To make the ray diagram, you need to know – The position of the object – The position of the center of curvature

• Three rays are drawn – They all start from the same position on the object

• The intersection of any two of the rays at a point locates the image – The third ray serves as a check of the construction

11

The Rays in a Ray Diagram

• Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F • Ray 2 is drawn through the focal point and is reflected parallel to the principle axis • Ray 3 is drawn through the center of curvature and is reflected back on itself

• The rays actually go in all directions from the object • The three rays were chosen for their ease of construction • The image point obtained by the ray diagram must agree with the value of q calculated

from the mirror equation

Ray Diagram for Concave Mirror, p > R

• The object is outside the center of curvature of the mirror • The image is real • The image is inverted • The image is smaller than the object

Ray Diagram for a Concave Mirror, p < f

• The object is between the mirror and the focal point • The image is virtual • The image is upright • The image is larger than the object

12

Ray Diagram for a Convex Mirror

• The object is in front of a convex mirror • The image is virtual • The image is upright • The image is smaller than the object

Formation of Images by Spherical Mirrors

Notes on Images

• With a concave mirror, the image may be either real or virtual – When the object is outside the focal point, the image is real – When the object is at the focal point, the image is infinitely far away

– When the object is between the mirror and the focal point, the image is virtual

13

• With a convex mirror, the image is always virtual and upright – As the object distance increases, the virtual image gets smaller

Index of Refraction

In general, light slows somewhat when traveling through a medium. The index of refraction of the medium is the ratio of the speed of light in vacuum to the speed of light in the medium:

Example 4: Light’s speed in diamond. Calculate the speed of light in diamond.

Willebrord Snell was an early seventeenth century Dutch mathematician who is best known for determining that transparent materials have different indices of refraction depending upon their composition.

Refraction: Snell’s Law

Light changes direction when crossing a boundary from one medium to another. This is called refraction, and the angle the outgoing ray makes with the normal is called the angle of refraction.

10.413

2.42

cv c

n

883.00 10 /

1.24 10 /2.42

x m sv x m s

14

We can now write the angle of refraction in terms of the index of refraction:

If light enters a medium of lower index of refraction, it will be bent away from the normal. If the angle of incidence is large enough, the angle of refraction is 90°; at larger incident angles the light will be totally reflected.

Basic properties of refraction:

Example 5: Refraction through flat glass. Light traveling in air strikes a flat piece of uniformly thick glass at an

incident of 60o , as shown. If the index of refraction of the glass is 1.50

a) What is the angle of refraction in the glass?

1.00sin sin 60 0.577 35.2

1.50

o o

A

15

b) What is the angle at which the ray emerges from the glass?

Since the faces of the glass are parallel, the incident angle at the second surface is just A , so

sin 0.577A . At this second interface, 1 21.50 and 1.00n n . Thus the ray re-enters the air

at an angle2( )B

Total Internal Reflection; Fiber Optics

If the angle of incidence is larger than this, no transmission occurs. This is called total internal reflection.

The Refraction of Light

Refraction can make objects immersed in water appear broken, and can create mirages.

Atmospheric Refraction

• There are many interesting results of refraction in the atmosphere

– Sunsets – Mirages

• Light rays from the sun are bent as they pass into the atmosphere

• It is a gradual bend because the light passes through layers of the atmosphere

– Each layer has a slightly different index of refraction • The Sun is seen to be above the horizon even after it has fallen below it

1.50sin sin 0.866 60

1.00

o

B A

16

Atmospheric Refraction and Mirages

• A mirage can be observed when the air above the ground is warmer than the air at higher elevations

• The rays in path B are directed toward the ground and then bent by refraction

• The observer sees both an upright and an inverted image

The Refraction of Light

This is called total internal reflection, and the incident angle at which the angle of refraction is 90°

is called the critical angle, C. Total internal reflection is used in some binoculars and in optical

fibers.

There is a special angle called Brewster’s angle; light reflected at this angle is totally polarized. Reflected light is completely polarized when the reflected and refracted beams are at right angles to one another. The direction of polarization is parallel to the reflecting surface.

Brewster’s angle can be calculated using the appropriate geometry:

17

Thin Lenses; Ray Tracing

Thin lenses are those whose thickness is small compared to their radius of curvature. They may be either converging (a) or diverging (b).

Ray Tracing for Lenses

Lenses are used to focus light and form images. There are a variety of possible types; we will consider only the symmetric ones, the double concave and the double convex.

If we think of a convex lens as consisting of prisms, we can see how light going through it converges at a focal point (assuming the lens is properly shaped).

18

A concave lens can also be modeled by prisms:

The three principal rays for lenses are similar to those for mirrors: • The P ray—or parallel ray—approaches the lens parallel to its axis. • The F ray is drawn toward (concave) or through (convex) the focal point. • The midpoint ray (M ray) goes through the middle of the lens. Assuming the lens is thin

enough, it will not be deflected. This is the thin-lens approximation.

These diagrams show the principal rays for both types of lenses:

As with mirrors, we use these principal rays to locate the image:

19

The convex lens forms different image types depending on where the object is located with respect to the focal point:

The Thin-Lens Equation

We derive the thin-lens equation in the same way we did the mirror equation, using these diagrams:

This gives us the thin-lens approximation, as well as the magnification:

Sign Conventions for Thin Lenses

20

The Thin-Lens Equation

Example 6: Image formed by converging lens. a) What is the position of the image of a 7.6-cm-high flower placed 1.00m from a +50mm focal length camera lens?

b) What is the size, of the image of the 7.6-cm-high flower placed 1.00m from a +50.0mm focal length camera lens?

Or 52.6mm behind the lens.

Thin Lenses; Ray Tracing

The power of a lens is the inverse of its focal length. Lens power is measured in diopters, D.

1 D = 1 m-1

1 1 1 1 1 20.0 1.0 19.0

5.00 100 100 100o od f d cm cm cm cm

1005.26

19.0i

cmd cm

5.260.0526

100

i

o

d cmm

d cm

( 0.0526)(7.6 ) 0.40i oh mh cm cm

21

Thin Lenses; Ray Tracing

Ray tracing for thin lenses is similar to that for mirrors. We have three key rays: 1. This ray comes in parallel to the axis and exits through the focal point. 2. This ray comes in through the focal point and exits parallel to the axis. 3. This ray goes through the center of the lens and is undeflected.

For a diverging lens, we can use the same three rays; the image is upright and virtual.

The Thin Lens Equation; Magnification The thin lens equation is the same as the mirror equation:

22

The Thin Lens Equation; Magnification

The magnification formula is also the same as that for a mirror:

The power of a lens is positive if it is converging and negative if it is diverging.

Combinations of Lenses

In lens combinations, the image formed by the first lens becomes the object for the second lens (this is where object distances may be negative).

Lensmaker’s Equation

This useful equation relates the radii of curvature of the two lens surfaces, and the index of refraction, to the focal length.

Dispersion and the Rainbow

The index of refraction varies slightly with the frequency of light; in general, the higher the frequency, the higher the index of refraction. This means that refracted light is “spread out” in a rainbow of colors; this phenomenon is known as dispersion.

23

Dispersion and the Rainbow

Rainbows are created by the dispersion of light as it refracts in a rain drop.

As the drop falls, all the colors of the rainbow arrive at the eye.

Sometimes a faint secondary arc can be seen.

24

CHAPTER 23

GEOMETRIC OPTICS

CONCEPTS

1. The diagram to the right represents an object 0.030 m high placed at point X, 0.60 m from the center of the lens. An image is formed at point Y, 0.30 m from the center of the lens. The image formed is real and inverted. 2. A plane mirror produces an image of an object. Compared to the object, the image appears reversed and the same size. 3. A convex (converging) lens can form images that are either real or virtual. 4. When the calculated image distance for an image formed using a curved mirror has a negative value, the image must be virtual. 5. The convex spherical mirror found on the passenger side of many cars contains the warning: “Objects are closer than they appear.” The best phrase that describes the image of an object viewed in this mirror is – virtual and smaller than the object. 6. The diagram to the right represents an object in front of a concave mirror. The image of the object formed by the mirror is virtual and larger than the object.

25

7. The diagram to the right shows a convex (converging) lens with focal length F. The object should be placed between F and the lens to produce a virtual image.

8. The diagram below that best represents the image of the letter “L” formed by the plane mirror MM is B.

9. An object is placed in front of a plane mirror as shown in the diagram to the right. The diagram below that best represents the image that is formed is C. 10. As the distance between a man and a plane mirror increases, the size of the image of the man produced by the mirror remains the same.

11. The diagram to the right shows light ray R parallel to the principal axis of a spherical concave (converging) mirror. Point F is the focal point of the mirror and C is the center of curvature. After reflecting, the light ray will pass through point F. 12. A searchlight consists of a high-density light source at the focal point of a concave (converging) mirror. The light reflected from the mirror will form a nearly parallel beam.

26

13. The graph below that best represents the relationship between image distance (di) and object distance (do) for a plane mirror is C. 14. A candle is located beyond the center of curvature, C, of a concave spherical mirror having principal focus F, as shown in the diagram to the right. The candle’s image is located between C and F. 15. The diagram on the right shows the principal axis of a concave spherical mirror. The focal point is F, and C is the center of the curvature of the mirror. The focal length of the mirror is 0.10 m. If an object is placed at point B, its image is virtual and erect. The diagram on the right shows a concave (converging) spherical mirror having principal focus F and center of curvature C. Point A lies on the principal axis. (Concepts 16 and 17) 16. When an object is placed at point A, its image is observed to the left of the mirror. 17. If an object is located at point A, its image is virtual and erect. 18. The phenomenon of light that accounts for the formation of images by a lens is refraction. 19. The phenomenon is represented by the diagram to the right is refraction. 20. The piece of glass that could be used to focus parallel rays of sunlight to a small spot of light is B. 21. An optical device that may form an enlarged image is a converging lens. 22. A student uses a magnifying glass to examine the crystals in a mineral specimen. The magnifying glass contains a convex (converging) lens.

27

23. The diagram below that best represents a lens being used to produce a real, enlarged image of object O is B. 24. The diagram below that shows the path of light rays as they pass from an object 2F through a converging lens to the image formed at 2F’ is B. 25. When light rays pass through the film in a movie projector, an image of the film is produced on a screen. In order to produce the image on the screen, the type of lens the projector uses and how far the lens must be placed from the film is converging lens, at a distance greater than the focal length. 26. The focal length of a lens is not dependent on the distance of an object from the lens. 27. The diagram to the right shows a lens with an object located at position A. As the object is moved from position A to position B, the image formed by lens will be inverted and increasing in size. 28. The diagram to the right represents a convex (converging) lens with focal point F. 29. If an object is placed at 2F, the image will be real, inverted, and the same size as the object. 30. A student’s solution to an optics problem had a negative value for the focal length. The optical device in the problem was most likely a diverging (concave) lens. 31. The image formed by a diverging lens is virtual.

28

32. The ray diagram below that is incorrect is B. 33. The diagram to the right shows the refraction of the blue and red components of a white light beam. The phenomenon illustrated is chromatic aberration. 34. The speed of light is 331 m/s in vacuum. 35. A convex (converging) lens can form images that are either real or virtual. 36. A medium that almost all light passes through is called translucent. 37. Light traveling at an angle into a denser medium is refracted toward the normal.

38. Lenses that are thicker at the center bend light rays to a point beyond the lens. 39. Light travels fastest in a vacuum. 40. For all transparent material substances, the index of refraction is equal to 1. 41. A plane mirror will form a virtual image.