Chapter 34B Chapter 34B -- Reflection Reflection and Mirrors II (Analytical)and Mirrors II (Analytical)

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:

•• Define and illustrate the following terms: Define and illustrate the following terms: realreal and and virtualvirtual images, images, convergingconverging and and divergingdiverging mirrors, mirrors, focal lengthfocal length, and , and magnificationmagnification..

•• Predict mathematically the Predict mathematically the naturenature, , sizesize, and , and locationlocation of images formed by spherical mirrors.of images formed by spherical mirrors.

•• Understand and apply the Understand and apply the sign conventionssign conventions that apply to focal lengths, image distances, that apply to focal lengths, image distances, image heights, and magnification.image heights, and magnification.

•• Determine mathematically the Determine mathematically the magnificationmagnification and/or the focal length of spherical mirrors.and/or the focal length of spherical mirrors.

Analytical OpticsAnalytical OpticsIn this unit, we will discuss analytical relationships to describe mirror images more accurately. But first we will review some graphical principles covered in Module 34a on light reflection.

In this unit, we will discuss analytical relationships In this unit, we will discuss analytical relationships to describe mirror images more accurately. But to describe mirror images more accurately. But first we will review some graphical principles first we will review some graphical principles covered in Module 34a on light reflection.covered in Module 34a on light reflection.

The Plane MirrorThe Plane Mirror

Object distance

Image distance=

p = q

ObjectObject ImageImage

pp qq

Object distance:Object distance: The straightThe straight--line distance line distance pp from the surface of a mirror to the object. from the surface of a mirror to the object.

Image distance:Image distance: The straightThe straight--line distance line distance qq from the surface of a mirror to the image. from the surface of a mirror to the image.

Image is virtual

Spherical MirrorsSpherical MirrorsA A spherical mirrorspherical mirror is is formed by the inside formed by the inside ((concaveconcave) or outside ) or outside ((convexconvex) surfaces of ) surfaces of a sphere.a sphere.

A A concave spherical concave spherical mirrormirror is shown here is shown here with parts identified.with parts identified.

The The axisaxis and and linear linear apertureaperture are shown.are shown.

Concave Mirror

Radius of Curvature RVertex V

Center of Curvature C

Linear aperture

V

C

R Axis

The Focal Length The Focal Length ff of a Mirrorof a Mirror

axis

Incident parallel ray

f

The focal length, f

The focal length f is equal to half the radius RThe focal length f is equal to half the radius R

Since Since ii = = rr , we , we find that find that FF is midis mid-- way between way between VV and and CC; we find:; we find:

The focal The focal length length f f is:is:

2Rf

C Vr

iR

F

Focal point

Converging and Diverging MirrorsConverging and Diverging MirrorsConcave Concave mirrors and mirrors and converging parallel rays converging parallel rays will be called will be called converging converging mirrorsmirrors..

ConvexConvex mirrors and mirrors and diverging parallel rays diverging parallel rays will be called will be called diverging diverging mirrorsmirrors..

CF

Converging Mirror

Concave

C F

Diverging Mirror

Convex

DefinitionsDefinitions

Focal length:Focal length: The straightThe straight--line distance line distance ff from from the surface of a mirror to focus of the mirror. the surface of a mirror to focus of the mirror.

Magnification:Magnification: The ratio of the size of the The ratio of the size of the image to the size of the object.image to the size of the object.

Real image:Real image: An image formed by real light An image formed by real light rays that can be projected on a screen. rays that can be projected on a screen.

Virtual image:Virtual image: An image that appears to be at An image that appears to be at a location where no light rays reach. a location where no light rays reach.

Converging and diverging mirrors:Converging and diverging mirrors: Refer to the Refer to the reflection of parallel rays from surface of mirror.reflection of parallel rays from surface of mirror.

Image Construction Summary:Image Construction Summary:Ray 1:Ray 1: A ray parallel to mirror axis passes A ray parallel to mirror axis passes through the focal point of a concave mirror through the focal point of a concave mirror or appears to come from the focal point of a or appears to come from the focal point of a convex mirror.convex mirror.

Ray 2:Ray 2: A ray passing through the focus of a A ray passing through the focus of a concave mirror or proceeding toward the concave mirror or proceeding toward the focus of a convex mirror is reflected parallel focus of a convex mirror is reflected parallel to the mirror axis. to the mirror axis.

Ray 3:Ray 3: A ray that proceeds along a radius is A ray that proceeds along a radius is always reflected back along its original path. always reflected back along its original path.

C F

Converging mirror

Examples of Image ConstructionExamples of Image ConstructionThe three principal rays for both converging The three principal rays for both converging (concave) and diverging (convex) mirrors.(concave) and diverging (convex) mirrors.

Ray 1

Ray 2

Ray 3

CC

Diverging mirror

F

Ray 1

Ray 2

Ray 3

Image

Review of Imaging FactsReview of Imaging FactsFor plane mirrors, the object distance equals For plane mirrors, the object distance equals the image distance and all images are erect and the image distance and all images are erect and virtual.virtual.

For converging mirrors and diverging mirrors, For converging mirrors and diverging mirrors, the focal length is equal to onethe focal length is equal to one--half the radius.half the radius.

All images formed from convex mirrors are All images formed from convex mirrors are erect, virtual, and diminished in size.erect, virtual, and diminished in size.

Except for objects located inside the focus Except for objects located inside the focus (which are erect and virtual), all images formed (which are erect and virtual), all images formed by converging mirrors are real and inverted. by converging mirrors are real and inverted.

Questions About ImagesQuestions About Images

3. Is it enlarged, diminished, or the same size?

2. Is the image real or virtual?

1. Is the image erect or inverted?

4. What are object and image distances p and q?

5. What is the height y’ or size of image?

6. What is the magnification M = y’/y of image?

Definition of SymbolsDefinition of SymbolsBy applying algebra and geometry to rayBy applying algebra and geometry to ray--tracing tracing diagrams, such as the one below, one can derive a diagrams, such as the one below, one can derive a relationship for predicting the location of images.relationship for predicting the location of images.

y

Y’

R

q

p

f

Object dist. p

Image dist. q

Focal length f

Radius R

Object size y

Image size y’ 2Rf

Mirror EquationMirror Equation

y

Y’

R

q

p

f

2Rf

1 1 1p q f

The following equations are given without derivation. They apply equally well for both converging and diverging mirrors.

The following equations are given without The following equations are given without derivation. They apply equally well for both derivation. They apply equally well for both converging and diverging mirrors.converging and diverging mirrors.

Sign ConventionSign Convention

1. Object distance p is positive for real objects and negative for virtual objects.

2. Image distance q is positive for real images and negative for virtual images.

3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

1 1 1p q f

Example 1.Example 1. A A 6 cm6 cm pencil is placed pencil is placed 50 cm50 cm from from the vertex of a the vertex of a 4040--cmcm diameter mirror. What diameter mirror. What are the location and nature of the image?are the location and nature of the image?

Sketch the rough image.Sketch the rough image.

p p = 50 cm; = 50 cm; RR = 40 cm= 40 cm

40 cm ; 20 cm2 2Rf f

1 1 1p q f 1 1 1

50 cm 20 cmq

C F

p

q

f

Example 1 (Cont.).Example 1 (Cont.). What are the location and What are the location and nature of the image? (nature of the image? (p p = 50 cm; = 50 cm; f f = 20= 20 cm)cm)

1 1 150 cm 20 cmq

1 1 120 cm 50 cmq

q = +33.3 cmq = +33.3 cm

The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C). The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C).

C F

p

q

f

Working With Reciprocals:Working With Reciprocals:The mirror equation can easily The mirror equation can easily be solved by using the reciprocal be solved by using the reciprocal button (button (1/x1/x) on most calculators:) on most calculators:

1 1 1p q f

P qP q1/x1/x ++ 1/x1/x == 1/x1/xFinding Finding f:f:

Same with reverse notation calculators might be:Same with reverse notation calculators might be:

Finding Finding f:f: P qP q1/x1/x ++1/x1/x 1/x1/xEnterEnter

Possible sequence for finding Possible sequence for finding ff on linear calculators:on linear calculators:

Be careful with substitution of signed numbers!Be careful with substitution of signed numbers!Be careful with substitution of signed numbers!

Alternative SolutionsAlternative SolutionsIt might be useful to solve the mirror equation It might be useful to solve the mirror equation algebraically for each of the parameters:algebraically for each of the parameters:

1 1 1p q f

qpfq p

qfpq f

pfqp f

Example 2:Example 2: An arrow is placed An arrow is placed 30 cm30 cm from the from the surface of a polished sphere of radius surface of a polished sphere of radius 80 cm80 cm. . What is the location and nature of image?What is the location and nature of image?

Draw image sketch:Draw image sketch:

p p = 30 cm; = 30 cm; RR = = --80 cm80 cm

-80 cm ; 40 cm2 2Rf f

Solve the mirror equation Solve the mirror equation for for qq, then watch signs , then watch signs carefully on substitution:carefully on substitution:

pfqp f

Example 2 (Cont.)Example 2 (Cont.) Find location and nature Find location and nature of image when of image when pp = 30 cm and = 30 cm and qq = = --40 cm.40 cm.

(30 cm)(-40 cm)30 cm - (-40 cm)

q

q = -17.1 cm

The image is virtual (-q), erect, and diminished. It appears to be located at a distance of 17.1 cm behind the mirror.

The image is The image is virtual virtual ((--q), q), erecterect, and , and diminisheddiminished. It appears to be located at a . It appears to be located at a distance of distance of 17.1 cm17.1 cm behindbehind the mirror.the mirror.

Magnification of ImagesMagnification of ImagesThe The magnificationmagnification MM of an image is the ratio of an image is the ratio of the of the image sizeimage size yy’’ to the to the object sizeobject size yy..

Magnification:

'y qMy p

yy and and yy’’ are positive when erect; negative inverted.are positive when erect; negative inverted.

Obj. Img. Obj. Img.

M = +2 M = -1/2

y y’ y y’

qq is positive when real; negative when virtual.is positive when real; negative when virtual.

MM is positive when image erect; negative inverted.is positive when image erect; negative inverted.

Example 3.Example 3. An An 88--cmcm wrench is placed wrench is placed 10 cm10 cm from a diverging mirror of from a diverging mirror of f = f = --20 cm20 cm. What . What is the location and size of the image? is the location and size of the image?

( 6.67 cm)10 cm

qMp

Magnification:

M = +0.667Since M = y’/y y’ = My or:

(10cm)(-20cm)10 cm - (-20 cm)

pfqp f

q = - 6.67 cmq = - 6.67 cm Virtual !Virtual !

y’ = +5.34 cm

Y’Y

p q

Virtual image

Converging mirror

FF CC

Wrench

Example 4.Example 4. How close must a girlHow close must a girl’’s face be s face be to a converging mirror of focal length 25 cm, to a converging mirror of focal length 25 cm, in order that she sees an erect image that is in order that she sees an erect image that is twice as large? (twice as large? (M = +2)M = +2)

2 ; 2qM q pp

Also, Also, pfq

p f

2pf pp f

Thus, Thus, f = f = --2(p 2(p -- f) = f) = --22pp + 2+ 2ff

f = f = --22pp + 2+ 2ff25 cm

2 2fp p = 12.5 cmp = 12.5 cm

SummarySummary

y

Y’

R

q

p

f

2Rf

1 1 1p q f

The following equations apply equally well for both converging and diverging mirrors. The following equations apply equally well The following equations apply equally well for both converging and diverging mirrors.for both converging and diverging mirrors.

Summary: Sign ConventionSummary: Sign Convention

1. Object distance p is positive for real objects and negative for virtual objects.

2. Image distance q is positive for real images and negative for virtual images.

3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

4. The image size y’ and the magnification M of images is positive for erect images and negative for inverted images.

1 1 1p q f

Summary: MagnificationSummary: MagnificationThe The magnificationmagnification MM of an image is the ratio of an image is the ratio of the of the image sizeimage size yy’’ to the to the object sizeobject size yy..

Magnification:

'y qMy p

yy and and yy’’ are positive when erect; negative inverted.are positive when erect; negative inverted.

Obj. Img. Obj. Img.

M = +2 M = -1/2

y y’ y y’

qq is positive when real; negative when virtual.is positive when real; negative when virtual.

MM is positive when image erect; negative inverted.is positive when image erect; negative inverted.

CONCLUSION: Chapter 34BCONCLUSION: Chapter 34B Reflection and Mirrors II Reflection and Mirrors II

(Analytical)(Analytical)