Chapter 36 - - Lenses - Saint Charles Preparatory School a concave surface of radius –40 cm 40 cm and a convex surface whose radius is +20 cm. What is the focal length of the lens?

Chapter 36 Chapter 36 -- LensesLensesA 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:

•• Determine the Determine the focal lengthfocal length of converging of converging and diverging lenses.and diverging lenses.

•• Use Use rayray--tracing techniquestracing techniques to construct images to construct images formed by converging and diverging lenses.formed by converging and diverging lenses.

•• Apply the Apply the lensmakerlensmaker’’s equations equation to find to find parameters related to lens construction.parameters related to lens construction.

•• Find the Find the locationlocation, , naturenature, and , and magnificationmagnification of of images formed by converging and diverging images formed by converging and diverging lenses.lenses.

Refraction in PrismsRefraction in Prisms

Two prisms base to base

If we apply the laws of refraction to two prisms, the rays bend toward the base, converging light.

Parallel rays, however, do not converge to a focus, leaving images distorted and unclear.

Refraction in Prisms (Cont.)Refraction in Prisms (Cont.)

Two prisms apex to apex

Similarly, inverted prisms cause parallel light rays to bend toward the base (away from the center).

Again there is no clear virtual focus, and once again, images are distorted and unclear.

Converging and Diverging LensConverging and Diverging LensIf a smooth surface replaces the prisms, a If a smooth surface replaces the prisms, a wellwell--defined focus produces clear images.defined focus produces clear images.

Converging Lens Diverging Lens

Double-convex Double-concave

Real focus

Virtual focus

The Focal Length of LensesThe Focal Length of LensesConverging Lens Diverging Lens

ff++

ff--

The focal length f is positive for a real focus (converging) and negative for a virtual focus.

Focal Focal length length ff

FF

The Principal FocusThe Principal FocusSince light can pass through a lens in either Since light can pass through a lens in either

direction, there are direction, there are two focal pointstwo focal points for each lens.for each lens.

The The principal principal focal focal point point FF is shown is shown here. Yellow here. Yellow F F is is the other one.the other one.

Now suppose light Now suppose light moves from right moves from right to left instead . . . to left instead . . .

Left to rightFFFF FFFF

Right to left

FF

FF FFFF

Types of Converging LensesTypes of Converging LensesIn order for a lens to converge light it must be In order for a lens to converge light it must be

thicker near the midpoint to allow more bending.thicker near the midpoint to allow more bending.

DoubleDouble-- convex lensconvex lens

PlanoPlano-- convex lensconvex lens

Converging Converging meniscus lens meniscus lens

Types of Diverging LensesTypes of Diverging LensesIn order for a lens to diverge light, it must be In order for a lens to diverge light, it must be

thinner near the midpoint to allow more bending.thinner near the midpoint to allow more bending.

DoubleDouble-- concave lensconcave lens

PlanoPlano-- concave lensconcave lens

Diverging Diverging meniscus lens meniscus lens

LensmakerLensmaker’’s Equations Equation

R1 R2

Surfaces of different radius

The Lensmaker’s Equation:

1 2

1 1 1( 1)nf R R

The focal length

f for a lens. The focal length

f for a lens.

Negative (Concave)

Positive (Convex)

Sign convention

R

Signs for LensmakerSigns for Lensmaker’’s Equations Equation

1. R1 and R2 are positive for convex outward surface and negative for concave surface.

2. Focal length f is positive for converging and negative for diverging lenses.

1.1. RR11 and and RR22 are positive for convex outward are positive for convex outward surface and negative for concave surface.surface and negative for concave surface.

2.2. Focal length Focal length ff is positive for converging and is positive for converging and negative for diverging lenses.negative for diverging lenses.

R1 R2

+

-

R1 and R2 are interchangeable

1 2

1 1 1( 1)nf R R

R1, R2 = Radiin= index of glassf = focal length

Example 1.Example 1. A glass meniscus lens (A glass meniscus lens (n = 1.5n = 1.5) ) has a concave surface of radius has a concave surface of radius ––40 cm40 cm and and a convex surface whose radius is a convex surface whose radius is +20 cm+20 cm. . What is the focal length of the lens?What is the focal length of the lens?

RR11 = 20 cm, R= 20 cm, R22 = = --40 cm40 cm

--40 cm40 cm

+20 cm+20 cm

n = 1.5n = 1.51 2

1 1 1( 1)nf R R

1 1 1 2 1(1.5 1)20 cm ( 40 cm 40 cmf

f = 20.0 cmf = 20.0 cm Converging (+) lens.Converging (+) lens.

ExampleExample 2:2: What must be the radius of the What must be the radius of the curved surface in a curved surface in a planoplano--convex lens in order convex lens in order that the focal length be 25 cm?that the focal length be 25 cm?

RR11 = = , R, R22 = 25 cm= 25 cm

2

1 1 1( 1)nf R

R1 =R2 =?

f = ?

00

2 2

1 1 0.500(1.5 1)25 cm R R

R2 = 12.5 cmR2 = 12.5 cm Convex (+) surface.Convex (+) surface.

RR22 = = 0.5(25 cm)0.5(25 cm)

Terms for Image ConstructionTerms for Image Construction


•• The The near focal pointnear focal point is the focus is the focus FF on the on the same side of the lens as the incident light.same side of the lens as the incident light.

•• The The far focal pointfar focal point is the focus is the focus FF on the on the opposite side to the incident light.opposite side to the incident light.

FFNear focus

FFNear focus

FF

Far focus

FF

Far focus

Image Construction:Image Construction:

Ray 1: A ray parallel to the lens axis passes through the far focus of a converging lens or appears to come from the near focus of a diverging lens.

Ray 1:Ray 1: A ray parallel to the lens axis passes through A ray parallel to the lens axis passes through the far focus of a converging lens or appears to the far focus of a converging lens or appears to come from the near focus of a diverging lens.come from the near focus of a diverging lens.


FF

Ray 1

FF

Ray 1

Image Construction:Image Construction:

Ray 2: A ray passing through the near focal point of a converging lens or proceeding toward the far focal point of a diverging lens is refracted parallel to the lens axis.

Ray 2:Ray 2: A ray passing through the A ray passing through the nearnear focal focal point of a converging lens or proceeding point of a converging lens or proceeding towardtoward the the farfar focal point of a diverging lens is focal point of a diverging lens is refracted parallel to the lens axis.refracted parallel to the lens axis.


FF

Ray 1

FF

Ray 1

Ray 2Ray 2

Ray 2Ray 2

Image Construction:Image Construction:Ray 3: A ray passing through the center of any lens continues in a straight line. The refraction at the first surface is balanced by the refraction at the second surface.

Ray 3:Ray 3: A ray passing through the center of any A ray passing through the center of any lens continues in a straight line. The refraction lens continues in a straight line. The refraction at the first surface is balanced by the at the first surface is balanced by the refraction at the second surface.refraction at the second surface.


FF

Ray 1

FF

Ray 1

Ray 2

Ray 2

Ray Ray 33

Ray Ray 33

ImagesImages’’ Tracing PointsTracing PointsDraw an arrow to represent the location of an object, then draw any two of the rays from the tip of the arrow. The image is where lines cross.

Draw an arrow to represent the location of an Draw an arrow to represent the location of an object, then draw any two of the rays from the object, then draw any two of the rays from the tip of the arrow. The image is where lines cross.tip of the arrow. The image is where lines cross.

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

2. Is the image real or virtual?1. Is the image erect or inverted?

• Real images are always on the opposite side of the lens. Virtual images are on the same side.

Object Outside 2FObject Outside 2F

1. The image is 1. The image is invertedinverted; ; i.e., opposite to the i.e., opposite to the object orientation.object orientation.

2. The image is 2. The image is realreal; i.e., ; i.e., formed by actual light formed by actual light rays in front of mirror. rays in front of mirror.

3. The image is 3. The image is diminished diminished in size; i.e., smaller in size; i.e., smaller than the object.than the object.

Image is located between F and 2F Image is located between F and 2F

FF

FF

2F2F

2F2F

Real; inverted; diminished

Object at 2FObject at 2F

FF

FF

2F2F

2F2F

Real; inverted; same size


2. The image is 2. The image is realreal; i.e., ; i.e., formed by actual light formed by actual light rays in front of the mirror. rays in front of the mirror.

3. The image is 3. The image is the same the same size size as the object.as the object. Image is located at

2F on other side Image is located at 2F on other side

Object Between 2F and FObject Between 2F and F

FF

FF

2F2F

2F2F

Real; inverted; enlarged


2. The image is 2. The image is realreal; ; formed by actual light formed by actual light rays on the opposite rays on the opposite sideside

3. The image is 3. The image is enlarged enlarged in size; i.e., larger than in size; i.e., larger than the object.the object.

Image is located beyond 2F Image is located beyond 2F

Object at Focal Length FObject at Focal Length F

FF

FF

2F2F

2F2F

When the object is located at the focal length, the rays of light are parallel. The lines never cross, and no image is formed.

When the object is located at the focal length, When the object is located at the focal length, the rays of light are parallel. The lines never the rays of light are parallel. The lines never cross, and no image is formed.cross, and no image is formed.

Parallel rays; no image formed

Object Inside FObject Inside F

FF

FF

2F2F

2F2F

Virtual; erect; enlarged

1. The image is 1. The image is erecterect; ; i.e., same orientation as i.e., same orientation as the object.the object.

2. The image is 2. The image is virtualvirtual; ; i.e., formed where light i.e., formed where light does does NOTNOT go. go.

3. The image is 3. The image is enlarged enlarged in size; i.e., larger than in size; i.e., larger than the object.the object.

Image is located on near side of lens Image is located on near side of lens

Review of Image FormationsReview of Image Formations

Object Outside 2F RegionObject Outside 2F Region

FF

FF

2F2F

2F2F

Real; inverted; diminished

FF

FF

2F2F

2F2F

Real; inverted; same size

FF

FF

2F2F

2F2F

Real; inverted; enlarged

FF

FF

2F2F

2F2F

Parallel rays; no image formed

FF

FF

2F2F

2F2F

Virtual; erect; enlarged

Diverging Lens ImagingDiverging Lens Imaging

Diverging Lens

FF

Diverging Lens

FF

All images formed by diverging lenses are erect, virtual, and diminished. Images get larger as object approaches.

All images formed by All images formed by divergingdiverging lenses are lenses are erecterect, , virtualvirtual, and , and diminisheddiminished. Images get larger as . Images get larger as object approaches.object approaches.

Analytical Approach to ImagingAnalytical Approach to Imaging

FF

FF

2F2F

2F2F

pf

q

y

-y’

1 1 1p q f

Lens Equation:'y qM

y p

Magnification:

Same Sign Convention as For MirrorsSame Sign Convention as For Mirrors

1. Object p and image q distances are positive for real and images negative for virtual images.

2. Image height y’ and magnification M are positive for erect negative for inverted 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

'y qMy p

Working With Reciprocals:Working With Reciprocals:The lens equation can easily be The lens equation can easily be solved by using the reciprocal 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 lens equation It might be useful to solve the lens 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 3.Example 3. A magnifying glass consists of a A magnifying glass consists of a converging lens of focal length 25 cm. A bug is converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What 8 mm long and placed 15 cm from the lens. What are the nature, size, and location of the image?are the nature, size, and location of the image?

FF

FF

p = 15 cm; f = 25 cm

1 1 1p q f

(15 cm)(25 cm)15 cm - 25 cm

pfqp f

q = -37.5 cm

The fact that q is negative means that the image is virtual (on same side as object).

The fact that The fact that qq is negative means that the is negative means that the image is image is virtualvirtual (on same side as object).(on same side as object).

Example 3 Cont.)Example 3 Cont.) A magnifying glass consists of A magnifying glass consists of a converging lens of focal length 25 cm. A bug is a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What 8 mm long and placed 15 cm from the lens. What is the size of the image?is the size of the image?

FF

FF

p = 15 cm; q = -37.5 cm

' ( 37.5 cm)8 mm 15 cm

y Y’ = +20 mm

The fact that y’ is positive means that the image is erect. It is also larger than object. The fact that The fact that yy’’ is positive means that the is positive means that the

image is image is erect. erect. It is also It is also largerlarger than object.than object.

'y qMy p

y’ y

Example 4:Example 4: What is the magnification of a What is the magnification of a diverging lens (diverging lens (ff = = --20 cm20 cm) if the object is ) if the object is located located 35 cm35 cm from the center of the lens?from the center of the lens?

FF

First we find q . . . then M

1 1 1p q f

'y qMy p

(35 cm)(-20 cm)35 cm - (-20 cm)

pfqp f

q = +12.7 cm

( 12.7 cm)35 cm

qMp

M = +0.364

Example 5:Example 5: Derive an expression for calculating Derive an expression for calculating the magnification of a lens when the object the magnification of a lens when the object distance and focal length are given.distance and focal length are given.

1 1 1p q f

'y qMy p

pfqp f

From last equation: From last equation: q = q = --pMpM

Substituting for Substituting for qq in second equation gives . . .in second equation gives . . .

pfpMp f

Thus, . . . Thus, . . . fMp f

Use this expression to verify answer in Example 4.Use this expression to verify answer in Example 4.

SummarySummaryA converging lens is one that refracts and converges parallel light to a real focus beyond the lens. It is thicker near the middle.

A A converging lensconverging lens is one that refracts and is one that refracts and converges parallel light to a real focus beyond converges parallel light to a real focus beyond the lens. It is thicker near the middle.the lens. It is thicker near the middle.

FFFF

A diverging lens is one that refracts and diverges parallel light which appears to come from a virtual focus in front of the lens.

A A diverging lensdiverging lens is one that refracts and is one that refracts and diverges parallel light which appears to come diverges parallel light which appears to come from a virtual focus in front of the lens.from a virtual focus in front of the lens.

The principal principal focusfocus is denoted

by the red FF..

FFFF

Summary: LensmakerSummary: Lensmaker’’s Equations Equation

1. R1 and R2 are positive for convex outward surface and negative for concave surface.

2. Focal length f is positive for converging and negative for diverging lenses.

1.1. RR11 and and RR22 are positive for convex outward are positive for convex outward surface and negative for concave surface.surface and negative for concave surface.

2.2. Focal length Focal length ff is positive for converging and is positive for converging and negative for diverging lenses.negative for diverging lenses.

R1 R2

+

-

R1 and R2 are interchangeable

1 2

1 1 1( 1)nf R R

R1, R2 = Radiin= index of glassf = focal length

Summary of Math ApproachSummary of Math Approach

FF

FF

2F2F

2F2F

pf

q

y

-y’

1 1 1p q f

Lens Equation:'y qM

y p

Magnification:

Summary of Sign Convention Summary of Sign Convention

1. Object p and image q distances are positive for real and images negative for virtual images.

2. Image height y’ and magnification M are positive for erect negative for inverted 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

'y qMy p

CONCLUSION: Chapter 36CONCLUSION: Chapter 36 LensesLenses

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Chapter 36 - - Lenses - Saint Charles Preparatory School a concave surface of radius –40 cm 40 cm and a convex surface whose radius is +20 cm. What is the focal length of the lens?