Chapter 36 - Lenses A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University A PowerPoint Presentation.

Chapter 36 - LensesChapter 36 - LensesA PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

Objectives: Objectives: After Completing After Completing This Module, You Should Be This Module, You Should Be

Able To:Able To:• Determine the Determine the focal lengthfocal length of of

converging and diverging lenses.converging and diverging lenses.

• Use Use ray-tracing techniquesray-tracing techniques to construct to construct images formed by converging and images formed by converging and diverging lenses.diverging lenses.

• Apply the Apply the lensmaker’s equationlensmaker’s equation to find to find parameters related to lens parameters related to lens construction.construction.

• Find the Find the locationlocation, , naturenature, and , and magnificationmagnification of images formed by of images formed by converging and diverging lenses.converging and diverging 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 Refraction in Prisms (Cont.)(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 Converging and Diverging LensLens

If a smooth surface replaces the If a smooth surface replaces the prisms, a well-defined focus produces prisms, a well-defined focus produces clear images.clear images.

Converging Lens

Diverging Lens

Double-convex

Double-concave

Real focus

Virtual focus

The Focal Length of lensesThe Focal Length of lenses

Converging 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 for each

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

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

Left to rightFFFF FFFF

Right to left

FF

FF FFFF

Types of Converging Types of Converging LensesLenses

In order for a lens to converge light it must In order for a lens to converge light it must be thicker near the midpoint to allow more be thicker near the midpoint to allow more

bending.bending.Double-Double-convex convex

lenslens

Plano-Plano-convex convex

lenslens

Converging Converging meniscus meniscus

lens lens

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

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

Double-Double-concave concave

lenslens

Plano-Plano-concave concave

lenslens

diverging diverging meniscus meniscus

lens lens

Lensmaker’s Lensmaker’s EquationEquation

R1 R2

Surfaces of different radius

The Lensmaker’s Equation:

1 2

1 1 1( 1)n

f R R

1 2

1 1 1( 1)n

f R R

The focal

length f for a lens.

The focal length f for a

lens.

Negative (Concav

e)

Positive (Conve

x)

Sign convention

R

Signs for Lensmaker’s Signs for Lensmaker’s EquationEquation

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

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

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

R1

R2

+

-

R1 and R2 are interchangeable

1 2

1 1 1( 1)n

f R R

1 2

1 1 1( 1)n

f R R

R1, R2 = Radii

n= index of glassf = focal length

Example 1.Example 1. A glass meniscus lens (A glass meniscus lens (n = n = 1.51.5) has a concave surface of radius ) has a concave surface of radius ––40 cm40 cm and a convex surface whose and a convex surface whose radius is radius is +20 cm+20 cm. What is the focal . What is the focal length of the lens.length of the lens.

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

-40 cm-40 cm

+20 cm+20 cm

n = 1.5n = 1.51 2

1 1 1( 1)n

f 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 plano-convex lens in curved surface in a plano-convex lens in order that the focal length be 25 cm?order that the focal length be 25 cm?

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

2

1 1 1( 1)n

f 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 0.5(25 cm)cm)

Terms for Image Terms for Image ConstructionConstruction

Converging Lens

Diverging Lens

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

FFNear focus

FFNear focus

FF

Far focus

FF

Far focus

Image Construction:Image Construction:

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

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

Converging Lens

Diverging Lens

FF

Ray 1

FF

Ray 1

Image Construction:Image Construction:

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

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

Converging Lens

Diverging Lens

FF

Ray 1

FF

Ray 1

Ray 2Ray 2

Ray 2Ray 2

Image Construction:Image Construction:

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

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

Converging Lens

Diverging Lens

FF

Ray 1

FF

Ray 1

Ray 2

Ray 2

Ray Ray 33

Ray Ray 33

Images Tracing PointsImages Tracing Points

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

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

2. The image is 2. The image is realreal, , i.e., formed by i.e., formed by actual light on the actual light on the opposite side of the opposite side of the lens. lens. 3. The image is 3. The image is

diminished diminished in size, in size, i.e., smaller than the i.e., smaller than the object.object.

Image is located between F and 2F

Image is located between F and 2F

FF

FF

22FF

2F2F

Real; inverted; diminished

Object at 2FObject at 2F

FF

FF

22FF

2F2F

Real; inverted; same size

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

2. The image is 2. The image is realreal, , i.e., formed by actual i.e., formed by actual light on the opposite light on the opposite side of lens. side of lens.

3. The image is 3. The image is the the same size same size as the as the object.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

22FF

2F2F

Real; inverted; enlarged

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

2. The image is 2. The image is realreal; ; formed by actual formed by actual light rays on light rays on opposite sideopposite side

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

Image is located beyond 2F

Image is located beyond 2F

Object at Focal Length FObject at Focal Length F

FF

FF

22FF

2F2F

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

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

Parallel rays; no image formed

Object Inside FObject Inside F

FF

FF

22FF

2F2F

Virtual; erect; enlarged

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

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

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

Image is located on near side of lens

Image is located on near side of lens

Review of Image Review of Image FormationsFormations

Object Outside 2F RegionObject Outside 2F Region

FF

FF

22FF

2F2F

Real; inverted; diminished

FF

FF

22FF

2F2F

Real; inverted; same size

FF

FF

22FF

2F2F

Real; inverted; enlarged

FF

FF

22FF

2F2F

Parallel rays; no image formed

FF

FF

22FF

2F2F

Virtual; erect; enlarged

Diverging Lens ImagingDiverging Lens Imaging

Diverging Lens

FF

Diverging Lens

FF

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

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

Analytical Approach to Analytical Approach to ImagingImaging

FF

FF

2F2F

2F2F

p

f

q

y

-y’

1 1 1

p q f

Lens Equation:

'y qM

y p

Magnification:

Same Sign Convention as For Same Sign Convention as For MirrorsMirrors

1. Object p and image q distances are positive for real and images negative for virtual images.2. Image height y’ and magnifi-cation M are positive for erect negative for inverted images3. The focal length f and the radius of curvature R is positive for converging lens or mirrors and negative for diverging lens or mirrors.

1 1 1

p q f

1 1 1

p q f

'y qM

y p

'y qM

y p

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

1 1 1

p q f

1 1 1

p q f

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

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

Finding Finding f:f: P qP q 1/x1/x ++1/x1/x 1/x1/xEnteEnterr

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

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

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

1 1 1

p q f

1 1 1

p q f

qpf

q p

qpf

q p

qf

pq f

qfp

q f

pf

qp f

pfq

p f

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

FF

FF

p = 15 cm; f = 25 cm

1 1 1

p q f

1 1 1

p q f

(15 cm)(25 cm)

15 cm - 25 cm

pfq

p f

q = -37.5 cm

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).

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

FF

FF

p = 15 cm; q = -37.5 cm

' ( 37.5 cm)

8 mm 15 cm

y Y’ = +20 mm

The fact that The fact that y’y’ is positive means that the is positive means that the image is image is erect. erect. It is also It is also largerlarger than than

object.object.

The fact that The fact that y’y’ is positive means that the is positive means that the image is image is erect. erect. It is also It is also largerlarger than than

object.object.

'y qM

y p

'y qM

y p

y’

y

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

FF

First we find q . . . then M

1 1 1

p q f

1 1 1

p q f 'y q

My p

'y qM

y p

(35 cm)(-20 cm)

35 cm - (-20 cm)

pfq

p f

q = +12.7

cm( 12.7 cm)

35 cm

qM

p

M = +0.364

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

1 1 1

p q f

1 1 1

p q f 'y q

My p

'y qM

y p

pf

qp f

pfq

p f

From last equation: From last equation: q = -q = -pMpMSubstituting for Substituting for qq in second equation in second equation

gives . . .gives . . .pf

pMp f

Thus, . . . Thus, . . . fM

p f

fM

p f

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

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

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

FFFF

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

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

The principal principal focusfocus is

denoted by the red FF..

FFFF

Summary: Lensmaker’s Summary: Lensmaker’s EquationEquation

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

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

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

R1

R2

+

-

R1 and R2 are interchangeable

1 2

1 1 1( 1)n

f R R

1 2

1 1 1( 1)n

f R R

R1, R2 = Radii

n= index of glassf = focal length

Summary of Math ApproachSummary of Math Approach

FF

FF

2F2F

2F2F

p

f

q

y

-y’

1 1 1

p 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 magnifi-cation M are positive for erect negative for inverted images3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

1 1 1

p q f

1 1 1

p q f

'y qM

y p

'y qM

y p

CONCLUSION: Chapter 36CONCLUSION: Chapter 36LensesLenses