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**Chapter 36 - Lenses A PowerPoint Presentation by**

Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007

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**Objectives: After Completing This Module, You Should Be Able To:**

Determine the focal length of converging and diverging lenses. Apply the lensmaker’s equation to find parameters related to lens construction. Use ray-tracing techniques to construct images formed by converging and diverging lenses. Find the location, nature, and magnification of images formed by converging and diverging lenses.

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Refraction 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.

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**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.

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**Converging and Diverging Lens**

If a smooth surface replaces the prisms, a well-defined focus produces clear images. Converging Lens Diverging Lens Real focus Virtual focus Double-convex Double-concave

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**The Focal Length of lenses**

Converging Lens Diverging Lens Focal length f F f - f + The focal length f is positive for a real focus (converging) and negative for a virtual focus.

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The Principal Focus Since light can pass through a lens in either direction, there are two focal points for each lens. Left to right The principal focal point F is shown here. Yellow F is the other one. F Right to left Now suppose light moves from right to left instead . . . F F

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**Types of Converging Lenses**

In order for a lens to converge light it must be thicker near the midpoint to allow more bending. Plano-convex lens Double-convex lens Converging meniscus lens

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**Types of Diverging Lenses**

In order for a lens to diverge light it must be thinner near the midpoint to allow more bending. Double-concave lens Plano-concave lens diverging meniscus lens

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**Lensmaker’s Equation The focal length f for a lens. R1 R2 R**

Surfaces of different radius The Lensmaker’s Equation: Negative (Concave) Positive (Convex) Sign convention R

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**Signs for Lensmaker’s Equation**

+ - R1 and R2 are interchangeable R1, R2 = Radii n= index of glass f = focal length R1 and R2 are positive for convex outward surface and negative for concave surface. Focal length f is positive for converging and negative for diverging lenses.

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**Example 1. A glass meniscus lens (n = 1**

Example 1. A glass meniscus lens (n = 1.5) has a concave surface of radius –40 cm and a convex surface whose radius is +20 cm. What is the focal length of the lens. -40 cm +20 cm n = 1.5 R1 = 20 cm, R2 = -40 cm f = 20.0 cm Converging (+) lens.

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Example 2: What must be the radius of the curved surface in a plano-convex lens in order that the focal length be 25 cm? R1= R2=? f = ? R1 = , R2 = 25 cm R2 = 0.5(25 cm) R2 = 12.5 cm Convex (+) surface.

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**Terms for Image Construction**

The near focal point is the focus F on the same side of the lens as the incident light. The far focal point is the focus F on the opposite side to the incident light. Converging Lens Diverging Lens F Far focus F Far focus F Near focus F Near focus

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

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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. Converging Lens Diverging Lens F Ray 1 Ray 2 Ray 2

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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. Converging Lens Diverging Lens F Ray 1 Ray 2 Ray 3 Ray 3

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Images Tracing Points Draw 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. 1. Is the image erect or inverted? 2. Is the image real or virtual? Real images are always on the opposite side of the lens. Virtual images are on the same side. 3. Is it enlarged, diminished, or same size?

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**Object Outside 2F F 2F Real; inverted; diminished**

1. The image is inverted, i.e., opposite to the object orientation. 2. The image is real, i.e., formed by actual light on the opposite side of the lens. 3. The image is diminished in size, i.e., smaller than the object. Image is located between F and 2F

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**Object at 2F F 2F Real; inverted; same size**

1. The image is inverted, i.e., opposite to the object orientation. 2. The image is real, i.e., formed by actual light on the opposite side of lens. 3. The image is the same size as the object. Image is located at 2F on other side

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**Object Between 2F and F F 2F Real; inverted; enlarged**

1. The image is inverted, i.e., opposite to the object orientation. 2. The image is real; formed by actual light rays on opposite side 3. The image is enlarged in size, i.e., larger than the object. Image is located beyond 2F

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**Object at Focal Length F**

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

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**Object Inside F F 2F Virtual; erect; enlarged**

1. The image is erect, i.e., same orientation as the object. 2. The image is virtual, i.e., formed where light does NOT go. 3. The image is enlarged in size, i.e., larger than the object. Image is located on near side of lens

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**Review of Image Formations**

Parallel rays; no image formed F 2F Real; inverted; enlarged F 2F Real; inverted; same size Object Outside 2F Region F 2F Real; inverted; diminished F 2F Virtual; erect; enlarged

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**Diverging Lens Imaging**

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

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**Analytical Approach to Imaging**

F 2F p f q y -y’ Lens Equation: Magnification:

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**Same 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 magnifi-cation M are positive for erect negative for inverted images 3. The focal length f and the radius of curvature R is positive for converging lens or mirrors and negative for diverging lens or mirrors.

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**Working With Reciprocals:**

The lens equation can easily be solved by using the reciprocal button (1/x) on most calculators: Possible sequence for finding f on linear calculators: P q 1/x + = Finding f: Same with reverse notation calculators might be: Finding f: P q 1/x + Enter

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**Alternative Solutions**

It might be useful to solve the lens equation algebraically for each of the parameters: Be careful with substitution of signed numbers!

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Example 3. A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are the nature, size, and location of image. F p = 15 cm; f = 25 cm q = cm The fact that q is negative means that the image is virtual (on same side as object).

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Example 3 Cont.) A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are size of image. F p = 15 cm; q = cm y y’ Y’ = +20 mm The fact that y’ is positive means that the image is erect. It is also larger than object.

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Example 4: What is the magnification of a diverging lens (f = -20 cm) the object is located 35 cm from the center of the lens? F First we find q then M q = cm M =

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**From last equation: q = -pM**

Example 5: Derive an expression for calculating the magnification of a lens when the object distance and focal length are given. From last equation: q = -pM Substituting for q in second equation gives . . . Thus, . . . Use this expression to verify answer in Example 4.

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**The principal focus is denoted by the red F.**

Summary A Converging lens is one that refracts and converges parallel light to a real focus beyond the lens. It is thicker near the middle. F The principal focus is denoted by the red F. F A diverging lens is one that refracts and diverges parallel light which appears to come from a virtual focus in front of the lens.

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**Summary: Lensmaker’s Equation**

+ - R1 and R2 are interchangeable R1, R2 = Radii n= index of glass f = focal length R1 and R2 are positive for convex outward surface and negative for concave surface. Focal length f is positive for converging and negative for diverging lenses.

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**Summary of Math Approach**

q y -y’ Lens Equation: Magnification:

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

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**CONCLUSION: Chapter 36 Lenses**

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