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**Image Formation by Lenses**

If a piece of glass or other transparent material takes on the appropriate shape, it will be capable of taking parallel rays of incident light and either converging them to a point or appear to diverge them from a point. Such a piece of glass is referred to as a lens.

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**Converging and Diverging Lens**

Properties of lenses are due to the refraction (bending) of light passing thru them according to the law of refraction. Convergent (Convex) Lenses—brings together rays of light which are traveling parallel to its principal axis. Divergent (Concave) Lenses—separates rays of light

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**Converging and Diverging Lens**

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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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. 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 rays in front of mirror. 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 rays in front of the mirror. 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 the 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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**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 the 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 is the size of the 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) if 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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**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 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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