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**Option G2: Optical Instruments**

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Thin Lenses A thin lens consists of a piece of glass or plastic, ground so that each of its two refracting surfaces is a segment of either a sphere or a plane Lenses are commonly used to form images by refraction in optical instruments (cameras, telescopes, etc.)

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**Thin Lens Shapes These are examples of converging lenses**

They have positive focal lengths They are thickest in the middle

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**More Thin Lens Shapes These are examples of diverging lenses**

They have negative focal lengths They are thickest at the edges

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Focal Length of Lenses The focal length, ƒ, is the image distance that corresponds to an infinite object distance This is the same as for mirrors A thin lens has two focal points, corresponding to parallel rays from the left and from the right A thin lens is one in which the thickness of the lens is negligible in comparison with the focal length

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**Focal Length of a Converging Lens**

The parallel rays pass through the lens and converge at the focal point F The parallel rays can come from the left or right of the lens f is positive

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**Focal Length of a Diverging Lens**

The parallel rays diverge after passing through the diverging lens The focal point is the point where the rays appear to have originated f is negative

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Thin Lenses When parallel rays pass through a converging lens, the rays converge at the focal point f of the lens. They form a real image of an object at infinity. When parallel rays pass through a diverging lens, the rays diverge away from the focal point f of the lens, and must be extrapolated back to find it. The image is virtual.

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Converging Lenses A thin lens is defined here as a lens with a thickness that is small relative to its focal length. We can approximate the lens behavior by assuming that the incident rays are bent as they pass through the plane of the lens. Parallel incident rays are brought to a focus beyond the lens at the downstream focal point f. Incident rays that pass through the upstream focal point f become parallel beyond the lens. Incident rays that pass through the center of the lens are not deflected.

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Real Images The rays diverge from point P on the object, are refracted by the lens, and converge to form a real inverted image at point P’ on the image. All points that lie in the same plane on the object (P, Q, R) will converge at corresponding points (P’, Q’, R’) on the image plane. The close-up view shows rays very near the image plane.

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**Ray Diagrams for Thin Lenses**

Ray diagrams are essential for understanding the overall image formation. Three rays are drawn: The first ray is drawn parallel to the first principle axis and then passes through (or appears to come from) one of the focal points. The second ray is drawn through the center of the lens and continues in a straight line. The third ray is drawn from the other focal point and emerges from the lens parallel to the principle axis There are an infinite number of rays, these are the convenient ones.

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**Converging Lens Ray Tracing**

1. Draw the lens. 2. Draw the optical axis through the center of the lens, with the focal points f placed symmetrically on both sides. 3. Represent the object with an upright arrow of height h at distance s. 4. Draw the three “special” rays from the tip of the object arrow: (a) A ray from the arrow tip parallel to the axis => right focus; (b) A ray from the arrow tip through the left focus => parallel; (c) A ray from the arrow tip through the lens center => straight. 5. Extend the rays until they converge (at the image arrow tip).

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Magnification 1. A positive value of M indicates that the image is upright relative to the object. A negative value of M indicates that the image is inverted relative to the object. 2. The absolute value of M gives the size ratio of image to object: h’/h = |M|. Note that even though M is called “magnification”, its magnitude can be less than 1, indicating that the image is smaller than the object, (i.e., it is demagnified).

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Virtual Images The rays diverge from point P on the object (which is inside f ), are refracted by the lens, and still diverge after the lens. However, if the downstream rays are extrapolated backwards, these extrapolated rays converge to a virtual image at P’. In the case shown, the virtual image is upright and has a magnification greater than 1. The rays reaching the eye appear to be coming from the virtual image.

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Diverging Lenses A lens that is thicker at the edges than at the center is called a diverging lens. Parallel incident rays, after passing through the lens, will diverge from a virtual focus point behind the lens, at the upstream focal point f. Incident rays converging toward the downstream focal point f become parallel after passing through the lens. Incident rays that pass through the center of the lens are not deflected.

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**Diverging Lens Ray Tracing**

1. Draw the lens. 2. Draw the optical axis through the center of the lens, with the focal points f placed symmetrically on both sides. 3. Represent the object with an upright arrow of height h at distance s. 4. Draw the three “special” rays from the tip of the object arrow: (a) A ray from the arrow tip parallel to the axis => from left focus; (b) A ray from the arrow tip toward the right focus => parallel; (c) A ray from the arrow tip through the lens center => straight. 5. Extrapolate these rays backwards until they converge.

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**Ray Diagram for Converging Lens, p > f**

The image is real The image is inverted

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**Ray Diagram for Converging Lens, p < f**

The image is virtual The image is upright

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**Ray Diagram for Diverging Lens**

The image is virtual The image is upright

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**Lens Equation tan q = PQ/f=h/f tan q = -h’/(q - f) h/f = - h’/(q -f)**

h’/h = -(q -f)/f, and with m = h’/h = - q/p it follows: Thin lens -q/p =-(q - f)/f → equation

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Lens Equation, cont. The equation can be used for both converging and diverging lenses A converging lens has a positive focal length A diverging lens has a negative focal length

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**Sign Conventions for Thin Lens**

Quantity Positive When Negative When Object location (p) Object is in front of the lens Object is in back of Image location (q) Image is in back of Image is in front of the lens Image height (h’) Image is upright Image is inverted R1 and R2 Center of curvature is in back of the lens in front of the lens Focal length (f ) Converging lens Diverging lens

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**The Thin Lens Equation Sign Conventions:**

do is positive for real objects (from which light diverges) do is negative for virtual objects (toward which light converges) di is positive for real images (on the opposite side of the lens from the object) di is negative for virtual images (same side as object) f is positive for converging (convex) lenses f is negative for diverging (concave) lenses m is positive for upright images m is negative for inverted images

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Example 1 An object is placed 20 cm in front of a converging lens of focal length 10 cm. Where is the image? Is it upright or inverted? Real or virtual? What is the magnification of the image? do = 20 cm f = 10 cm Real image

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Example 2 An object is placed 5 cm in front of a converging lens of focal length 10 cm. Where is the image? Is it upright or inverted? Real or virtual? What is the magnification of the image? do = 5 cm f = 10 cm Virtual image

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Example 3 An object is placed 8 cm in front of a diverging lens of focal length 2.67 cm. Where is the image? Is it upright or inverted? Real or virtual? What is the magnification of the image? do = 8 cm f = cm Virtual

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**Combinations of Thin Lenses**

The key point to remember is that the image produced by one lens serves as the object for the next lens. The total magnification of a compound lens system is the product of the individual magnification factors: mtotal=m1 m2 m3….

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Example 4 (a)Determine the distance from lens 1 to the final image for the system shown in the figure. (b) What is the magnification of this image? When you have two lenses, the image of the first lens is the object for the second lens.

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Example 4 Solved

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