College Physics by Serway and Faughn Chapter 23 and 25 Mirrors and Lenses

Notation for Mirrors and Lenses The object distance is the distance from the object to the mirror or lens Denoted by p The image distance is the distance from the image to the mirror or lens Denoted by q The lateral magnification of the mirror or lens is the ratio of the image height to the object height Denoted by M

Types of Images for Mirrors and Lenses A real image is one in which light actually passes through the image point Real images can be displayed on screens A virtual image is one in which the light does not pass through the image point The light appears to diverge from that point Virtual images cannot be displayed on screens

More About Images To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror

Atmospheric Refraction There are many interesting results of refraction in the atmosphere Sunsets Mirages

Atmospheric Refraction and Sunsets Light rays from the sun are bent as they pass into the atmosphere It is a gradual bend because the light passes through layers of the atmosphere Each layer has a slightly different index of refraction The Sun is seen to be above the horizon even after it has fallen below it

Atmospheric Refraction and Mirages A mirage can be observed when the air above the ground is warmer than the air at higher elevations The rays in path B are directed toward the ground and then bent by refraction The observer sees both an upright and an inverted image

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

Thin Lens Shapes These are examples of converging lenses They have positive focal lengths They are thickest in the middle

More Thin Lens Shapes These are examples of diverging lenses They have negative focal lengths They are thickest at the edges

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 distance between the surface of the lens and the center of the lens is negligible

Focal Length of a Converging Lens The parallel rays pass through the lens and converge at the focal point The parallel rays can come from the left or right of the lens

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

Lens Equations The geometric derivation of the equations is very similar to that of mirrors

Lens Equations The equations 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

Sign Conventions for Thin Lenses Quantity Positive When Negative When Object location (p) Object is in front of the lens Object is in back of the lens Image location (q) Image is in back of the lens 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 Center of curvature is in front of the lens Focal length (f) Converging lens Diverging lens

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 lengths 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 convenient

Ray Diagram for Converging Lens, p > f The image is real The image is inverted

Ray Diagram for Converging Lens, p < f The image is virtual The image is upright

Ray Diagram for Diverging Lens The image is virtual The image is upright

A plastic sandwich bag filled with water can act as a crude converging lens in air. If the bag is filled with air and placed under water, is the effective lens (a) converging or (b) diverging? QUICK QUIZ 23.4

QUICK QUIZ 23.4 ANSWER (b). In this case, the index of refraction of the lens material is less than that of the surrounding medium. Under these conditions, a biconvex lens will be divergent.

QUICK QUIZ 23.5 In the figure below, the blue object arrow is replaced by one that is much taller than the lens. How many rays from the object will strike the lens?

QUICK QUIZ 23.5 ANSWER Although a ray diagram only uses 2 or 3 rays (those whose direction is easily determined using only a straight edge), an infinite number of rays leaving the object will always pass through the lens.

QUICK QUIZ 23.6 An object is placed to the left of a converging lens. Which of the following statements are true and which are false? (a) The image is always to the right of the lens. (b) The image can be upright or inverted. (c) The image is always smaller or the same size as the object. Justify your answers with ray diagrams.

QUICK QUIZ 23.6 ANSWER (a) False. A virtual image is formed on the left side of the lens if p < f. (b) True. An upright, virtual image is formed when p < f, while an inverted, real image is formed when p > f. (c) False. A magnified, real image is formed if 2f > p > f, and a magnified, virtual image is formed if p > f.

Problem Solving Strategy Be very careful about sign conventions Do lots of problems for practice Draw confirming ray diagrams

Combinations of Thin Lenses The image produced by the first lens is calculated as though the second lens were not present The light then approaches the second lens as if it had come from the image of the first lens The image of the first lens is treated as the object of the second lens The image formed by the second lens is the final image of the system

Combination of Thin Lenses, 2 If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens p will be negative The overall magnification is the product of the magnification of the separate lenses

Combination of Thin Lenses, example

Lens and Mirror Aberrations One of the basic problems is the imperfect quality of the images Largely the result of defects in shape and form Two common types of aberrations exist Spherical aberration Chromatic aberration

Spherical Aberration Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis For a mirror, parabolic shapes can be used to correct for spherical aberration

Chromatic Aberration Different wavelengths of light refracted by by a lens focus at different points Violet rays are refracted more than red rays The focal length for red light is greater than the focal length for violet light Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses

Chapter 25 Optical Instruments

Optical Instruments Analysis generally involves the laws of reflection and refraction Analysis uses the procedures of geometric optics To explain certain phenomena, the wave nature of light must be used

Diopters Optometrists and ophthalmologists usually prescribe lenses measured in diopters The power of a lens in diopters equals the inverse of the focal length in meters P = 1/ƒ

Simple Magnifier A simple magnifier consists of a single converging lens This device is used to increase the apparent size of an object The size of an image formed on the retina depends on the angle subtended by the eye

The Size of a Magnified Image When an object is placed at the near point, the angle subtended is a maximum The near point is about 25 cm When the object is placed near the focal point of a converging lens, the lens forms a virtual, upright, and enlarged image

Angular Magnification Angular magnification is defined as The angular magnification is at a maximum when the image formed by the lens is at the near point of the eye q = - 25 cm Calculated by

Magnification by a Lens With a single lens, it is possible to achieve angular magnification up to about 4 without serious aberrations With multiple lens, magnifications of up to about 20 can be achieved The multiple lens can correct for aberrations

Compound Microscope A compound microscope consists of two lenses Gives greater magnification than a single lens The objective lens has a short focal length, ƒo<1 cm The ocular lens (eyepiece) has a focal length, ƒe of a few cm

Compound Microscope, cont The lens are separated by a distance L L is much greater than either focal length The approach to analysis is the same as for any two lenses in a row The image formed by the first lens becomes the object for the second lens The image seen by the eye, I2, is virtual, inverted and very much enlarged

Magnifications of the Compound Microscope The lateral magnification of the microscope is The angular magnification of the eyepiece of the microscope is The overall magnification of the microscope is the product of the individual magnifications

Other Considerations with a Microscope The ability of an optical microscope to view an object depends on the size of the object relative to the wavelength of the light used to observe it For example, you could not observe an atom (d  0.1 nm) with visible light (λ 500 nm)

Telescopes Two fundamental types of telescopes Refracting telescope uses a combination of lens to form an image Reflecting telescope uses a curved mirror and a lens to form an image Telescopes can be analyzed by considering them to be two optical elements in a row The image of the first element becomes the object of the second element

Refracting Telescope The two lenses are arranged so that the objective forms a real, inverted image of a distance object The image is near the focal point of the eyepiece The two lenses are separated by the distance ƒo + ƒe which corresponds to the length of the tube The eyepiece forms an enlarged, inverted image of the first image

Angular Magnification of a Telescope The angular magnification depends on the focal lengths of the objective and eyepiece Angular magnification is particularly important for observing nearby objects Very distance objects still appear as a small point of light

Disadvantages of Refracting Telescopes Large diameters are needed to study distant objects Large lenses are difficult and expensive to manufacture The weight of large lenses leads to sagging which produces aberrations

Reflecting Telescope Helps overcome some of the disadvantages of refracting telescopes Replaces the objective lens with a mirror The mirror is often parabolic to overcome spherical aberrations In addition, the light never passes through glass Except the eyepiece Reduced chromatic aberrations

Reflecting Telescope, Newtonian Focus The incoming rays are reflected from the mirror and converge toward point A At A, a photographic plate or other detector could be placed A small flat mirror, M, reflects the light toward an opening in the side and passes into an eyepiece

Examples of Telescopes Reflecting Telescopes Largest in the world are 10 m diameter Keck telescopes on Mauna Kea in Hawaii Largest single mirror in US is 5 m diameter on Mount Palomar in California Refracting Telescopes Largest in the world is Yerkes Observatory in Wisconsin Has a 1 m diameter