Image Formation with Concave Spherical Mirrors

 The figure shows a concave mirror, a mirror in which the edges curve toward the light source.  Rays parallel to the optical axis reflect and pass through the focal point of the mirror.

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A Spherical Mirror: Central Rays A few rays are easy to figure out where they go.

All rays satisfy the “angle of incidence = angle of reflection” measured to the normal to the surface All rays through the center strike the mirror perpendicular to the surface and bounce back along their incoming path.

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A Spherical Mirror: Central Ray A few rays are easy to figure out where they go.

All rays satisfy the “angle of incidence = angle of reflection” measured to the normal to the surface The ray hitting the central line of the diagram is particularly simple.

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A Spherical Mirror: Parallel Rays A few rays are easy to figure out where they go.

All rays satisfy the “angle of incidence = angle of reflection” measured to the normal to the surface All rays parallel to and near an axis of the sphere reflect through a single point on the axis (the focal point)

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A Real Image Formed by a Concave Mirror

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Images in a Spherical Mirror: 1 Physical

center of sphere

focal point

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Tactics: Ray Tracing for a Spherical Mirror

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The Mirror Equation For a spherical mirror with negligible thickness, the object and image distances are related by:

where the focal length f is related to the mirror’s radius of curvature by:

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You see an upright, magnified image of your face when you look into magnifying “cosmetic mirror.” The image is located

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A. In front of the mirror’s surface. B. On the mirror’s surface. C. Behind the mirror’s surface. D. Only in your mind because it’s a virtual image.

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Example 23.17 Analyzing a Concave Mirror

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Example 23.17 Analyzing a Concave Mirror

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Example 23.17 Analyzing a Concave Mirror

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Example 23.17 Analyzing a Concave Mirror

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Image Formation with Spherical Mirrors A city skyline is reflected in this polished sphere.

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Image Formation with Convex Spherical Mirrors  The figure shows parallel light rays approaching a mirror in which the edges curve away from the light source.  This is called a convex mirror.  The reflected rays appear to come from a point behind the mirror.

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A Real Image Formed by a Convex Mirror

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Lenses  The photos below show parallel light rays entering two different lenses.  The left lens, called a converging lens, causes the rays to refract toward the optical axis.  The right lens, called a diverging lens, refracts parallel rays away from the optical axis.

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Converging Lenses  A converging lens is thicker in the center than at the edges.  The focal length f is the distance from the lens at which rays parallel to the optical axis converge.  The focal length is a property of the lens, independent of how the lens is used. 4/28/2014

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Diverging Lenses  A diverging lens is thicker at the edges than in the center.  The focal length f is the distance from the lens at which rays parallel to the optical axis appear to diverge.  The focal length is a property of the lens, independent of how the lens is used. 4/28/2014

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You can use the sun’s rays and a lens to start a fire. To do so, you should use

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A. A converging lens. B. A diverging lens. C. Either a converging or a diverging lens will work if you use it correctly. PHYS 132

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin converging lens.  Situation 1: A ray initially parallel to the optic axis will go through the far focal point after passing through the lens. 4/28/2014

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin converging lens.  Situation 2: A ray through the near focal point of a thin lens becomes parallel to the optic axis after passing through the lens. 4/28/2014

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin converging lens.  Situation 3: A ray through the center of a thin lens is neither bent nor displaced but travels in a straight line.

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Thin Lenses: Ray Tracing Rays from an object point P are refracted by the lens and converge to a real image at point P′.

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A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is removed?

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A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. A sharp, upright image. D. A blurry, upright image. E. No image at all.

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A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if a piece of dark paper is lowered to cover the top half of the lens?

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A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. Only the top half of the image. D. Only the bottom half of the image. E. No image at all.

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A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is covered by a dark mask having only a small hole in the center?

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A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. Only the top half of the image. D. Only the bottom half of the image. E. No image at all.

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Image Formation  The figure is a close-up view of the rays very near the image plane.  To focus an image, you must either move the screen to coincide with the image plane or move the lens or object to make the image plane coincide with the screen. 4/28/2014

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Tactics: Ray Tracing for a Converging Lens

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Tactics: Ray Tracing for a Converging Lens

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A lens creates an image as shown. In this situation, the object distance s is

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A lens creates an image as shown. In this situation, the image distance s′ is

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33

Lateral Magnification  The image can be either larger or smaller than the object, depending on the location and focal length of the lens.  The lateral magnification m is defined as:

 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.  The absolute value of m gives the size ratio of the image and object: h′/h = |m|. 4/28/2014

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Virtual Images

 Consider a converging lens for which the object is inside the focal point, at distance s < f.  You can see all three rays appear to diverge from point P′.  Point P′ is an upright, virtual image of the object point P. 4/28/2014

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Virtual Images

 You can “see” a virtual image by looking through the lens.  This is exactly what you do with a magnifying glass, microscope or binoculars.

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Example 23.9 Magnifying a Flower

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Example 23.9 Magnifying a Flower

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Example 23.9 Magnifying a Flower

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin diverging lens.  Situation 1: A ray initially parallel to the optic axis will appear to diverge from the near focal point after passing through the lens.

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin diverging lens.  Situation 2: A ray directed along a line toward the far focal point becomes parallel to the optic axis after passing through the lens. 4/28/2014

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Thin Lenses: Ray Tracing  Three situations form the basis for ray tracing through a thin diverging lens.  Situation 3: A ray through the center of a thin lens is neither bent nor displaced but travels in a straight line.

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Light rays are converging to point 1. The lens is inserted into the rays with its focal point at point 1. Which picture shows the rays leaving the lens?

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C

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Tactics: Ray Tracing for a Diverging Lens

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Example 23.10 Demagnifying a Flower

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Example 23.10 Demagnifying a Flower

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Example 23.10 Demagnifying a Flower

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Wave Model

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The second model for light: Electromagnetic wave

• Light is an oscillating electromagnetic wave. (Long story) • A “close-up” of a ray: a plane wave

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It’s hard to picture EM waves in 3D • Let’s build some intuition by working through a simpler example.

Waves on the surface of water (treating the height of the surface only – that moves up and down – transvers to the wave motion: the actual bits of water move in small circles) 4/28/2014

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http://www.falstad.com/ripple/

Ripple tank analogy Can two sources lead to both “bright spots” and “dark spots”?

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Chapter 22 Preview

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100 micron slit

Spot actually gets wider… Does this mean light has a “size”? 4/28/2014

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Chapter 22 Preview

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What a difference a slit makes

The big deal here is that opening an additional slit makes it darker in some places. No way this happens in either the ray or photon model. 4/28/2014 PHYS 132

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Diffraction of Light  When red light passes through an opening that is only 0.1 mm wide, it does spread out.  Diffraction of light is observable if the hole is sufficiently small.

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Young’s Double-Slit Experiment

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Young’s Double-Slit Experiment

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Analyzing Double-Slit Interference

 The figure shows the “big picture” of the double-slit experiment.  The next slide zooms in on the area inside the circle.

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Analyzing Double-Slit Interference  The figure shows a magnified portion of the double-slit experiment.  The wave from the lower slit travels an extra distance.

 Bright fringes (constructive interference) will occur at angles θm such that ∆r = mλ, where m = 0, 1, 2, 3, … 4/28/2014

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Analyzing Double-Slit Interference  The mth bright fringe emerging from the double slit is at an angle:

where θm is in radians, and we have used the smallangle approximation.  The y-position on the screen of the mth bright fringe on a screen a distance L away is:

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A laboratory experiment produces a double-slit interference pattern on a screen. The point on the screen marked with a dot is how much farther from the left slit than from the right slit?

A. B. C. D. E.

1.0 λ. 1.5 λ. 2.0 λ. 2.5 λ. 3.0 λ.

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A laboratory experiment produces a double-slit interference pattern on a screen. If the screen is moved farther away from the slits, the fringes will be

A. closer together. B. in the same positions. C. farther apart. D. fuzzy and out of focus. 4/28/2014

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A laboratory experiment produces a double-slit interference pattern on a screen. If green light is used, with everything else the same, the bright fringes will be

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A. B. C. D.

and green light has a shorter wavelength.

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A laboratory experiment produces a double-slit interference pattern on a screen. If the slits are moved closer together, the bright fringes will be

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The figure shows what happens if you put white light through the same slit-screen system. Why are the different colors separated on either side of the center?

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Intensity of the Double-Slit Interference Pattern

The intensity of the double-slit interference pattern at position y is:

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Intensity of the Double-Slit Interference Pattern

The actual intensity from a double-slit experiment slowly decreases as |y| increases.

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Single-Slit Diffraction

 Diffraction through a tall, narrow slit is known as single-slit diffraction.  A viewing screen is placed distance L behind the slit of width a, and we will assume that L >> a.

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Huygens’ Principle: Plane Waves

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Huygens’ Principle: Spherical Waves

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Analyzing Single-Slit Diffraction

 The figure shows a wave front passing through a narrow slit of width a.  According to Huygens’ principle, each point on the wave front can be thought of as the source of a spherical wavelet. 4/28/2014

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Single-Slit Diffraction

 The light pattern from a single slit consists of a central maximum flanked by a series of weaker secondary maxima and dark fringes.  The dark fringes occur at angles:

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The Width of a Single-Slit Diffraction Pattern  The central maximum of this single-slit diffraction pattern is much brighter than the secondary maximum.  The width of the central maximum on a screen a distance L away is twice the spacing between the dark fringes on either side:

 The farther away from the screen (larger L), the wider the pattern of light becomes.  The narrower the opening (smaller a), the wider the pattern of light becomes! 4/28/2014

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25%

25%

D

25%

C

A

25%

B

A laboratory experiment produces a double-slit interference pattern on a screen. If the left slit is blocked, the screen will look like

75