Physics E-1ax March 31, 2015 Light, Photons, and MRI
When light hits an object, some of it will be reflected. The reflected light can form an image. We usually want to be able to characterize the image given what we know about the mirror and about the object. The image can be: real or virtual upright or inverted enlarged or reduced We can make a qualitative prediction of the location of an image by tracing three principal rays from the object to the image: 1. perpendicular ray 2. parallel ray 3. focal ray We can make a quantitative prediction of the location of an image by using the mirror equation: 1 1 1 d = + m = − i f do di do
The sign conventions can be summarized as follows: any real quantity is positive; any virtual quantity is negative When light passes through a curved surface (e.g. a lens), the light rays will be bent in a predictable fashion. For a convex spherical surface, as found on a converging lens, incoming parallel rays will be focused on the far side of the lens at a point called the focal point. For a concave spherical surface, as found on a diverging lens, incoming parallel rays will diverge as if they had emerged from the focal point on the near side of the lens. For any object—that is, a source of light—we can predict the image by tracing three principal rays through the lens. The principal rays are: the parallel ray, which enters the lens parallel to the optical axis; the focal ray, which enters the lens along a line through the focal point; and the central ray, which passes straight through the center of the lens. 1 1 1 The thin lens equation is identical to the mirror equation: = + f do di where f is the focal length, do is the distance to the object, and di is the distance to the image. Sign conventions are important, and can be summarized as: if light rays actually converge or diverge from a point, the corresponding quantity will be positive. If light rays only appear to converge or diverge from a point, the corresponding quantity will be negative.
The magnification of an image is given by m = −di do . It is positive if the image is upright, and negative if the image is inverted.
The human eye has a curved cornea and adjustable lens. Together these two elements allow it to take light rays from an object and focus them on the retina. Normally, the eye can focus on any object from 25 cm to infinity. When this is not the case, we use corrective lenses (glasses or contacts) to adjust the range of focus.
1 Physics E-1ax March 31, 2015 • Learning objectives: After this lecture, you will be able to… 1. Explain the basic properties of ray optics: • object vs. image • real vs. virtual • upright vs. inverted • enlarged vs. reduced 2. Define the focal length and the center of curvature of a curved mirror. 3. Define the three principal rays used with mirrors: the perpendicular ray, the parallel ray, and the focal ray. 4. Use ray tracing to predict the properties of the image formed by a planar, concave, or convex mirror. 5. Use the mirror equation to predict the properties of the image formed by a planar, concave, or convex mirror, and show that the mirror equation and ray tracing give the same results. 6. Describe the different kinds of lenses (converging, diverging) and explain qualitatively how a lens focuses light using Snell’s Law. 7. Define the focal point, focal length, and focal plane of a lens. Explain why every lens has two focal points, and how both focal points are used to predict the paths of light rays. 8. Given an object and a single lens, use ray tracing to determine the location of the image, and the image properties:
• real vs. virtual • upright vs. inverted • enlarged vs. reduced
9. Use the lens equation to make a quantitative prediction of the image location and image properties 10. Given an object and two optical elements (lenses and/or mirrors), determine the location and properties of the intermediate image, and of the final image. Explain how this scenario might require a virtual object. 11. Describe how the eye focuses light from an object to form a clear image on the retina. 12. Explain how corrective lenses can correct for various vision problems, and determine the lenses required to correct for nearsighted or farsighted vision.
2 Physics E-1ax March 31, 2015 Mirrors
• Two big questions in any situation involving a mirror: Where is the image? What kind of image is it? Can answer using ray tracing (geometry) or the mirror equation (math). • What do we mean by: the object
the image
There are two kinds of image: real image virtual image
3 Physics E-1ax March 31, 2015 Activity 1: Plane Mirrors
1. On the diagram below, draw three light rays that emanate from the tip of the arrow (the “object”) and strike the mirror (the dark line). Using the relationship θi = θr, trace the paths of these rays after they bounce off of the mirror.
2. What will we see? Where is the image? Is it real or virtual?
4 Physics E-1ax March 31, 2015 Activity 2: Concave Mirrors
• A concave mirror will focus incoming rays that are parallel to the axis. The point where parallel rays converge is called the focal point (F). • The focal point is located halfway between the surface of the mirror and the center of curvature (C). 1. On the diagram below, draw a ray that passes through C, goes through the tip of the arrow, and strikes the mirror. In principle, it will strike the mirror at normal incidence. Thus, this ray will bounce straight back along its incoming path. This is called the perpendicular ray. Extend the perpendicular ray as far as it will go. 2. Now draw a ray that passes through the tip of the arrow and is parallel to the optical axis. This is called the parallel ray. It will bounce off the mirror and travel through the focal point. Extend this ray as far as it will go. 3. Now draw a ray that passes through the tip of the arrow and through the focal point. This is called the focal ray. It will bounce off the mirror and travel parallel to the optical axis. Extend this ray as far as it will go.
4. Where do the three rays intersect? This is the image! Is it upright or inverted? Larger or smaller? Real or virtual?
C F
5 Physics E-1ax March 31, 2015 Activity 3: More Concave Mirrors
1. Draw the three principal rays for the following object. Do the rays intersect?
C F
2. What can you say about the image in this case?
• We can summarize the properties of concave mirrors as follows:
6 Physics E-1ax March 31, 2015 Activity 4: Convex Mirrors
1. Now do the ray tracing for a convex mirror (the shiny side is the convex side). In this case points F and C are on the opposite side (the “dark side”) of the mirror. - The perpendicular ray will head towards C and be reflected away from it. - The parallel ray will head in parallel and be reflected away from F. - The focal ray will head towards F and be reflected parallel.
F C
2. What can you say about the image in this case?
7 Physics E-1ax March 31, 2015 The Mirror Equation
• You can also find out all the information about an image using the mirror equation:
1 1 1 = + f do di
where we need certain definitions and (very important!) sign conventions: f = focal length Sign conventions: (+) on “shiny side”
(–) on “dark side”
do = object distance Sign conventions: (+) on “shiny side” (this is the usual case!)
(–) on “dark side” (can only happen combined with lenses)
di = image distance Sign conventions: (+) on “shiny side”
(–) on “dark side”
d We can also define the magnification m = − i do
m = magnification Sign conventions: (+) upright (same as object)
(–) inverted (“upside down”) Let’s see how this works…
8 Physics E-1ax March 31, 2015 Activity 5: Lenses and Refraction
• What do you see when you look through a lens?
• Two big questions in any situation involving a lens: Where is the image? What kind of image is it? 1. How can glass “bend” a ray of light? On the following images, sketch the approximate paths of each of the indicated light rays. The dashed line is normal to each surface. (Hint: remember Snell’s law? When n2 > n1, the light ray will bend…)
prism planar block
9 Physics E-1ax March 31, 2015 Activity 6: Converging Lenses
• If we stack two prisms on top of a block, we can ensure that three rays of incoming parallel light will all focus at a point:
• And if we have a curved surface, we can ensure that all parallel incoming rays will focus on the other side of the lens at a point known as the focal point:
1. The paths of light rays are reversible. So what will happen to light rays that all emerge from a focal point and head towards the lens?
2. I have a converging lens made of glass (n = 1.5). What do you predict will happen to the focal length (distance to the focal point) if I submerge the lens under water (n = 1.33)? Focal length will (circle): get shorter stay the same get longer
• Note that parallel rays will always be focused; if they come in at an angle the focus will simply be shifted within the focal plane: • We can define the focusing power of a lens as:
1 P = f
Optometrists give the focusing power of a lens in diopters, which are units of (meters)–1.
10 Physics E-1ax March 31, 2015 Activity 7: Ray Tracing for Converging Lens
1. Trace the three principal rays for a converging lens: The central ray goes straight through the center of the lens (within the “thin lens” approximation). The parallel ray goes in parallel to the axis and is deflected towards the far focal point. The focal ray goes through the near focal point and emerges parallel to the axis.
2. Characterize the image. (real / virtual, upright / inverted, enlarged / reduced)
Bonus! A converging lens can have a variety of shapes— what do they all have in common?
11 Physics E-1ax March 31, 2015 Activity 8: Ray Tracing for Diverging Lens
1. Trace the three principal rays for a diverging lens: The central ray goes straight through the center of the lens (within the “thin lens” approximation). The parallel ray goes in parallel to the axis and is deflected away from the near focal point. The focal ray heads towards the far focal point and emerges parallel to the axis.
2. Characterize the image. (real / virtual, upright / inverted, enlarged / reduced)
Bonus! A diverging lens can have a variety of shapes— what do they all have in common?
12 Physics E-1ax March 31, 2015 The Lens Equation
• You can also find out all the information about an image using the lens equation:
1 1 1 = + f do di
where we need certain definitions and (very important!) sign conventions: f = focal length Sign conventions: (+) converging lens (“real focus”)
(–) diverging lens (“virtual focus”)
do = object distance Sign conventions: (+) light rays actually emerge from the object (“real”)
(–) light rays only appear to emerge from object (“virtual”)
di = image distance Sign conventions: (+) light rays actually emerge from image (“real”)
(–) light rays only appear to emerge from image (“virtual”)
d We can also define the magnification m = − i do
m = magnification Sign conventions: (+) upright (image same as object)
(–) inverted (image is “flipped” compared with object) Let’s see how this works…
13 Physics E-1ax March 31, 2015 Am I getting it?
1. An electric eel is swimming underwater. Several meters away, an apple hangs just over the surface of the water. Which of the following statements accurately describes the appearance of the apple from the eel’s perspective? A) The image of the apple appears lower than the apple’s actual position. B) The image of the apple appears at the apple’s actual position. C) The image of the apple appears higher than the apple’s actual position. D) The eel cannot see the apple at all due to total internal reflection.
2. Which of the following optical devices, when used in isolation, is capable of creating an upright real image of a bright object (such as a candle)? Choose all that apply. A) converging lens B) diverging lens C) concave mirror D) convex mirror E) none of the above
3. Shown at right is a ray diagram for an object and a diverging lens. Which of the rays drawn is not a principal ray?
A) ray a B) ray b C) ray c D) ray d
4. A concave mirror forms an inverted, real image of a person, as shown in the top figure at right. What will happen to the image if an opaque shield is used to block the top part of the mirror, as shown in the second figure? A) The head of the image will disappear. B) The feet of the image will disappear. C) The image will become dimmer. D) Nothing will happen to the image.
14 Physical Sciences 3: Lecture 6c April 7, 2015
8 Physics E-1ax March 31, 2015 Activity 9: Combinations of Lenses
• Suppose we combine two or more lenses (and/or mirrors): The image from one will become the object for the next Just be careful to remember that distances are always measured with respect to the lens or mirror in question. 1. Try solving the following problem using ray tracing: Two converging lenses spaced a distance of 80.0 cm apart.
fA = 20.0 cm fB = 25.0 cm do,A = 60.0 cm Where is the final image, and what is the overall magnification?
2. Now use the lens equation… do you get the same result?
15 Physics E-1ax March 31, 2015 The Human Eye
• What is the structure of the human eye? How does it allow us to see?
• How is light focused on the retina? Why is there a “blind spot”?
16 Physics E-1ax March 31, 2015 Normal Vision
• The eye can adjust the shape of the lens to focus on objects that are near or far. What should it do to focus on a very distant object? What is the normal limit of focus for distant objects?
What should it do to focus on a near object? What is the normal limit of focus for near objects?
• Is the image on the retina upright or inverted? Try this experiment at home: 1. Take two index cards and poke a hole in one with a tack. 2. Look through the hole (using one eye) at a bright object. 3. Slowly cover the hole using the other card, placing the card behind the hole. Which way does the image disappear? 4. Try it again, this time with the card in front of the hole (between the hole and your eye). Which way does the image disappear? 5. Draw a ray diagram and explain what’s going on…
17 Physics E-1ax March 31, 2015 Corrective Lenses
• Many people do not have “normal” vision, and instead require corrective lenses to help them see clearly. If you cannot focus clearly on distant objects, you are nearsighted. What kind of lens is needed to correct nearsightedness?
• If you cannot focus clearly on nearby objects, you are farsighted. What kind of lens is needed to correct farsightedness?
18 Physics E-1ax March 31, 2015 One-Minute Paper
Your name: ______
Names of your group members: ______
______
• Please tell us any questions that came up for you today during lecture. Write “nothing” if no questions(s) came up for you between 6–9pm (or while viewing it online).
• What single topic left you most confused after today’s class?
• Any other comments or reflections on today’s class?
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