Physics 4 Geometric Optics Mirrors and Thin Lenses

Physics 4 Geometric Optics Mirrors and Thin Lenses Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

We have already learned the basics of Reflection and Refraction. Reflection - angle of incidence = angle of reflection Refraction - light bends toward the normal according to Snell’s Law Now we apply those concepts to some simple types of mirrors and lenses. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

We have already learned the basics of Reflection and Refraction: Reflection - angle of incidence = angle of reflection Refraction - light bends toward the normal according to Snell’s Law Now we apply those concepts to some simple types of mirrors and lenses. Flat Mirror This is the simplest mirror – a flat reflecting surface. The light rays bounce off and you see an image that seems to be behind the mirror. This is called a VIRTUAL IMAGE because the light rays do not actually travel behind the mirror. The image will appear reversed, but will be the same size and the same distance from the mirror. A typical light ray entering the eye of the viewer is shown. The object distance is labeled do and the image distance is labeled di. Real Object Virtual Image do di Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Spherical Mirrors For curved mirrors we will assume that the shape is spherical (think of a big shiny ball, and slice off any piece of that – there’s your spherical mirror). This will make our math relatively simple, with only a couple of formulas. The hard part will be to get the negative signs correct. The radius of curvature describes the shape of the mirror. This is the same as the radius of the big shiny ball that the mirror was cut from. We will have two types of mirrors, depending on which direction they curve: CONCAVE mirrors curve toward you, and have POSITIVE R (like the inside of the sphere). CONVEX mirrors curve away from you, and have NEGATIVE R (think of the outside of the ball). There is a point called the FOCAL POINT which is halfway between the mirror and the center. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? Before we answer this let’s look at a few basic formulas for spherical mirrors. 1) The focal length is half the radius. Remember the sign convention – if the mirror is concave R is positive. If convex, R is negative. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? Before we answer this let’s look at a few basic formulas for spherical mirrors. 1) The focal length is half the radius. Remember the sign convention – if the mirror is concave R is positive. If convex, R is negative. 2) This formula relates the object (do) and image (di) positions to the focal length (f) of the mirror. Here do is always positive for mirrors, and di is positive if the image is on the same side as the object (a REAL image). To remember this, just follow the light – a real (positive) image will have light rays passing through it. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? Before we answer this let’s look at a few basic formulas for spherical mirrors. 1) The focal length is half the radius. Remember the sign convention – if the mirror is concave R is positive. If convex, R is negative. 2) This formula relates the object (do) and image (di) positions to the focal length (f) of the mirror. Here do is always positive for mirrors, and di is positive if the image is on the same side as the object (a REAL image). To remember this, just follow the light – a real (positive) image will have light rays passing through it. 3) The magnification (m) of the image is related to the relative positions of the object and image. Don’t forget the negative sign in this formula. The sign of m tells you if the image is upright (+) or inverted (-) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? focal length OK, back to the problem. We have given information: object distance Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? focal length OK, back to the problem. We have given information: Now we can use formula 2 to locate the image (di) object distance Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? focal length OK, back to the problem. We have given information: Now we can use formula 2 to locate the image (di) object distance This means the image will be located 1 m BEHIND the mirror. This is a VIRTUAL image. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? focal length OK, back to the problem. We have given information: Now we can use formula 2 to locate the image (di) object distance This means the image will be located 1 m BEHIND the mirror. This is a VIRTUAL image. For the magnification, just use formula 3. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? focal length OK, back to the problem. We have given information: Now we can use formula 2 to locate the image (di) object distance This means the image will be located 1 m BEHIND the mirror. This is a VIRTUAL image. For the magnification, just use formula 3. So the image is upright (+) and 5 times as large as the object. We could also draw the ray diagram… Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example using the Formula Method: A concave makeup mirror with radius of curvature 0. 5 m is held 0. 2 m from a woman’s face. Where is her image and how large is it? Notice the 3 rays in the diagram. They all start at the object and go toward the mirror. Ray 1 through the center is easy to draw. So is ray 2, which starts out flat, then bounces off the mirror and goes through the focal point (f). Ray 3 is the tricky one. Since the object is inside the focal point (closer to the mirror, or do<f) we can’t draw the ray through the focal point. Instead we pretend the ray came from the focal point and passed through the object on its way to the mirror, then bounced off flat. The outgoing rays do not intersect! So we have to trace them backwards to find their intersection point behind the mirror. This is what your brain does for you every time you look in a mirror. The virtual image appears at the point where the outgoing light rays seem to be coming from. 3 Object 1 f Image do di 2 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Convex Mirrors These will work the same way as concave, but R and f are negative. Take a look at where the center of the sphere is – it is behind the mirror. There are no light rays there. This is why the radius is negative. Because the light rays do not go there. The 3 typical light rays are shown. • Ray 1 points toward the center and bounces straight back. • Ray 2 starts flat and bounces off as if it is coming from the focal point. • Ray 3 starts toward the focal point and bounces off flat. object R f 3 C Image (this is a virtual image behind the mirror, so di is negative) 1 2 Convex Mirror – R is negative Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

SPHERICAL MIRROR EQUATIONS AND SIGN CONVENTION Concave Mirror Illustrated Light In Side do < 0 Virtual Object do > 0 Real Object C F Light Out Side di > 0 Real Image C This Side, R > 0 V Optic Axis di < 0 Virtual Image C This Side, R < 0 C – Center of Curvature R – Radius of Curvature F – Focal Point (Same Side as C) V – Vertex Equations: Paraxial Approximation Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

REFRACTION AT SPHERICAL INTERFACE BETWEEN TWO OPTICAL MATERIALS Light In Side do > 0 Real Object di < 0 Virtual Image C This Side, R < 0 na – Index of Refraction Light Out Side do < 0 Virtual Object di > 0 Real Image C This Side, R > 0 nb – Index of Refraction Illustrated Interface Has C, Center of Curvature, On The Light Out Side, Thus R > 0 A Flat Interface Has R = ∞ Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part a) consider the fish to be the light source, and calculate the image position for light rays exiting the bowl. We will be using this formula: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part a) consider the fish to be the light source, and calculate the image position for light rays exiting the bowl. We will be using this formula: Here is the given information: This radius is negative because the center of the bowl is on the same side as the light source (the fish) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part a) consider the fish to be the light source, and calculate the image position for light rays exiting the bowl. We will be using this formula: Here is the given information: This radius is negative because the center of the bowl is on the same side as the light source (the fish) A negative value for S’ means the image is on the same side of the interface as the object (i. e. inside the bowl in this case). So the observer will see a virtual image of the fish at the center of the bowl. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part a) consider the fish to be the light source, and calculate the image position for light rays exiting the bowl. Magnification can be found from this formula: We will be using this formula: Here is the given information: This radius is negative because the center of the bowl is on the same side as the light source (the fish) A negative value for S’ means the image is on the same side of the interface as the object (i. e. inside the bowl in this case). So the observer will see a virtual image of the fish at the center of the bowl. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part a) consider the fish to be the light source, and calculate the image position for light rays exiting the bowl. Magnification can be found from this formula: We will be using this formula: Here is the given information: The fish appears larger by a factor of 1. 33 This radius is negative because the center of the bowl is on the same side as the light source (the fish) A negative value for S’ means the image is on the same side of the interface as the object (i. e. inside the bowl in this case). So the observer will see a virtual image of the fish at the center of the bowl. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part b) the light source is the sun, which is really far away (i. e. object distance is infinity). This is what they mean by “parallel rays from the sun”. The focal point will be where the sun’s rays converge, so we need to find the sunlight image distance di. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part b) the light source is the sun, which is really far away (i. e. object distance is infinity). This is what they mean by “parallel rays from the sun”. The focal point will be where the sun’s rays converge, so we need to find the sunlight image distance di. Our given information becomes: This radius is positive because the center of the bowl is on the opposite side as the light source (the sun) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? For part b) the light source is the sun, which is really far away (i. e. object distance is infinity). This is what they mean by “parallel rays from the sun”. The focal point will be where the sun’s rays converge, so we need to find the sunlight image distance di. Our given information becomes: Image#1 This radius is positive because the center of the bowl is on the opposite side as the light source (the sun) This would be the focal point if the light stayed in the water until it got there. Since the light rays travel back into the air at the other side of the bowl, we will have to do a second calculation to find the final position of the image. The first image becomes the object for the interface at the right side of the. Prepared bowl. by Vince Zaccone For Campus Learning Assistance Services at UCSB

Problem 34. 17 A small tropical fish is at the center of a water-filled (n=1. 33) spherical fishbowl 28 cm in diameter. a) Find the apparent position and magnification of the fish to an observer outside the bowl. b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl? The first calculation gave us an image position to the right of the bowl, a distance of 28 cm to the right. Thus the new object distance will be do=-28 cm. The values we know for the 2 nd calculation are: sunlight Image #2 This radius is negative because the center of the bowl is on incoming light side (inside the bowl) This final image is 14 cm to the right of the bowl, and is the focal point of the “lens” formed by the bowl of water. Since this focal point is outside the bowl, our fish is safe. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Two Basic Types of Lenses CONVERGING Focal Point • f is positive • Thicker in middle • Object outside focal point = real image • Object inside focal point = virtual image DIVERGING • f is negative • Thinner in middle • Real object always gives a virtual image Focal Point

Ray Diagrams for Lenses There are 3 convenient rays to draw for a thin lens. The profile of the lens is just shown for reference – all the bending of the light occurs at the plane through the center of the lens.

THIN LENS EQUATIONS AND SIGN CONVENTION Surface 1 Light In Side do > 0 Real Object di < 0 Virtual Image C 1 This Side, R 1 < 0 C 2 This Side, R 2 < 0 Surface 2 Light Out Side do < 0 Virtual Object di > 0 Real Image C 1 This Side, R 1 > 0 C 2 This Side, R 2 > 0 n – Index of Refraction C 1 – Center of Curvature, Surface 1 C 2 – Center of Curvature, Surface 2 Illustrated Lens is Double Convex Converging With C 1 on the Light Out Side and C 2 on the Light In Side Equations: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. Radius=20 cm Radius=15 cm Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. To find the focal length we use thin lens equation: Radius=20 cm Radius=15 cm Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. To find the focal length we use thin lens equation: R 1=+20 cm R 2=+15 cm Light traveling this direction The difficult part is to get the signs correct for the radii. We can suppose the light is coming from the left, so the light encounters the 20 cm side first. Since the center of that 20 cm-radius circle is on the other side (where the light rays are going to end up) we call this radius positive – so R 1=+20 cm. Similarly, the 15 cm-radius circle has its center on the other side, so this is also positive: R 2=+15 cm Your basic rule of thumb is this: follow the light rays – they end up on the positive side. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. To find the focal length we use thin lens equation: R 1=+20 cm R 2=+15 cm Light traveling this direction For extra bonus fun, try calculating the focal length when the light comes from the other side – so the 15 cm radius is encountered first. You ought to get the same answer for the focal length. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm For part b) we can use the formula: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm di=-33. 3 cm For part b) we can use the formula: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm di=-33. 3 cm For part b) we can use the formula: The height of the image comes from our magnification formula: The image is virtual, upright, and 8 cm tall. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm di=-33. 3 cm For part b) we can use the formula: The height of the image comes from our magnification formula: The image is virtual, upright, and 8 cm tall. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm di=-33. 3 cm For part b) we can use the formula: The height of the image comes from our magnification formula: The image is virtual, upright, and 8 cm tall. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Sample Problem a) Find the focal length of the thin lens shown. The index of refraction is 1. 6. b) A 12 cm-tall object is placed 50 cm away from this lens. Find the image location and height. Is this image real or virtual? Draw the ray diagram. f=-100 cm do=+50 cm di=-33. 3 cm For part b) we can use the formula: The red ray in our diagram is initially headed for the focal point on the other side of the lens at x=+100 cm. The lens deflects it parallel to the axis, and we trace it back to find the image (at the intersection with the other 2 rays) The height of the image comes from our magnification formula: The image is virtual, upright, and 8 cm tall. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB