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Announcements  Today’s material WILL NOT be covered on Exam 3! Material for Exam 3: Chapters 8, 9, 10  Friday of this week is the last drop day.  Exam 3 is one week from yesterday. I will need to know by the end of lectures today of any students who have special needs different than for exam 2.

Homework Hints If a problem involving a plane mirror is assigned: R=  for a flat surface. If Problem is assigned: you are looking at the outside bottom of the bowl. If the syllabus says do a ray diagram, then do a ray diagram!

Today’s agenda: Plane Mirrors. You must be able to draw ray diagrams for plane mirrors, and be able to calculate image and object heights, distances, and magnifications. Spherical Mirrors: concave and convex mirrors. You must understand the differences between these two kinds of mirrors, be able to draw ray diagrams for both kinds of mirrors, and be able to solve the mirror equation for both kinds of mirrors.

Mirrors Plane mirrors form virtual images; no light actually comes from the image. The solid red rays show the actual light path after reflection; the dashed black rays show the perceived light path. Images Formed by Plane Mirrors

*The object distance and image distance are equal: s=-s’. The object height and image height are equal: y=y’. The magnification of a plane mirror is therefore one. ss’ y’y The image is upright and virtual. The image is reversed front-to-back relative to the object. *The – sign is needed because of sign conventions that go with the mirror equation—see later. These do not appear to be on your starting equation sheet, but they really are! Just use the mirror equation (coming soon) with f= .

ss’ Example: how tall must a full-length mirror be? A light ray from the top of your head reflects directly back from the top of the mirror.

ss’ y/2    To reach your eye, a light ray from your foot must reflect halfway up the mirror (because  I =  R =  ).

ss’ y/2    The mirror needs to be only half as tall as you. This calculation assumed your eyes are at the top of your head.

Example: where is the image located (top view)?

Example: distant object in a small mirror. Find some similar triangles! (Only one is fully shown in the figure.) The last time I tried to do this, the object was too big to “fit in” the mirror, and my head got in the way.

Today’s agenda: Plane Mirrors. You must be able to draw ray diagrams for plane mirrors, and be able to calculate image and object heights, distances, and magnifications. Spherical Mirrors: concave and convex mirrors. You must understand the differences between these two kinds of mirrors, be able to draw ray diagrams for both kinds of mirrors, and be able to solve the mirror equation for both kinds of mirrors.

Images Formed by Spherical Mirrors Spherical mirrors are made from polished sections cut from a spherical surface. The center of curvature, C, is the center of the sphere, of which the mirror is a section. Of course, you don’t really make these mirrors by cutting out part of a sphere of glass. C

The radius of curvature, R, is the radius of the sphere, or the distance from V to C. CV R The center of the mirror is often called the vertex of the mirror.

The principal axis (or optical axis) is the line that passes through the center of curvature and the center of the mirror. CV R Principal Axis The center of the mirror is often called the vertex of the mirror.

Paraxial rays are parallel to the principal axis of the mirror (from an object infinitely far away). Reflected paraxial rays pass through a common point known as the focal point F. CV F

The focal length f is the distance from P to F. Your text shows that f = R/2. CP F f R

Reality check: paraxial rays don’t really pass exactly through the focal point of a spherical mirror (“spherical aberration”). CV F

If the mirror is small compared to its radius of curvature, or the object being imaged is close to the principal axis, then the rays essentially all focus at a single point. CV F We will assume mirrors with large radii of curvature and objects close to the principal axis.

In “real life” you would minimize aberration by using a parabolic mirror. CV F

Today’s agenda: Plane Mirrors. You must be able to draw ray diagrams for plane mirrors, and be able to calculate image and object heights, distances, and magnifications. Spherical Mirrors: concave and convex mirrors. You must understand the differences between these two kinds of mirrors, be able to draw ray diagrams for both kinds of mirrors, and be able to solve the lens equation for both kinds of mirrors.

Concave and Convex Mirrors There are two kinds of spherical mirrors: concave and convex. F concaveconvex

Ray Diagrams for Mirrors We can use three “principal rays” to construct images. In this example, the object is “outside” of F. Ray 1 is parallel to the axis and reflects through F. Ray 2 passes through F before reflecting parallel to the axis. Ray 3 passes through C and reflects back on itself. F C A fourth principal ray is the one directed at the vertex V. V

We use three “principal rays” to construct images. C An image is formed where the rays converge. The image from a concave mirror, object outside the focal point, is real, inverted, and smaller than the object. “Real” image: you could put a camera there and detect the image. F Ray Diagrams for Concave Mirrors Two rays would be enough to show us where the image is. We include the third ray for “safety.” You don’t have to use principal rays, but they are easiest to trace.

C The image from a concave mirror, object inside the focal point, is virtual, upright, and larger than the object. Ray 1: parallel to the axis then through F. Ray 2: “through” F then parallel to the axis. Ray 3: “through” C. F With this size object, there was a bit of spherical aberration present, and I had to “cheat” my C a bit to the left to make the diagram look “nice.” What do you mean, this is a virtual image? I can see it!

C You could show that if an object is placed at the focal point, reflected rays all emerge parallel, and *no image is formed. Ray 1: parallel to the axis then through F. Ray 2: “through” F then parallel to the axis. Can’t do! Ray 3: through C. F no image *Actually, the image is formed at infinity. Worth thinking about: what if the object is placed between F and C?

With a bit of geometry, you can show that C The magnification is the ratio of the image to the object height: F The Mirror Equation f s s’ y y’ f=  for a flat mirror.

Sign conventions for the mirror equation: C F f s s’ y y’ When the object, image, or focal point is on the reflecting side of the mirror, the distance is positive. When the object, image, or focal point is “behind” the mirror, the distance is negative. The image height is positive if the image is upright, and negative if the image is inverted relative to the object.

“When the object, image, or focal point is “behind” the mirror, the distance is negative.” If the image distance is negative, can the image be real?

Handy quick reference card from Dr. Hale:

Example: a dime (height is 1.8 cm) is placed 100 cm away from a concave mirror. The image height is 0.9 cm and the image is inverted. What is the focal length of the mirror. y=1.8 y’=-0.9 s=100

Example: a dime (height is 1.8 cm) is placed 100 cm away from a concave mirror. The image height is 0.9 cm and the image is inverted. What is the focal length of the mirror. C F f s s’ y y’ s’, s, or f on reflecting side are + y is – if image is inverted

Applications of concave mirrors. Shaving mirrors. Makeup mirrors. Solar cookers. Ant fryers. Flashlights, headlamps, stove reflectors. Satellite dishes (when used with electromagnetic radiation). Kid scarers.

Today’s agenda: Plane Mirrors. You must be able to draw ray diagrams for plane mirrors, and be able to calculate image and object heights, distances, and magnifications. Spherical Mirrors: concave and convex mirrors. You must understand the differences between these two kinds of mirrors, be able to draw ray diagrams for both kinds of mirrors, and be able to solve the lens equation for both kinds of mirrors.

C F Ray Diagrams for Convex Mirrors Ray 1: parallel to the axis then “from” F. Ray 2: “through” Vertex. Ray 3: “from” C. The image is virtual, upright, and smaller than the object.

C F Instead of sending ray 2 “through” V, we could have sent it “through” F. The ray is reflected parallel to the principal axis. Your text talks about all four of the “principal rays” we have used.

C F Because they are on the “other” side of the mirror from the object, s’ and f are negative. The mirror equation still works for convex mirrors. s f s’ y y’

The ray diagram looks like the one on the previous slide, but with the object much further away (difficult to draw). On reflecting side  positive. Not on reflecting side  negative. Example: a convex rearview car mirror has a radius of curvature of 40 cm. Determine the location of the image and its magnification for an object 10 m from the mirror.

…algebra… Remind me… what does it say on passenger side rear view mirrors?

Applications of convex mirrors. Passenger side rear-view mirrors. Grocery store aisle mirrors. Anti-shoplifting (surveillance) mirrors. Christmas tree ornaments. Railroad crossing mirrors.

Sign Conventions Introduced Today When the object, image, or focal point is on the reflecting side of the mirror, the distance is positive. When the object, image, or focal point is “behind” the mirror, the distance is negative. The image height is positive if the image is upright, and negative if the image is inverted relative to the object. Hint: download Dr. Hale’s quick reference card.quick reference card

Summary of Sign Conventions Object Distance. When the object is on the same side as the incoming light, the object distance is positive (otherwise is negative). Here’s a compact way of expressing mirror and lens (coming soon) sign conventions all at once. Image Distance. When the image is on the same side as the outgoing light, the image distance is positive (otherwise is negative). (negative image distance  virtual image) Radius of Curvature. When the center of curvature C is on the same side as the outgoing light, R is positive (otherwise is negative). (positive image distance  real image)

I really don’t understand these ray diagrams! If, like many students, you don’t understand ray diagrams, the following supplementary (will not be covered in lecture) slides may help.

For a concave lens, the center of curvature and focal point are on the same side of the lens as the object. C F Concave Mirror

Light coming from the object parallel to the axis will always reflect through F. C F Concave Mirror

Light coming from the object and passing through F before it hits the mirror will always reflect parallel to the axis. C F Concave Mirror

Light coming from the object and passing through C before it hits the mirror will always reflect back on itself. C F Concave Mirror

Light coming from the object and striking the vertex will reflect off with an outgoing angle equal to the incoming angle. This is often more difficult to draw (unless you measure the angle). C   Concave Mirror

C F For a convex lens, the center of curvature and focal point are on the opposite side of the lens as the object. Light from the object will never actually pass through C or F. Convex Mirror

C F Light coming from the object parallel to the axis will always reflect back as if it had come from F. Follow the path of the light back “through” the mirror to see where it appears to have come from. Convex Mirror

C F Light coming from the object and directed at F will always reflect back parallel to the axis. Follow the path of the light back “through” the mirror to see where it appears to have come from. Convex Mirror

C F Light coming from the object and directed at C will always reflect back on itself. Follow the path of the light back “through” the mirror to see where it appears to have come from. Convex Mirror

C F Light coming from the object and striking the vertex will reflect back with an outgoing angle equal to the incoming angle. This is often more difficult to draw (unless you measure the angle). Follow the path of the light back “through” the mirror to see where it appears to have come from.   Convex Mirror