Spherical Mirrors Printer Friendly Version
To create a spherical mirror envision a large, silver, mylar beach ball - silly, I know, but stay with me here - that is reflective on both its inside and outside surfaces. After applying the reflective film, the ball hardens. Now, take a knife and cut out a section of the ball's surface. That section is a spherical mirror. If you hold the mirror so that you look into its outer, convex surface, then it will be a convex mirror and the only images you will see will those "trapped inside (behind) the mirror," that is: virtual, reduced, upright images. I call this a "shoplifting" mirror. You see them in the stores located at upper rear corners so that store managers can see what customers are doing in between tall aisle dividers. You may easily experience this by looking into the convex side of a serving spoon, your image is virtual, upright and reduced in size. These mirrors are considered to be negative mirrors since their mirrored surface faces "away from the center of the sphere (our mylar beach ball).

In the diagram shown below, the convex surface is the front of the mirror. The radius of the "mylar beach ball" is behind the mirror and is represented by the length of the line VC where V is the vertex of our convex mirror and C is the center of our ball. The focal length, f, is half of the radius, or the length of the line VFF is called the focal point.

Back to our mylar, beach ball example. I want you to now hold up the mirrored segment so that you look into its inner, concave surface. That will be a concave mirror and you can see a variety of images depending on how far you stand from the mirror. If the light rays, once reflected off the mirror, converge back together, then the image will be real. It is called a real image because the actual, real, rays of light form the image. Real images can be projected onto screens. (Although not formed by mirrors, but by a lens, the image produced by an overhead projector is a real image that is projected onto the video screen for everyone to view.)  Real images have the property that they are always inverted and left-right reversed. (You may have experienced this if you have ever placed slides into a slide projector. You quickly learned that the slides have to be inverted and flipped left-to-right so that the projected image is seen correctly - again an example of a real image produced by a lens, but the same principle applies to real images produced by mirrors. They are inverted and reversed left-to-right.)

In the diagram shown below, the concave surface is the front of the mirror. The radius of the "mylar beach ball" is in front of the mirror and is represented by the length the length of the line CV where C is the center of the ball and V is the vertex of our concave mirror. The focal length, f, is half of the radius, or the length of the line FVF is called the focal point.

When you were drawing mirrors, you take a compass and sweep out an arc. You then bisect the distance from the center of the mirror to the vertex, that is the radius of the mirror, to find the position of the focus, F, so that CF = FV.

Open the following physlet and activate a mirror and a source. Notice that when the source is placed at the center of the mirror, the rays cross when they reflect from the mirror; also notice if the source can be placed directly on the focus, the rays reflect parallel to each other and to the axis.

Converging (Concave) Mirrors

There are three primary rays which are used to locate the images formed by converging mirrors. Each ray starts from the top of the object.

 Ray #1(pink) runs parallel to the axis until it reaches the mirror; then it reflects off the mirror and leaves along a path that passes through the mirror's focus Ray #2(gold) runs straight through the center of the mirror, reflects off the mirror, and reflects through the center, never bending Ray #3(aqua) first passes through the focal point until it reaches the mirror; then it reflects off the mirror and leaves parallel to the mirror's axis

Remember that ALL rays must have arrows! When all three rays meet, they will form the image. Also, the larger the aperture (or opening) of the mirror, the less aberration you will experience with your ray diagrams, that is, larger radii mirrors make better ray diagrams. In some textbooks, the mirror is actually represented by a vertical line labeled as a mirror.

Six Special Cases

In each of the following illustrations:

• Region I is greater than two focal lengths in front of the mirror.
• Region II is between one and two focal lengths in front of the mirror,
• Region III is within one focal length in front of the mirror; and, conversely
• Region IV is within one focal length behind the mirror,
• Region V is between one and two focal lengths behind the mirror, and
• Region VI is beyond two focal lengths behind the mirror.

Case #1: object located infinitely far away

Case #2: object is located in region 1

Case #3: object is located on the line between regions I and II, exactly two focal lengths in front of the mirror

Case #4: object is located in region II

Case #5: object is located on the line between regions II and III, exactly one focal length in front of the mirror

Case #6: object is located in region III

Take a moment and answer the following questions designed to test your conceptual understanding of the properties of the images formed by concave spherical mirrors. You might also like to reopen this physlet and activate a concave mirror and an object. Place the object at the far side of the optical bench and move it closer and closer towards the mirror. Notice what happens to the images that are formed.

 In which cases is a real image formed?

 On which side of the mirror are real images formed?

 Are real images formed by converging or diverging rays?

 In which case(s) is the image smaller than the object?

 In which case is a virtual image formed?

 On which side of the mirror are virtual images formed?

 Are virtual images formed by converging or diverging rays?

 Which type of image (real or virtual) is upright?

 Which type of image (real or virtual) could be called a "hot" image?

 In which case(s) is the image larger than the object?

 In which case is no image formed?

 Diverging (Convex) Mirrors       The same three rays are used to locate images formed by convex mirrors. Since F and C are located "behind the mirror" the rays must now be dotted back to form the image.   Ray #1 (pink) still starts on the top of the object, runs parallel to the axis until it strikes the mirror. Since F is now behind the mirror, the ray "pivots" to align with F and you draw in the reflected ray as a solid pink line and dot back its reflection to F.   Ray #2 (gold) still starts on the top of the object and runs straight towards the center of the mirror. Since C is now behind the mirror, the ray is reflected back on itself. You draw in the reflected ray as a solid gold line and dot back is reflection to C.   Ray #3 (aqua) starts on the top of the object, "aims" for F, strikes the mirror, and reflects parallel to the axis. You draw in the reflected ray as a solid aqua line and dot back is reflection.   All three reflected rays diverge. Their "dotted back reflections" always form virtual, upright, reduced images located between V and F.   Reopen the optics physlet one last time and activate a convex mirror and an object. Place the object at the far side of the optical bench and move it closer and closer towards the mirror. Notice what happens to the images that are formed. Also note that there is never a location for the object that does NOT result in the production of an image.

Summary

For concave (converging) mirrors, as long as the object is placed greater than one focal length in front of the mirror (our first four cases), a real image is produced. When the object is placed exactly one focal length (Case #5) in front of the mirror, no image is formed since the rays reflected from the mirror are parallel and can never intersect either in front of or behind the mirror. When the image is placed within one focal length of the mirror (Case #6), a virtual, enlarged image is formed when the reflected, diverging rays, are "dotted back" behind the mirror. Concave spherical mirrors are considered to be positive mirrors since their mirrored surface faces "towards the center of the sphere" (our mylar beach ball).

For convex (diverging) mirrors, no matter when the object is placed in front of the mirror, a virtual, upright, reduced image is formed "behind the mirror" between F and V. Convex spherical mirrors are considered to be negative mirrors since their mirrored surface faces "away from the center of the sphere" (our mylar beach ball).

Always remember, virtual images are formed by diverging rays; while real images are always formed by converging rays.

It is also important to be aware that mirrors can be classified according to the characteristics of the virtual images they form:

• plane mirrors: virtual images are the same size as their objects
• concave spherical mirrors: virtual images are larger that their objects
• convex spherical mirrors: virtual images are smaller than their objects