Spherical Mirrors

When a mirror has a curved surface, the image reflected from the mirror takes on some different properties. The law of reflection still applies: the angle of incidence for a ray of light equals the angle of reflection as measured from the normal. But in a curved mirror, the normal line is changing along the face of the mirror. We will look at two examples: Concave mirrors and convex mirrors.

A concave mirror is a mirror that curves inward as you look at it. A line drawn though the center of the mirror is called the principal axis. Two important points lie on this axis. The first is the center of curvature (C). The distance from C to any point along the mirror is the radius of curvature of the mirror. If you consider the mirror to be part of the surface of a sphere, the center of the sphere would be at C. The second point that is important is the focal point (F). The focal point is defined as the point of intersection of all rays that arrive at the mirror from an infinite distance (i.e. they are all parallel to the principal axis.) The distance from the focal point to the mirror along the principal axis is called the focal length (f). The focal length is one half of the radius of curvature, or

The image that is reflected from a concave mirror is a real image. It can be projected onto a screen. If you hold a piece of paper up to the image point on a concave mirror, you will see the image formed on the paper.

We use ray tracing to determine to location of images for spherical mirrors. Let's start with rays for a concave mirror. Three rays can be used to define the location of the mirror.

  1. Ray 1 passes from the object, parallel to the principal axis, and is reflected back through the focal point.
  2. Ray 2 passes from the object through the focal point, and is reflected back parallel to the principal axis.
  3. Ray 3 passes from the center of curvature, C, through the object and is reflected back along that same line.

The distance from the object to the mirror is do. The distance from the image to the mirror is di. The height of the object is given by ho, and the height of the object is hi. Algebraically, we can find the location of our image by using the mirror equation:

Lateral Magnification (m) relates the height of the image to the height of the object. The equation for magnification is:

When magnification is negative, it means that the image is inverted with respect to the object. Whether or not the image is larger or smaller, or inverted or not, depends on the location of the image with respect to the C and F. Although the ray tracing remains the same, each case provides a different solution:

When the Object is beyond the Center of Curvature, the image is Real, Inverted, Smaller

When the Object is between Center of Curvature and the Focal Point, the image is ambiguous. If the object is on the focal point, the image will "focus" at an infinite location and will not be seen. (The rays leave roughly parallel to the mirror and will not converge.)

When the Object is inside the Focal Point, the Image is Virtual, Upright, and Larger

A convex mirror is a mirror that is curved outward on the reflective surface. Here, as far as the viewer is concerned, the center of curvature and the focal point are both on the other side of the mirror. A convex mirror will yield a virtual, upright image that is smaller than the object - such mirrors include the mirror on the passenger side of your car and the security mirrors used in stores.

Ray tracing for convex mirrors is a little different and only two rays are required. The rays to draw are:

- Ray 1 extends from the object parallel to the principal axis and appears to pass through the focal point.

- Ray 2 passes from the object through the center of curvature and is reflected back along that path.

The equations for determining the characteristics of the image are the same as for a concave mirror. No matter what the location of the object is with respect to the mirror, the image will always be upright, virtual, and smaller in a concave mirror. Sign convention is important to maintain track of the characteristics of our image.  When the object, image, or focal point is on the reflecting side of the mirror, the corresponding distance is positive. If any of these move to the other side, it becomes negative.  The image height is positive if the image is upright with
respect to the object. If it is inverted,, the image height is negative. 

 A parabolic mirror is a special mirror with a small diameter that is shaped so that all light rays form exactly at the focal point. This takes care of aberrations, or a characteristic of spherical mirrors that causes light rays from far distances to focus at points slightly different from the focal point, causing a blurred image. Reflecting telescopes use parabolic mirrors.

Also try:

http://www.physicsclassroom.com/Class/refln/u13l3c.cfm (concave mirrors)

http://www.physicsclassroom.com/Class/refln/U13L4a.html (convex mirrors)

For Practice Problems, Try:Giancoli Multiple Choice PracticeQuestions (Questions 1-14)