Spherical Mirrors
Optical Bench	Light Source
Mirror (concave)	Mirror (convex)
Screen

Purpose: to examine the geometry involved in the forming of virtual and real images in spherical mirrors.

Theory

	A spherical mirror is a portion of the surface of a hollow sphere as shown in figure 1.
	If the sphere reflects from its inner surface, it will reflect light as shown in Figure 2. Such a mirror is called a concave spherical mirror.
	If the sphere reflects light from its outer surface, it will reflect light as shown in Figure 3. This type of mirror is called a convex spherical mirror.
Unlike the images formed by plane mirrors, the spherical mirrors produce images of bigger or smaller size. This is due to the converging and diverging of the light that falls on these mirrors. Most of the telescopes are made with concave mirrors. These mirrors also find a great many uses in optical instruments.

Definitions

Study the following definitions which are illustrated in preceding Figures.

Center of Curvature (C) -- The center of the sphere, of which the spherical mirror is a part, is known as the center of curvature.

Radius of Curvature (R) -- The radius of the sphere, of which the mirror is a part, is known as the radius of curvature.

Principal Axis (PA) -- The principal axis is the line passing through the center of the mirror and is perpendicular to the surface. It is also the axis of symmetry.

Focal Point and Focal Length -- If a ray of light, which is parallel to the principal axis (PA), is incident on a concave mirror (Fig. 2), it will be reflected through point F which is known as the focal point. The distance, OF, is known as the focal length of the concave mirror. The focal length is half the radius of curvature (R)

If a ray of light, which is parallel to the principal axis (PA), is incident on a convex mirror (Fig 3), the ray will be reflected away from the axis and appear to emerge from point F, which is the focal point. The distance is the focal length.

Virtual and Real Focus -- In the case of a concave mirror, the rays of light converge and pass through F. This point is known as a real focus. In the case of the diverging mirror the parallel rays of light appear to diverge from F and it is known as a virtual focus.

Object Distance (d_o) -- The distance from the center of the mirror to the position of the object is known as the object distance.

Image Distance (d_i) The distance from the center of the mirror to the position of the image is known as the image distance.

Magnification (M) -- Magnification is defined as the height of the image (h_i) divided by the height of the object (h_o).

Real Image -- At a real image, the light actually passes through the point to reproduce the object.

Virtual Image -- At a virtual image, the light appears to emerge from the position of the image.

Locating Images

We can use the law of reflection, (angle of incidence = angle of reflection), to locate the images that mirrors form. We can use the following rays to locate the image.

Concave Mirror

	1.	If the ray goes parallel to the axis, then it will be reflected through the focal point.
	2.	If the ray goes through the focal point, it is reflected parallel to the optical axis.
	3	If the ray enters along the radius of curvature, then it is reflected back along the same line.

Convex Mirror

	1.	If the ray goes parallel to the axis, then it will be reflected as to appear that it came from the focal point.
	2.	If the ray would go through the focal point (if it kept going), it would be reflected parallel to the optical axis.
	3.	If the ray enters along the radius of curvature, then it is reflected back along the same line.

We can obtain an equation which relates the focal length (f), the object distance (d_o) and the image distance (d_i).

However, in using this equation one must pay attention to the sign (positive or negative) of the various quantities involved. This is due to the fact that real images are formed in front of the mirror and virtual images are formed behind the mirror.

Sign Convention

1.	Object distances (d_o) are positive if the object is on the reflecting side of the mirror.
2.	Image distances (d_i) are positive if the image is on the reflecting side of the mirror.
3.	The focal length of a concave mirror is positive, and it is negative for a convex mirror.

Procedure

A. Finding the focal length of a concave mirror.

1.	Place a blank sheet of white paper on the table.
2.	Place the light box and the concave mirror on the sheet of paper.
3.	Adjust the number of beams from the light box to 4.
4.	Arrange the light box and the concave mirror so that the beams are parallel to the principle optical axis of the mirror.
5.	With a pencil, plot a point at each end of all four original beams and their reflections.
6.	Draw in all eight beams and the mirror.
7.	Then measure the focal length

B. Finding the focal length of a convex mirror.

1.	Place a blank sheet of white paper on the table.
2.	Place the light box and the convex mirror on the sheet of paper.
3.	Adjust the number of beams from the light box to 4.
4.	Arrange the light box and the convex mirror so that the beams are parallel to the principle optical axis of the mirror.
5.	With a pencil, plot a point at each end of all four original beams and their reflections.
6.	Draw in all eight beams and the mirror.
7.	Extend the reflections in order to locate the virtual focal point.
8.	Then measure the focal length.

C. Observing your image in a concave mirror

1.	Hold the concave mirror (with the black plastic holder) at arms length and locate your image in the mirror.
2.	Slowly bring the mirror in until it is very close to your eye.
3.	Be able to describe any changes in the image.

D. Observing your image in a convex mirror.

1.	Hold the convex mirror (with the black plastic holder) at arms length and locate your image in the mirror.
2.	Slowly bring the mirror in until it is very close to your eye.
3.	Be able to describe any changes in the image.

E. Locating the image between the object and the concave mirror.

1.	Locate the image with the screen when the mirror is at the following locations: 15, 20, 25, 30, 35 and 40 cm.
2.	Complete the data table and the implied calculations.

F. Viewing the image when d_i > d_o.

1.	Use a white sheet of paper instead of the little screen in the holder.
2.	By adjust the angle of the light source and the lens holder, you should be able to project the enlarged image on the sheet of paper above and behind the light source.
3.	Can you verify that the image distance and the object distance are the same for d_o = 2f?

Laboratory Report

Data Table and Calculations

Object Distance cm	Observed Image Distance cm	Focal Length cm
15		calc
20		calc
25		calc
30		calc
35		calc
40		calc
Average Experimental Focal Length		calc
Focal Length Printed on the holder		_______
% Error		calc

Questions

1.	What is the focal length in procedure A?
2.	What is the focal length in procedure B?
3.	Describe and interpret what you saw in procedure C.
4.	Describe and interpret what you saw in procedure D.
5.	Give the full range of possible object distances for the condition in procedure E.
6.	Give the full range of possible object distances for the condition in procedure F.
7.	Why can't you screen the image for object distances less than the focal length in procedure F?

last updated April 08, 2002