THE FOCAL LENGTH OF SPHERICAL MIRRORS
Section 1: The Focal length of a concave mirror
Section 2: The focal length of a convex mirror
24 Oct. 11
Section 1: To determine the focal length of a Concave Mirror by locating the centre of curvature. -------------------------------------------------
In this paper we want to discuss the focal point of a concave mirror by locating the centre of curvature. The focal point is a point in space at which light incident towards the mirror and travelling parallel to the principal axis will meet after reflection. The diagram at the right depicts this principle. In fact, if some light from the sun were collected by a concave mirror, then it would converge at the focal point! When a ray of light is incident on a polished surface, it is reflected in such a way that the angle of reflection is equal to the angle of incidence. A ray through the centre of curvature of a concave mirror is reflected back through the same path. This is because it is at an angle of 90 degrees. The radius of curvature is the straight line from the centre of curvature, to the arc of the mirror. The focal length of a concave mirror is the length between the focal point, where parallel rays converge after reflection, and the centre of the mirror. Focal length, object distance and image distance are all related by the equation 1/f = 1/u + 1/v (where u and v are object and image distance respectively). As the object distance increases, the image distance approaches the value of the focal length (f). The image distance when the object distance equals the radius of curvature; where 2f = R. 2/R = 1/R + 1/v
1/R (2-1) = 1/v
1/R = 1/v
R = v (image distance)
Proving that the image distance is equal to radius of curvature.
We see an object because light from the object travels to our eyes as we sight along a line at the object. Similarly, we see an image of an object because light from the object reflects off a mirror and travel to our eyes as we sight at the image location of the object. From these two basic premises, we want to define the image location as the location in space where light appears to diverge from. Ray diagrams have been a valuable tool for determining the path taken by light from the object to the mirror to our eyes. The two rules of reflection for concave mirrors: * Any incident ray traveling parallel to the principal axis on the way to the mirror will pass through the focal point upon reflection. * Any incident ray passing through the focal point on the way to the mirror will travel parallel to the principal axis upon reflection.
If a concave mirror were thought of as being a slice of a sphere, then there would be a line passing through the centre of the sphere and attaching to the mirror in the exact centre of the mirror. This line is known as the principal axis. The point in the centre of the sphere from which the mirror were sliced is known as the centre of curvature and is denoted by the letter C in the diagram below. The point on the mirror's surface where the principal axis meets the mirror is known as the vertex and is denoted by the letter A in the diagram below. The vertex is the geometric centre of the mirror. Midway between the vertex and the centre of curvature is a point known as the focal point; the focal point is denoted by the letter F in the diagram below. The distance from the vertex to the centre of curvature is known as the radius of curvature (represented by R). The radius of curvature is the radius of the sphere from which the mirror was cut. Finally, the distance from the mirror to the focal point is known as the focal length (represented by f). Since the focal point is the midpoint of the line segment adjoining the vertex and the centre of curvature, the focal length would be one-half the radius of curvature. (Mossin, 2005)
To be able to figure out how an image will be formed in one of these...
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