![]() To find the location of an image formed by a spherical mirror, we first use ray tracing, which is the technique of drawing rays and using the law of reflection to determine the reflected rays (later, for lenses, we use the law of refraction to determine refracted rays). In this case, their angles θ θ of reflection are small angles, so sin θ ≈ tan θ ≈ θ sin θ ≈ tan θ ≈ θ. In this approximation, all rays are paraxial rays, which means that they make a small angle with the optical axis and are at a distance much less than the radius of curvature from the optical axis. In this chapter, we assume that the small-angle approximation (also called the paraxial approximation) is always valid. Inserting this into the equation for the radius R, we get If the angle θ θ is small (so that sin θ ≈ θ sin θ ≈ θ this is called the “small-angle approximation”), then F X ≈ F P F X ≈ F P or C F ≈ F P C F ≈ F P. Thus, triangle CXF is an isosceles triangle with C F = F X C F = F X. The law of reflection tells us that angles OXC and CXF are the same, and because the incident ray is parallel to the optical axis, angles OXC and XCP are also the same. We want to find how the focal length FP (denoted by f) relates to the radius of curvature of the mirror, R, whose length is R = C F + F P R = C F + F P. Note that all incident rays that are parallel to the optical axis are reflected through the focal point-we only show one ray for simplicity. The point at which the reflected ray crosses the optical axis is the focal point. The incident ray is parallel to the optical axis. How does the focal length of a mirror relate to the mirror’s radius of curvature? Figure 2.8 shows a single ray that is reflected by a spherical concave mirror. (credit b: modification of work by Jenny Downing) (b) Photograph of a virtual image formed by a convex mirror. The focal point is virtual because no real rays pass through it. The distance along the optical axis from the mirror to the focal point is called the focal length of the mirror.įigure 2.7 (a) Rays reflected by a convex spherical mirror: Incident rays of light parallel to the optical axis are reflected from a convex spherical mirror and seem to originate from a well-defined focal point at focal distance f on the opposite side of the mirror. This mirror is a good approximation of a parabolic mirror, so rays that arrive parallel to the optical axis are reflected to a well-defined focal point. ![]() Part (c) shows a spherical mirror that is small compared to its radius of curvature. This is called spherical aberration and results in a blurred image of an extended object. For this mirror, the reflected rays do not cross at the same point, so the mirror does not have a well-defined focal point. Part (b) of this figure shows a spherical mirror that is large compared with its radius of curvature. Following the law of reflection, these rays are reflected so that they converge at a point, called the focal point. A concave mirror has silvering on the interior surface (think “cave”), and a convex mirror has silvering on the exterior surface.Ĭonsider rays that are parallel to the optical axis of a parabolic mirror, as shown in part (a) of Figure 2.6. For a spherical mirror, the optical axis passes through the mirror’s center of curvature and the mirror’s vertex, as shown in Figure 2.5.įigure 2.5 A spherical mirror is formed by cutting out a piece of a sphere and silvering either the inside or outside surface. The symmetry axis of such optical elements is often called the principal axis or optical axis. Symmetry is one of the major hallmarks of many optical devices, including mirrors and lenses. ![]() If the inside surface is the reflecting surface, it is called a concave mirror. If the reflecting surface is the outer side of the sphere, the mirror is called a convex mirror. We can define two general types of spherical mirrors. We will concentrate on spherical mirrors for the most part, because they are easier to manufacture than mirrors such as parabolic mirrors and so are more common. In general, any curved surface will form an image, although some images may be so distorted as to be unrecognizable (think of fun house mirrors).īecause curved mirrors can create such a rich variety of images, they are used in many optical devices that find many uses. A curved mirror, on the other hand, can form images that may be larger or smaller than the object and may form either in front of the mirror or behind it. The image in a plane mirror has the same size as the object, is upright, and is the same distance behind the mirror as the object is in front of the mirror.
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