Step 7. Figure $$\PageIndex{4}$$ shows a single ray that is reflected by a spherical concave mirror. Answer: Radius of curvature (R) = 20 cm. 1 do + 1 di = 2 R. No approximation is required for this result, so it is exact. i Example $$\PageIndex{1}$$: Solar Electric Generating System. The area for a length of 1.00 m is then, \begin{align*} A&=\dfrac{\pi}{2}R(1.00m) \\[4pt] &=\dfrac{(3.14)}{2}(0.800\,m)(1.00\,m) \\[4pt] &=1.26\,m^2. Recall that the small-angle approximation holds for spherical mirrors that are small compared to their radius. Equation \ref{eq61} in fact describes the linear magnification (often simply called “magnification”) of the image in terms of the object and image distances. A ray traveling along a line that goes through the focal point of a spherical mirror is reflected along a line parallel to the optical axis of the mirror (ray 2 in Figure $$\PageIndex{5}$$). To do so, we draw rays from point $$Q$$ that is on the object but not on the optical axis. If the vertex lies to the right of the center of curvature, the radius of curvature is negative. ). Thus, these rays are not focused at the same point as rays that are near the optical axis, as shown in the figure. 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. The principal axis is a line that is perpendicular to the center of the mirror. Radius of curvature (ROC) has specific meaning and sign convention in optical design. The vertex of the lens surface is located on the local optical axis. The distance from the pole to the focal point is called the focal length ( f ). Such distortion is called aberration. For this mirror, the reflected rays do not cross at the same point, so the mirror does not have a well-defined focal point. For example, we show, as a later exercise, that an object placed between a concave mirror and its focal point leads to a virtual image that is upright and larger than the object. Here we briefly discuss two specific types of aberrations: spherical aberration and coma. Step 3. If $$|m|>1$$, the image is larger than the object, and if $$|m|<1$$, the image is smaller than the object. If the light source is 12 cm from the cornea and the image magnification is 0.032, what is the radius of curvature of the cornea? Coma is similar to spherical aberration, but arises when the incoming rays are not parallel to the optical axis, as shown in Figure $$\PageIndex{8b}$$. By the end of this section, you will be able to: 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. Combining ray tracing with the mirror equation is a good way to analyze mirror systems. A ray traveling along a line that goes through the center of curvature of a spherical mirror is reflected back along the same line (ray 3 in Figure $$\PageIndex{5}$$). We will discuss both coma and spherical aberration later in this chapter, in connection with telescopes. ) f= R/2. Symmetry is one of the major hallmarks of many optical devices, including mirrors and lenses. An analogous scenario holds for the angles $$θ$$ and $$θ′$$. The radius of curvature found here is reasonable for a cornea. Part (a) is related to the optics of spherical mirrors. Related Questions & Answers. Figure $$\PageIndex{7}$$ shows such a working system in southern California. The distance from cornea to retina in an adult eye is about 2.0 cm. The distance between the center of curvature and pole of a spherical mirror is called radius of curvature. The desired image distance is $$d_i=40.0\,cm$$. If you find the focal length of the convex mirror formed by the cornea, then you know its radius of curvature (it’s twice the focal length). In this case, their angles $$θ$$ of reflection are small angles, so, \[\sin θ≈ \tan θ≈ θ. Have questions or comments? {\displaystyle K} The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Do the signs of object distance, image distance, and focal length correspond with what is expected from ray tracing? Step 6. {\displaystyle K} ( \begin{array}{rcl} \tanϕ=\dfrac{h_o}{d_o-R} \\ \tanϕ′=−\tanϕ=\dfrac{h_i}{R-d_i} \end{array}\right\} =\dfrac{h_o}{d_o-R}=−\dfrac{h_i}{R-d_i}, $−\dfrac{h_o}{h_i}=\dfrac{d_o-R}{R-d_i}. \label{eq61}$. As in the case of lenses, the cartesian sign convention is used here, and that is the origin of the negative sign above. Also, the real image formed by the concave mirror in Figure $$\PageIndex{6}$$ is on the opposite side of the optical axis with respect to the object. The radius r for a concave mirror is a negative quantity (going left from the surface), and this gives a positive focal length, implying convergence. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. However, as discussed above, in the small-angle approximation, the focal length of a spherical mirror is one-half the radius of curvature of the mirror, or $$f=R/2$$. \label{smallangle}\]. Missed the LibreFest? The vertex of the lens surface is located on the local optical axis. This is called spherical aberration and results in a blurred image of an extended object. In this case, the image height should have the opposite sign of the object height. The rules for ray tracing are summarized here for reference: We use ray tracing to illustrate how images are formed by mirrors and to obtain numerical information about optical properties of the mirror. Example $$\PageIndex{2}$$: Image in a Convex Mirror. The distance along the optical axis from the mirror to the focal point is called the focal length of the mirror. Understanding the sign convention allows you to describe an image without constructing a ray diagram. This heated fluid is pumped to a heat exchanger, where the thermal energy is transferred to another system that is used to generate steam and eventually generates electricity through a conventional steam cycle. Principal ray 1 goes from point $$Q$$ and travels parallel to the optical axis. The focal length of a spherical mirror is then approximately half its radius of curvature. \nonumber \], c. The increase in temperature is given by $$Q=mcΔT$$. [3] Care should be taken when using formulas taken from different sources. A ray that strikes the vertex of a spherical mirror is reflected symmetrically about the optical axis of the mirror (ray 4 in Figure $$\PageIndex{5}$$). When this approximation is violated, then the image created by a spherical mirror becomes distorted. An array of such pipes in the California desert can provide a thermal output of 250 MW on a sunny day, with fluids reaching temperatures as high as 400°C. K The image, however, is below the optical axis, so the image height is negative. Assume that all solar radiation incident on the reflector is absorbed by the pipe, and that the fluid is mineral oil. For the convex mirror, the extended image forms between the focal point and the mirror. 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.