Related Questions & Answers. Hence, the focal length of the given spherical mirror is 10 cm. 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. Figure \(\PageIndex{2b}\) shows a spherical mirror that is large compared with its radius of curvature. In the problems and exercises, you will show that, for a fixed object distance, a smaller radius of curvature corresponds to a smaller the magnification. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). \label{mag}\]. First identify the physical principles involved. Figure \(\PageIndex{2c}\) shows a spherical mirror that is small compared to its radius of curvature. The sun is the object, so the object distance is essentially infinity: \(d_o=\infty\). In other words, in the small-angle approximation, the focal length \(f\) of a concave spherical mirror is half of its radius of curvature, \(R\): In this chapter, we assume that the small-angle approximation (also called the paraxial approximation) is always valid. These are the objects whose images we want to locate by ray tracing. For more information contact us at or check out our status page at If ray tracing is required, use the ray-tracing rules listed near the beginning of this section. In fact, we already used ray tracing above to locate the focal point of spherical mirrors, or the image distance of flat mirrors. A sketch is very useful even if ray tracing is not specifically required by the problem. The image, however, is below the optical axis, so the image height is negative. Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. Inserting this into Equation \ref{eq57} gives the mirror equation: \[\underbrace{ \dfrac{1}{d_o}+\dfrac{1}{d_i}=\dfrac{1}{f}}_{\text{mirror equation}}. f= R/2. For a plane mirror, the image distance has the opposite sign of the object distance. The reflection of this ray must pass through the focal point, as discussed above. In this case, all four principal rays run along the optical axis, reflect from the mirror, and then run back along the optical axis. First find the image distance \(d_i\) and then solve for the focal length \(f\). Radius of curvature (ROC) has specific meaning and sign convention in optical design. The focal length of a spherical mirror is then approximately half its radius of curvature. Determine whether ray tracing, the mirror equation, or both are required. Focal length is half of the radius of curvature. 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 \(\PageIndex{5}\)). Step 7. from the axis. The insolation is 900 W/m2. K Consider the object \(OP\) shown in Figure \(\PageIndex{6}\). PROBLEM-SOLVING STRATEGY: SPHERICAL MIRRORS. One of the solar technologies used today for generating electricity involves a device (called a parabolic trough or concentrating collector) that concentrates sunlight onto a blackened pipe that contains a fluid. For a spherical mirror, the optical axis passes through the mirror’s center of curvature and the mirror’s vertex, as shown in Figure \(\PageIndex{1}\). Use the examples as guides for using the mirror equation. In this case, spherical mirrors are good approximations of parabolic mirrors. Step 5. 2 For the convex mirror, the backward extension of the reflection of principal ray 1 goes through the focal point (i.e., a virtual focus). The four principal rays intersect at point \(Q′\), which is where the image of point \(Q\) is located. ( 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). The result is, \[ \begin{align*} \dfrac{1}{d_o}+\dfrac{1}{d_i} &= \dfrac{1}{f} \\[4pt] f &= \left(\dfrac{1}{d_o}+\dfrac{1}{d_i}\right)^{−1}  \\[4pt] &= \left(\dfrac{1}{12cm}+\dfrac{1}{-0.384cm}\right)^{−1} \\[4pt] &=-40.0 \,cm \end{align*} \], The radius of curvature is twice the focal length, so. Therefore, the image of the base of the object is on the optical axis directly above the image of the tip, as drawn in the figure. {\displaystyle R} To completely locate the extended image, we need to locate a second point in the image, so that we know how the image is oriented. Finally, principal ray 4 strikes the vertex of the mirror and is reflected symmetrically about the optical axis. The vertex of the lens surface is located on the local optical axis. {\displaystyle K} {\displaystyle z(r)} . \label{smallangle}\]. The point at which the reflected rays intersect, either in real space or in virtual space, is where the corresponding point of the image is located. z Solve the mirror equation for the focal length \(f\) and insert the known values for the object and image distances. a. {\displaystyle \alpha _{2}} Thus, the focal point is virtual because no real rays actually pass through it; they only appear to originate from it. So f = 24/2 = + 12 cm … Because the angles \(ϕ\) and \(ϕ′\) are alternate interior angles, we know that they have the same magnitude. 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. Although a spherical mirror is shown in Figure \(\PageIndex{8b}\), comatic aberration occurs also for parabolic mirrors—it does not result from a breakdown in the small-angle approximation (Equation \ref{smallangle}). The center of curvature of the mirror is labeled \(C\) and is a distance \(R\) from the vertex of the mirror, as marked in the figure. From the geometry of the spherical mirror, note that the focal length is half the radius of curvature: Show. This is called spherical aberration and results in a blurred image of an extended object. The symmetry axis of such optical elements is often called the principal axis or optical axis. {\displaystyle r} Are the object and image distances reasonable. The distance from the pole to the center of curvature is called (no surprise, I hope) the radius of curvature ( r ). 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.