Solar Electric Generating System

To solve an Integrated Concept Problem we must first identify the physical principles involved. Part (a) is related to the current topic. Part (b) involves a little math, primarily geometry. Part (c) requires an understanding of heat and density.

To a good approximation for a concave or semi-spherical surface, the point where the parallel rays from the sun converge will be at the focal point, so $R=2f=\text{80.0 cm}$.

The insolation is $\text{900 W}{\text{/m}}^2$. We must find the cross-sectional area $\mathrm{A}$ of the concave mirror, since the power delivered is $\text{900 W}{\text{/m}}^2\times \text{A}$. The mirror in this case is a quarter-section of a cylinder, so the area for a length $\mathrm{L}$ of the mirror is $\mathrm{A}=\frac{1}{4}\mathrm{(2}\mathrm{\pi R}\mathrm{)L}$. The area for a length of 1.00 m is then

The insolation on the 1.00-m length of pipe is then

The increase in temperature is given by $Q=\mathrm{mc}\mathrm{\Delta }T$. The mass $m$ of the mineral oil in the one-meter section of pipe is

Therefore, the increase in temperature in one minute is

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 $\text{400º}\text{C}$. We are considering only one meter of pipe here, and ignoring heat losses along the pipe.

Figure 25.44 Parabolic trough collectors are used to generate electricity in southern California. (credit: kjkolb, Wikimedia Commons)

What happens if an object is closer to a concave mirror than its focal length? This is analogous to a case 2 image for lenses ( $d_{\text{o}}<f$ and $f$ positive), which is a magnifier. In fact, this is how makeup mirrors act as magnifiers. Figure 25.45(a) uses ray tracing to locate the image of an object placed close to a concave mirror. Rays from a common point on the object are reflected in such a manner that they appear to be coming from behind the mirror, meaning that the image is virtual and cannot be projected. As with a magnifying glass, the image is upright and larger than the object. This is a case 2 image for mirrors and is exactly analogous to that for lenses.

Figure 25.45 (a) Case 2 images for mirrors are formed when a converging mirror has an object closer to it than its focal length. Ray 1 approaches parallel to the axis, ray 2 strikes the center of the mirror, and ray 3 approaches the mirror as if it came from the focal point. (b) A magnifying mirror showing the reflection. (credit: Mike Melrose, Flickr)

All three rays appear to originate from the same point after being reflected, locating the upright virtual image behind the mirror and showing it to be larger than the object. (b) Makeup mirrors are perhaps the most common use of a concave mirror to produce a larger, upright image.

A convex mirror is a diverging mirror ($f$ is negative) and forms only one type of image. It is a case 3 image—one that is upright and smaller than the object, just as for diverging lenses. Figure 25.46(a) uses ray tracing to illustrate the location and size of the case 3 image for mirrors. Since the image is behind the mirror, it cannot be projected and is thus a virtual image. It is also seen to be smaller than the object.

Figure 25.46 Case 3 images for mirrors are formed by any convex mirror. Ray 1 approaches parallel to the axis, ray 2 strikes the center of the mirror, and ray 3 approaches toward the focal point. All three rays appear to originate from the same point after being reflected, locating the upright virtual image behind the mirror and showing it to be smaller than the object. (b) Security mirrors are convex, producing a smaller, upright image. Because the image is smaller, a larger area is imaged compared to what would be observed for a flat mirror (and hence security is improved). (credit: Laura D’Alessandro, Flickr)