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  1. If we wish to place the fluid-carrying pipe 40.0 cm from the concave mirror at the mirror’s focal point, what will be the radius of curvature of the mirror?
  2. Per meter of pipe, what will be the amount of sunlight concentrated onto the pipe, assuming the insolation (incident solar radiation) is 0.900 k W/m 2 size 12{"0.900"" W/m" rSup { size 8{2} } } {} ?
  3. If the fluid-carrying pipe has a 2.00-cm diameter, what will be the temperature increase of the fluid per meter of pipe over a period of one minute? Assume all the solar radiation incident on the reflector is absorbed by the pipe, and that the fluid is mineral oil.

Strategy

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.

Solution to (a)

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 = 2 f = 80.0 cm size 12{R=2f="80"" cm"} {} .

Solution to (b)

The insolation is 900 W /m 2 size 12{"900"" W/m" rSup { size 8{2} } } {} . We must find the cross-sectional area A of the concave mirror, since the power delivered is 900 W /m 2 × A . The mirror in this case is a quarter-section of a cylinder, so the area for a length L of the mirror is A = 1 4 (2 πR )L . The area for a length of 1.00 m is then

A = π 2 R (1.00 m) = (3.14) 2 (0.800 m) (1.00 m) = 1.26 m 2 .

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

( 9.00 × 10 2 W m 2 ) ( 1.26 m 2 ) = 1130 W.

Solution to (c)

The increase in temperature is given by Q = mc Δ T size 12{Q=mcDT} {} . The mass m size 12{m} {} of the mineral oil in the one-meter section of pipe is

m = ρ V = ρπ ( d 2 ) 2 (1.00 m) = ( 8.00 × 10 2 kg/m 3 ) (3.14) (0.0100 m) 2 (1.00 m) = 0.251 kg.

Therefore, the increase in temperature in one minute is

Δ T = Q / m c = (1130 W) (60.0 s) (0.251 kg) (1670 J·kg/ºC) = 162ºC .

Discussion for (c)

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 400º C size 12{"400"°C} {} . We are considering only one meter of pipe here, and ignoring heat losses along the pipe.

A parabolic trough solar thermal electric power plant located at Kramer Junction, California
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 o < f and f size 12{f} {} positive), which is a magnifier. In fact, this is how makeup mirrors act as magnifiers. [link] (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 (a) shows three incident rays, 1, 2, and 3, falling on a concave mirror. Ray 1 falls parallel, ray 2 falls making an angle with the axis, and ray 3 is from focal point F. These rays after reflection appear to come from a point above the axis. The image is erect and enlarged, and falls above the axis behind the mirror. Here, the distance from the center of the mirror to F is focal length small f, distances of the object and the image from the mirror are d sub o  and d sub I, respectively. The heights of the object and the image are h sub o and h sub i, respectively. Figure (b) shows a woman applying makeup looking into her magnified reflection in the concave mirror.
(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)
Practice Key Terms 3

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Source:  OpenStax, Physics subject knowledge enhancement course (ske). OpenStax CNX. Jan 09, 2015 Download for free at http://legacy.cnx.org/content/col11505/1.10
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