Parabolic Reflector

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Near the bottom of the parabola the perpendicular line is nearly vertical, meaning an incoming beam barely changes its angle after being reflected, allowing it to reach the focus above the bottom part of the parabola. Further up the parabola the perpendicular becomes more horizontal, allowing a light beam to undergo the greater change in angle needed to reach the focus.
Near the bottom of the parabola the perpendicular line is nearly vertical, meaning an incoming beam barely changes its angle after being reflected, allowing it to reach the focus above the bottom part of the parabola. Further up the parabola the perpendicular becomes more horizontal, allowing a light beam to undergo the greater change in angle needed to reach the focus.
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Parabolic reflectors can also work in reverse: if a light emitter is placed at the focus and shined inward towards the parabola, the light that reaches the parabola will be reflected straight out, with the beams of light parallel to each other. Headlights on cars often use this effect to shine light directly forward.
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[[Image:Parabdiagram3.JPG|thumb|400px|right|Diagram for the proof]]
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[[Image:Parabdiagram3.JPG|thumb|400px|right|Figure 1]]
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[[Image:Parabdiagram4.JPG|thumb|400px|right|'''A''' represents equal angles: the line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis.]]
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[[Image:Parabdiagram4.JPG|thumb|400px|right|Figure 2: '''A''' represents equal angles: the line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis.]]
The fact that a parabolic reflector can collect light in this way can be proven. A proof follows:
The fact that a parabolic reflector can collect light in this way can be proven. A proof follows:
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::Taking the derivative with respect to x gives the slope of the tangent at any point on the parabola:
::Taking the derivative with respect to x gives the slope of the tangent at any point on the parabola:
::*<math> \frac{x}{2p} = \frac{dy}{dx} </math>
::*<math> \frac{x}{2p} = \frac{dy}{dx} </math>
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::The slope of this tangent line is relative to the x-axis. When the slope is zero, the tangent line is parallel to the x-axis. The line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis, as shown in the diagram.
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::The slope of this tangent line is relative to the x-axis: when the slope is zero, the tangent line is parallel to the x-axis. The line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis, as shown in Figure 2.
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::We can use this slope to find the angle between the normal and the y-axis, which is the same as the angle between the normal and an incoming beam of light. The desired angle <math> \omega </math> can be expressed as:
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::We can use this slope to find the angle between the normal and the y-axis, which is the same as the angle between the normal and an incoming beam of light. The desired angle <math> \theta</math> can be expressed as:
::*<math> \tan\theta= \frac{x}{2p} </math> (Equation 1)
::*<math> \tan\theta= \frac{x}{2p} </math> (Equation 1)
::As mentioned previously, a beam of light is reflected 'over' the normal. This means that the angle the beam of light takes relative to a vertical line is equal to two times the angle the normal makes with the same vertical line.
::As mentioned previously, a beam of light is reflected 'over' the normal. This means that the angle the beam of light takes relative to a vertical line is equal to two times the angle the normal makes with the same vertical line.
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::We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus. Notice that geometrically, this angle <math> 2\omega </math> satisfies the relationship
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::We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus; that is, we want to show that the angle the light travels at after being reflected is the angle needed to hit the focus.
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::Notice that geometrically, this angle <math> 2\theta</math> satisfies the relationship
::* <math>\tan2\theta= \frac{x}{p-x^2/4p}</math>
::* <math>\tan2\theta= \frac{x}{p-x^2/4p}</math>
::Which, by a trigonometric identity, also implies
::Which, by a trigonometric identity, also implies
::* <math>\tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} </math> (Equation 2)
::* <math>\tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} </math> (Equation 2)
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::We now manipulate our Equation 1 for <math> \tan\omega </math> to show its equivalence to Equation 2:
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::We now manipulate Equation 1 for <math> \tan\theta</math> to show its equivalence to Equation 2:
::* <math> 2tan\theta= 2\frac{x}{2p} = \frac{x}{p} </math> , and
::* <math> 2tan\theta= 2\frac{x}{2p} = \frac{x}{p} </math> , and
::* <math> 1 - \tan\theta^2 = 1 - (\frac{x}{2p})^2 = 1 - \frac{x^2}{4p^2} </math> . Combining these two expressions gives
::* <math> 1 - \tan\theta^2 = 1 - (\frac{x}{2p})^2 = 1 - \frac{x^2}{4p^2} </math> . Combining these two expressions gives
::* <math> \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p} </math>
::* <math> \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p} </math>
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::Which is the same as the former expression for
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::Which is the same as the former expression in Equation 2.
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:Therefore, a beam of light will hit the parabola's focus after being reflected. ■
|AuthorName=Energy Information Administration
|AuthorName=Energy Information Administration

Revision as of 14:18, 17 June 2009

Image:inprogress.png
Parabolic Reflector Dish
Solar Dishes such as the one shown use a paraboloid shape to focus the incoming light into a single collector.

Basic Description

Incoming beams of light perpendicular to the directrix bounce off the dish directly towards the focus.
Incoming beams of light perpendicular to the directrix bounce off the dish directly towards the focus.
Note that incoming beams reflect 'over' the line perpendicular to the parabola at the point of contact.
Note that incoming beams reflect 'over' the line perpendicular to the parabola at the point of contact.

The geometry of a parabola makes this shape useful for solar dishes. If the dish is facing the sun, beams of light coming from the sun are essentially parallel to each other when they hit the dish. Upon hitting the surface of the dish, the beams are reflected directly towards the focus of the parabola, where a device to absorb the sun's energy would be located.

We can see why beams of light hitting the parabola-shaped dish will reflect towards the same point. A beam of light reflects 'over' the line perpendicular to the parabola at the point of contact. In other words, the angle the light beam makes with the perpendicular when it hits the parabola is equal to the angle it makes with same perpendicular after being reflected.

Near the bottom of the parabola the perpendicular line is nearly vertical, meaning an incoming beam barely changes its angle after being reflected, allowing it to reach the focus above the bottom part of the parabola. Further up the parabola the perpendicular becomes more horizontal, allowing a light beam to undergo the greater change in angle needed to reach the focus.

Parabolic reflectors can also work in reverse: if a light emitter is placed at the focus and shined inward towards the parabola, the light that reaches the parabola will be reflected straight out, with the beams of light parallel to each other. Headlights on cars often use this effect to shine light directly forward.

A More Mathematical Explanation

Figure 1
Figure 1
[[Image:Parabdiagram4.JPG|thumb|400px|right|Fi [...]

Figure 1
Figure 1
Figure 2: A represents equal angles: the line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis.
Figure 2: A represents equal angles: the line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis.

The fact that a parabolic reflector can collect light in this way can be proven. A proof follows:

Begin with the equation of a parabola with focus at (0,p):
  •  x^2=4py
Taking the derivative with respect to x gives the slope of the tangent at any point on the parabola:
  •  \frac{x}{2p} = \frac{dy}{dx}
The slope of this tangent line is relative to the x-axis: when the slope is zero, the tangent line is parallel to the x-axis. The line normal to the parabola has the same slope relative to the y-axis as the line tangent to the parabola has relative to the x-axis, as shown in Figure 2.
We can use this slope to find the angle between the normal and the y-axis, which is the same as the angle between the normal and an incoming beam of light. The desired angle  \theta can be expressed as:
  •  \tan\theta= \frac{x}{2p} (Equation 1)
As mentioned previously, a beam of light is reflected 'over' the normal. This means that the angle the beam of light takes relative to a vertical line is equal to two times the angle the normal makes with the same vertical line.
We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus; that is, we want to show that the angle the light travels at after being reflected is the angle needed to hit the focus.
Notice that geometrically, this angle  2\theta satisfies the relationship
  • \tan2\theta= \frac{x}{p-x^2/4p}
Which, by a trigonometric identity, also implies
  • \tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} (Equation 2)
We now manipulate Equation 1 for  \tan\theta to show its equivalence to Equation 2:
  •  2tan\theta= 2\frac{x}{2p} = \frac{x}{p} , and
  •  1 - \tan\theta^2 = 1 - (\frac{x}{2p})^2 = 1 - \frac{x^2}{4p^2} . Combining these two expressions gives
  •  \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p}
Which is the same as the former expression in Equation 2.
Therefore, a beam of light will hit the parabola's focus after being reflected. ■




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