Parabolic Reflector

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The fact that a parabolic reflector can collect light in this way can be proven. We can prove this by showing that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus.
The fact that a parabolic reflector can collect light in this way can be proven. We can prove this by showing that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus.
-
:'''Step 1''' We begin with the equation of a parabola with focus at (0,p):
+
:'''Step 1'''
-
::<math> x^2=4py </math>
+
::We begin with the equation of a parabola with focus at (0,p):
 +
::*<math> x^2=4py </math>
:'''Step 2'''
:'''Step 2'''
::We take the derivative with respect to x, giving the slope of the tangent at any point on the parabola:
::We take the derivative with respect to x, giving the slope of the tangent at any point on the parabola:
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::<math> \frac{x}{2p} = \frac{dy}{dx} </math>
+
::*<math> \frac{x}{2p} = \frac{dy}{dx} </math>
::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.
::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.
:'''Step 3'''
:'''Step 3'''
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::*<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.
 +
:'''Step 4'''
::We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus.
::We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus.
::Notice from Figure 1 that geometrically, the angle needed to hit the focus is equal to <math> 2\theta</math>, and satisfies the relationship
::Notice from Figure 1 that geometrically, the angle needed to hit the focus is equal to <math> 2\theta</math>, and satisfies the relationship
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::* <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
+
:'''Step 5'''
-
::* <math>\tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} </math> (Equation 2)
+
::We use a trigonometric identity to rewrite the equation in Step 4:
 +
::*<math>\tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} </math> (Equation 2)
 +
:'''Step 6'''
::We now manipulate Equation 1's expression for <math> \tan\theta</math> to show its equivalence to Equation 2 (that is, to show the angle <math> \theta </math> in Equation 1 is the same as the angle <math> \theta </math> in Equation 2).:
::We now manipulate Equation 1's expression for <math> \tan\theta</math> to show its equivalence to Equation 2 (that is, to show the angle <math> \theta </math> in Equation 1 is the same as the angle <math> \theta </math> in 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> .
-
::* <math> \frac{2\tan\theta}{1-\tan\theta^2} = \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p} </math>
+
:'''Step 7'''
 +
::We combine the two expressions in Step 6, giving:
 +
::*<math> \frac{2\tan\theta}{1-\tan\theta^2} = \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p} </math>
::Which is the same as the expression in Equation 2.
::Which is the same as the expression in Equation 2.
:Therefore, a beam of light will hit the parabola's focus after being reflected. ■
:Therefore, a beam of light will hit the parabola's focus after being reflected. ■

Revision as of 11:44, 22 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 will be reflected straight out of the parabola, with the beams of light traveling 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. We can prove this by showing that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus.

Step 1
We begin with the equation of a parabola with focus at (0,p):
  • x^2=4py
Step 2
We take the derivative with respect to x, giving 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.
Step 3
We 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.
Step 4
We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus.
Notice from Figure 1 that geometrically, the angle needed to hit the focus is equal to 2\theta, and satisfies the relationship
  • \tan2\theta= \frac{x}{p-x^2/4p}
Step 5
We use a trigonometric identity to rewrite the equation in Step 4:
  • \tan2\theta= \frac{x}{p-x^2/4p} =\frac{2\tan\theta}{1-\tan\theta^2} (Equation 2)
Step 6
We now manipulate Equation 1's expression for \tan\theta to show its equivalence to Equation 2 (that is, to show the angle \theta in Equation 1 is the same as the angle \theta in 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} .
Step 7
We combine the two expressions in Step 6, giving:
  • \frac{2\tan\theta}{1-\tan\theta^2} = \frac{\frac{x}{p}}{1 - \frac{x^2}{4p^2}} = \frac{x}{p-x^2/4p}
Which is the same as the expression in Equation 2.
Therefore, a beam of light will hit the parabola's focus after being reflected. ■




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Parabolic Reflector Dish
Field: Geometry
Created By: Energy Information Administration
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