Parabolic Reflector

From Math Images

(Difference between revisions)
Jump to: navigation, search
Line 21: Line 21:
::The angle that this perpendicular line makes with the horizontal (made positive for simplicity) is:
::The angle that this perpendicular line makes with the horizontal (made positive for simplicity) is:
::*<math> \theta = \arctan \frac{2p}{x} </math>
::*<math> \theta = \arctan \frac{2p}{x} </math>
 +
::The angle that this perpendicular line makes with the vertical is then
 +
::*<math> \frac{\pi}{2} - \arctan \frac{2p}{x} = \psi</math>
 +
|AuthorName=Energy Information Administration
|AuthorName=Energy Information Administration
|SiteURL=http://www.eia.doe.gov/cneaf/solar.renewables/page/solarthermal/solarthermal.html
|SiteURL=http://www.eia.doe.gov/cneaf/solar.renewables/page/solarthermal/solarthermal.html

Revision as of 11:28, 16 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

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.

A More Mathematical Explanation

The fact that a parabolic reflector can collect light in this way can be proven. A rough proof follo [...]

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

Begin with the equation of a parabola in terms of the location of the focus:
  •  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 line normal to the parabola at any point is perpendicular to the tangent line, having slope
  •  -\frac{2p}{x}
The angle that this perpendicular line makes with the horizontal (made positive for simplicity) is:
  •  \theta = \arctan \frac{2p}{x}
The angle that this perpendicular line makes with the vertical is then
  •  \frac{\pi}{2} - \arctan \frac{2p}{x} = \psi




Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.









If you are able, please consider adding to or editing this page!

Have questions about the image or the explanations on this page?
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.






Personal tools