Difference between revisions of "Parabolic Reflector"
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[[Image:Parabdiagram3.JPG|thumb|400px|right|Figure 3]] | [[Image:Parabdiagram3.JPG|thumb|400px|right|Figure 3]] | ||
− | [[Image:Parabdiagram4.JPG|thumb|400px|right|Figure 4: '''A''' represents equal angles: the line normal to the parabola | + | [[Image:Parabdiagram4.JPG|thumb|400px|right|Figure 4: '''A''' represents equal angles: the line normal to the parabola makes the same angle with 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 show that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus, as follows: | The fact that a parabolic reflector can collect light in this way can be proven. We can show that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus, as follows: | ||
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::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: | ||
::{{hide|1=::*<math> \frac{x}{2p} = \frac{dy}{dx} </math> | ::{{hide|1=::*<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 | + | ::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 makes the same angle with the y-axis as the line tangent to the parabola has relative to the x-axis, as shown in Figure 4.}} |
:'''Step 3''' | :'''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.{{hide|1= ::The desired angle <math> \theta</math> can be expressed as: | ::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.{{hide|1= ::The desired angle <math> \theta</math> can be expressed as: |
Revision as of 14:26, 10 July 2009
Parabolic Reflector Dish |
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Parabolic Reflector Dish
- Solar Dishes such as the one shown use a parabolic 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, as shown in Figure 2.
Near the bottom of the parabola the perpendicular line is nearly vertical, meaning an incoming beam is still nearly vertical after being reflected. Being nearly vertical allows 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 take on the more horizontal angles needed to reach the focus.
Parabolic reflectors are used in many modern application because of their ability to collect incoming information efficiently. For example, TV dishes reflect incoming television signals towards a receiver centered at the focus of the dish.
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
- Note: understanding of this explanation requires: *Elementary Trigonometry and Calculus.
The fact that a parabolic reflector can collect light in this way can be proven. We can show that any beam of light coming straight down into a parabola will reflect at exactly the angle needed to hit the focus, as follows:
- Step 1
- We begin with the equation of a parabola with focus at (0,p).
- We take the derivative with respect to x, giving the slope of the tangent at any point on the parabola:
- 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.
- We now must show that the direction the light takes after being reflected is exactly the angle needed to hit the focus.
- We now manipulate Equation 1's expression for to show its equivalence to Equation 2 (that is, to show the angle in Equation 1 is the same as the angle in Equation 2).
- We combine the two expressions in Step 6.
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