# Parabolic Reflector

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Parabolic Reflector Dish
Field: Geometry
Image Created By: Energy Information Administration
Website: [1]

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 vertical (made positive for simplicity) can be expressed in terms of tangent:

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