*Notes to a video lecture on http://www.unizor.com*

__Parabolic Reflector__

Consider now a more complicated case of a curved reflection surface.

Any smooth surface can be considered as an infinite set of infinitesimally small flat pieces with each piece reflecting light in a direction that can be determined by a plane tangential to a surface at that point.

So, to determine the reflected light at some point of a surface we can just replace a surface with a tangential plane at that point and use the

**Laws of Reflection**presented in the previous lecture.

Let's illustrate this on a concrete example of a

*paraboloid*as a surface reflecting the light. Paraboloid is a surface obtained by rotating a parabola

*in the XZ-plane around the Z-axis.*

**z=k·x²**As a result of this rotation, the three-dimensional formula for a paraboloid is

**z = k·(x² + y²)**We will examine how vertically going down rays of light are reflected by this surface.

Assume, a light ray falls down parallel to the Z-axis of a paraboloid within XZ-plane at distance

*from this axis and hits paraboloid at point*

**a***(vertical blue line on the picture below).*

**B**After the reflection off the surface of paraboloid, which we will analyze as if reflected off the tangential plane to paraboloid at point

*, the reflected ray of light crosses the Z-axis of this paraboloid at point*

**B***(black line*

**C***on the picture below).*

**BC**The reflected ray of light should cross the Z-axis at some point

*from the considerations of rotational symmetry of the paraboloid.*

**C**We will analyze this using a two-dimensional cut along the plane going through a point

*on the surface of paraboloid, where the light ray hits its surface and the vertical axis of this paraboloid with Z-axis coinciding with the axis of paraboloid*

**B**The light blue colored line represents the light going down at a distance

*from the Z-axis. It hits a point*

**a=OA***on a parabola*

**B***(red curved line) and the solid black line represents the reflected ray of light that hits the Z-axis at point*

**z=k·x²***.*

**C**The green line is tangential to a parabola at point

*and should be used to determine the direction of the reflected light by establishing a normal to a parabola line (a purple line perpendicular to a green tangential line) and using the law of reflection about equality between an incidence angle*

**B***and the reflection angle*

**θ**_{i}*.*

**θ**_{r}Our task is to determine a distance

*from the origin of coordinates to point*

**OC***, where the reflected ray of light intersects the Z-axis of this parabola.*

**C**The analysis of this task, going from what is to be found back to what's known, is:

1. Find

*as the difference between*

**OC***(known to be the value of*

**AB***at*

**z=k·x²***, that is*

**x=a***) and an unknown*

**k·a²***.*

**BD**2. To find

*, we will use the formula*

**BD***,*

**BD = CD·cot(∠CBD)**where

**CD=OA=a**3. Angle

*is the difference between*

**∠CBD***and angle*

**π***, that is*

**∠θ**_{i}+∠θ_{r}(since

*)*

**θ**_{i}=θ_{r}=θ

**∠CBD = π−2θ**4. Since

*and*

**∠BEA=∠θ***is a tangential to our parabola*

**BE***, tangent of*

**z=k·x²***equals to a derivative of*

**∠BEA***at point*

**z=k·x²***, from which follows:*

**x=a**

**tan(∠θ) = 2k·a**Based on this analysis, we derive the following:

(a)

**tan(∠BEA) =**

= tan(∠θ) = 2k·a= tan(∠θ) = 2k·a

(b)

**tan(∠CBD) =**

= tan(π−2θ) = −tan(2θ) =

= −2tan(θ)/(1−tan²(θ)) =

= 4k·a/(4k²·a²−1)= tan(π−2θ) = −tan(2θ) =

= −2tan(θ)/(1−tan²(θ)) =

= 4k·a/(4k²·a²−1)

(c)

**cot(∠CBD) =**

= 1/tan(∠CBD) =

= (4k²·a²−1)/(4k·a) =

= k·a − 1/(4k·a)= 1/tan(∠CBD) =

= (4k²·a²−1)/(4k·a) =

= k·a − 1/(4k·a)

(d)

**BD = a·cot(∠CBD) =**

= k·a² −1/(4k)= k·a² −1/(4k)

(e)

**OC = AB − BD =**

= k·a² − (k·a² −1/(4k)) =

= 1/(4k)= k·a² − (k·a² −1/(4k)) =

= 1/(4k)

So, as we see, the reflected ray of light will intersect the Z-axis at point

*at a distance*

**C***from the bottom of a paraboloid.*

**OC=1/(4k)**What's remarkable about this result is that the

**location of point**.

*does not depend on the value of parameter***C***- the distance of the incident light from the Z-axis***a**So, any vertically directed ray of light will be reflected by a paraboloid towards the same point on its axis - its

*focal*point - located at distance

*from the bottom, where parameter*

**f=1/(4k)***defines the "steepness" of a paraboloid.*

**k**Using the parabolic mirror, we can "gather" the sun rays into a focal point and boil the water positioned there to use the steam to generate electricity.

If the source of light is positioned at the focal point of a parabolic mirror, all its emitted light will be directed in one direction parallel to the axis of a paraboloid. That's the principle of work of a projector.

The dish-like parabolic antenna, directed towards a stationary satellite broadcasting some radio signals, collects all the radio waves falling into it, reflecting all these signals towards its focal point, where a radio receiver is located. This allows to catch even a relatively weak radio signal.

When we don't hear a distant sound, we make a sort of a "dish" with our hand, directing the reflected sound towards the ear to hear better.

All the above examples and many others are the usages of a principle of focusing the waves by parabolic (or almost parabolic) reflectors.

## No comments:

Post a Comment