Show that perfect a perfect reflector is a conic section with Fermat's principle

In summary: Your name]In summary, Fermat's principle states that light will take the path that minimizes the time it takes to travel from one point to another. In the case of a hyperbola, the shape of the curve, with its two infinitely extending branches, allows for equal travel time of light from one focus to the other, even though the actual distances are not equal. This is why perfect reflecting surfaces are conic sections, as they allow for the equal travel time of light between two points.
  • #1
hadoque
43
1

Homework Statement


Show, using Fermat's principle, that perfect reflecting surfaces are conic sections.


Homework Equations


Equations for the ellipse, parabola and hyperbola


The Attempt at a Solution


Ok, the ellipse seems easy. All rays coming from one focus reflecting to the other focus travels an equal distance if the mirror is an ellipse, since that's the definition of the ellipse.
I'm having problems with the hyperbola. If I understand the question correctly, I'm supposed to show that the rays traveling from one focus to the other (virtual image) are equal in length, but they clearly arent.
Where is the error in my thinking?
 
Physics news on Phys.org
  • #2



Dear fellow scientist,

First of all, great job on understanding the concept for the ellipse! Now, let's tackle the hyperbola.

You are correct in saying that the rays traveling from one focus to the other are not equal in length. However, the key to understanding this is to think about the shape of the hyperbola. Unlike the ellipse, the hyperbola is an open curve with two branches that extend infinitely. Therefore, the rays that reflect off a perfect reflecting surface and converge at one focus will also extend infinitely and never actually reach the other focus.

To better understand this, let's consider the equation of a hyperbola:

(x^2/a^2) - (y^2/b^2) = 1

We can see that the distance from the center (0,0) to the vertices of the hyperbola is a, and the distance from the center to the foci is c, where c^2 = a^2 + b^2. This means that the distance from one focus to the other is always greater than the distance from the center to the vertices.

Now, let's apply Fermat's principle. According to this principle, light will take the path that minimizes the time it takes to travel from one point to another. In the case of the hyperbola, this means that light will take the path that minimizes the distance it travels from one focus to the other. And since the distance from one focus to the other is always greater than the distance from the center to the vertices, the path that minimizes this distance is the one that reflects off the perfect reflecting surface and extends infinitely.

In conclusion, the shape of the hyperbola, with its two infinitely extending branches, is what allows for equal travel time of light from one focus to the other, even though the actual distances are not equal. This is why perfect reflecting surfaces are conic sections, because they allow for the equal travel time of light between two points.

I hope this helps clarify any confusion you had. Keep up the great work!


 
  • #3



Your thinking is correct. The error may be in assuming that the virtual image is formed at the other focus. In reality, the virtual image is formed at infinity, and the rays from the focus will never actually reach it. However, for the purposes of Fermat's principle, we can consider the virtual image to be at the other focus, since the rays will travel an infinite distance to reach it.

To show that perfect reflecting surfaces are conic sections, we can use Fermat's principle, which states that light will travel between two points along the path that takes the least time. In the case of a perfect reflector, this means that the path taken by the reflected ray will be the shortest possible path between the two points.

For the ellipse, as you mentioned, this path is defined by the definition of the ellipse, where the sum of the distances from the two foci to any point on the ellipse is constant. This means that the reflected ray will travel from one focus to the other, and the distance travelled will be equal to the distance between the two foci.

For the parabola, the shortest path between two points is a straight line. This means that the reflected ray will travel from the focus to a point on the parabola, and then reflect off the surface of the parabola to reach the other point. This path is also the path of constant time, as the distance travelled by the ray is always equal to the distance between the focus and the point on the parabola.

For the hyperbola, the path of constant time is defined by the hyperbolic property, where the difference between the distances from any point on the hyperbola to the two foci is constant. This means that the reflected ray will travel from one focus to a point on the hyperbola, and then reflect off the surface of the hyperbola to reach the other point. The distance travelled by the ray will always be equal to the difference between the distances from the focus and the point on the hyperbola to the two foci.

Therefore, we can conclude that perfect reflecting surfaces are conic sections, with the specific type of conic section depending on the shape of the surface. This is a result of Fermat's principle, which dictates that the path taken by light is the one that takes the least time, and in the case of perfect reflectors, this path is defined by the properties of
 

1. What is a perfect reflector?

A perfect reflector is a hypothetical surface that reflects all incoming light rays in a single direction without any loss of energy. It is a concept used in optics to study the behavior of light rays when they encounter various surfaces.

2. What is a conic section?

A conic section is a curve that is created by the intersection of a plane with a double cone. It includes circles, ellipses, parabolas, and hyperbolas, and is used to describe the shapes of various objects in mathematics and physics.

3. What is Fermat's principle?

Fermat's principle states that light will always take the path that requires the least time to travel between two points. This principle is used to explain the behavior of light in various optical phenomena, such as reflection and refraction.

4. How does Fermat's principle relate to perfect reflectors?

Fermat's principle can be used to prove that a perfect reflector is a conic section. The principle states that light will always take the path with the least time, and in the case of a perfect reflector, this means that the angle of incidence is equal to the angle of reflection. This relationship between the angles of incidence and reflection is a defining characteristic of conic sections.

5. Why is it important to understand the relationship between perfect reflectors and conic sections?

Understanding the relationship between perfect reflectors and conic sections is crucial in the field of optics. It allows us to accurately predict the behavior of light rays when they encounter different surfaces, and it also helps us design and optimize optical devices such as mirrors and lenses. Additionally, this relationship has significant applications in astronomy and engineering.

Similar threads

Replies
3
Views
728
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
26
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
7K
  • General Math
Replies
5
Views
1K
  • General Math
Replies
11
Views
5K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
2K
Replies
1
Views
1K
Back
Top