# Prove involving parabolic mirrors

• alejandro7
In summary, the conversation is discussing the problem of showing that when rays of light parallel to the $z$ axis pass through the same point when reflected by a parabolic mirror. The solution involves considering the equations of two straight lines and following a specific path.
alejandro7
I'm having trouble with the following problem:

"Consider the parabolic mirror given by the equation $z=x^2+y^2$. Show that when the rays of light that travel parallel to the $z$ axis pass through the same point when reflected."

I'm familiar with the law of reflection but I'm stuck because I don't know how to apply vector calculus to this situation. I have tried to apply the law of reflection but I can't seem to get anywhere.

Thanks.

You simply need to consider the equations of two straight lines:

- a certain ray parallel to the axis, and see where it cuts the mirror
- the ray reflected from there, according the the law of reflexion

The picture below suggests what you have to do: simply follow the red path.

Last edited:

## 1. How does a parabolic mirror work?

A parabolic mirror works by reflecting light rays that are parallel to the axis of the parabola to a single focal point. This focal point is also where the object being reflected appears to be located, creating a magnified or inverted image.

## 2. What are some common uses for parabolic mirrors?

Parabolic mirrors are commonly used in telescopes, satellite dishes, and solar collectors. They are also used in headlights and flashlights to reflect and focus light.

## 3. What is the difference between a parabolic mirror and a spherical mirror?

A parabolic mirror has a parabolic shape, meaning the surface curves evenly in all directions, while a spherical mirror has a spherical shape, meaning the surface curves more towards the center. This results in a different focal point and image distortion in spherical mirrors.

## 4. How is the focal length of a parabolic mirror determined?

The focal length of a parabolic mirror is determined by the distance from the center of the mirror to the focal point. This can be calculated using the equation: f = (1/4) * D^2 / a, where f is the focal length, D is the diameter of the mirror, and a is the depth or curvature of the parabola.

## 5. Can a parabolic mirror be used to create a perfect image?

No, a parabolic mirror cannot create a perfect image as it suffers from spherical aberration, which results in distortion and blurring of the image. This can be reduced by using a smaller mirror or adding a secondary mirror to correct the curvature. However, it is impossible to completely eliminate spherical aberration in a parabolic mirror.

• Optics
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
9
Views
1K
• Special and General Relativity
Replies
26
Views
846
• Introductory Physics Homework Help
Replies
16
Views
454
• Optics
Replies
9
Views
2K
• Optics
Replies
8
Views
1K
• Special and General Relativity
Replies
2
Views
752
• Aerospace Engineering
Replies
6
Views
2K