How can you derive the expression for ds from a light ray's path?

In summary, the conversation discusses the path followed by light rays and how they travel through different mediums. Factors that affect the path of a light ray are also mentioned, including the angle of incidence, refractive index, and surface encountered. It is noted that light rays can change direction through refraction, and this bending of light is utilized in lenses to form images and correct vision.
  • #1
Fernando Rios
96
10
Homework Statement
Find the path followed by a light ray if the index of refraction (in polar coordinates) is proportional to r^-2.
Relevant Equations
∫nds
This is an example from a textbook. They show the following:

∫nds = ∫r-2*ds = ∫r-2*√((dr)2+r2*(dθ)2)

How do you obtain the expression for ds?
 
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  • #2
The arc length in Cartesian coordinate is given by :

##ds^2 = dx^2 + dy^2##
Which is Pythagoras theorem for an infinitesimal part of curve.

The transformation function to polar coordinate has form:

## x= r cos\theta##
##y=r sin\theta##Substituting in above equation and manipulating gives desired result.
 
  • #3
Got it! Thank you for your help.
 

1. What is the path followed by a light ray?

The path followed by a light ray is known as a straight line or a ray. It is the trajectory of light as it travels through a medium or vacuum.

2. How does light travel through different mediums?

Light travels through different mediums by changing its speed and direction. When light enters a denser medium, it slows down and bends towards the normal. When it exits a denser medium, it speeds up and bends away from the normal.

3. What is the law of reflection?

The law of reflection states that the angle of incidence is equal to the angle of reflection. This means that the incoming light ray and the reflected light ray will make equal angles with the normal line at the point of incidence.

4. Can light rays change direction without hitting a surface?

Yes, light rays can change direction without hitting a surface. This phenomenon is known as refraction and occurs when light travels through a medium with varying densities, such as air and water.

5. What is the difference between reflection and refraction?

Reflection occurs when light bounces off a surface and changes direction, while refraction occurs when light passes through a medium and changes direction. Reflection follows the law of reflection, while refraction follows Snell's law.

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