Question

Use Fermat's principle to find the path of a light ray through a medium of index of refraction proportional to the given function. \((x+y)^{1 / 2} \quad\) Hint: Make the change of variables \((45^{\circ}\) rotation). $$X=\frac{1}{\sqrt{2}}(x+y), \quad Y=\frac{1}{\sqrt{2}}(x-y) ; \quad \text { what is } \quad d X^{2}+d Y^{2} ?$$

Short answer

The change of variables results in \(dX^2 + dY^2 = dx^2 + dy^2\).

Step-by-step solution

Understand Fermat's Principle

Fermat's principle states that the path taken between two points by a light ray is the path that can be traversed in the least time. For a medium with a varying index of refraction, this principle can be used to determine the path of the light ray.

Change of Variables

Introduce a change of variables using a 45° rotation:- Let \(X = \frac{1}{\sqrt{2}}(x + y)\)- Let \(Y = \frac{1}{\sqrt{2}}(x - y)\)

Compute the Differential Elements

Find the differentials for the new variables X and Y:- Compute \( dX\): \(dX = \frac{\partial X}{\partial x}dx + \frac{\partial X}{\partial y}dy = \frac{1}{\sqrt{2}}(dx + dy) \)- Compute \( dY\): \(dY = \frac{\partial Y}{\partial x}dx + \frac{\partial Y}{\partial y}dy = \frac{1}{\sqrt{2}}(dx - dy)\)

Calculate \(dX^2 + dY^2\)

Use the expressions for \(dX\) and \(dY\) to find \(dX^2 + dY^2\):- First, square both expressions: \(dX^2 = \left(\frac{1}{\sqrt{2}}(dx + dy)\right)^2 = \frac{1}{2}(dx^2 + 2dxdy + dy^2)\)- \(dY^2 = \left(\frac{1}{\sqrt{2}}(dx - dy)\right)^2 = \frac{1}{2}(dx^2 - 2dxdy + dy^2)\)- Add them together: \(dX^2 + dY^2 = \frac{1}{2}(dx^2 + 2dxdy + dy^2) + \frac{1}{2}(dx^2 - 2dxdy + dy^2) = dx^2 + dy^2\)

Conclusion

The calculated value of \(dX^2 + dY^2\) confirms that it simplifies to \(dx^2 + dy^2\), as required for the application of Fermat's principle.

Key concepts

index of refraction

The term 'index of refraction' is fundamental when discussing the behavior of light as it passes through different mediums. The index of refraction, denoted as 'n', measures how much the speed of light is reduced inside a medium compared to its speed in a vacuum.
The mathematical definition is given by: \[ n = \frac{c}{v} \]
where 'c' is the speed of light in a vacuum and 'v' is the speed of light in the medium.

This concept is key to understanding why light bends or changes direction when moving from one medium to another. For example, light slows down and bends when it enters water from air due to water's higher index of refraction.
In this problem, the index of refraction is proportional to \( (x+y)^{1/2} \)
which indicates a varying medium where the path and speed of light depend on the spatial coordinates. By using Fermat's principle, we aim to find the path light would take in such an environment.

change of variables

The exercise suggests a 45° rotation change of variables to simplify the problem. This transformation, characterized by:
\(X = \frac{1}{\frac{\beta}}(x + y), \ Y = \frac{1}{\frac{\beta}}(x - y)\)
is useful because it helps align the problem with coordinate axes that make the math more manageable.

Here, \( X \) and \( Y \) are new variables that represent a rotated coordinate system, making it easier to visualize and calculate the light's path. This rotation helps in converting the light's path problem into a simpler form without changing the physics of light propagation.
Think of this as turning a complex maze (our original coordinate system) into a more straightforward path to navigate (new coordinate system) by rotating it so the paths match up with our natural way of movement (aligned with axes).

differential elements

Differential elements are tiny segments used in calculus to approximate changes in a function. Here, we compute the differentials \(dX\) and \(dY\) for our new variable system to set up our calculations. These differentials represent small changes in the new coordinates 'X' and 'Y'.

Using the chain rule, compute:

• \(dX = \frac{\beta}{2}(dx + dy)\)
• \(dY = \frac{\beta}{2}(dx - dy)\)

where \( dx \) and \( dy \) are the differentials of the original coordinates.
Squaring these and adding gives us
\[dX^2 + dY^2 = \frac{1}{2}(dx^2 + 2dxdy + dy^2) + \frac{1}{2}(dx^2 - 2dxdy + dy^2) = dx^2 + dy^2 \]

Thus, the expression for \( dX^2 + dY^2 \) simplifies back to the familiar format in the new variable system, making it easier to apply Fermat's principle and solve for the path of light in the varying medium.