Light Wave Polarization (Trig ID)

[pillanoid]

Homework Statement

Suppose that we write the E_x and E_y components of a light wave generally as:

E_x=E_xocos([wt-kz) and E_y=E_yocos(wt-kz+p)

Show that at any instant E_x and E_y satisfy the ellipse equation on the E_y vs. E_x coordinate system:

(E_x/E_xo)²+(E_y/E_yo)²-2(E_x/E_xo)(E_y/E_yo)cos(p)=sin²(p)

E=electric field strength
E_{(x or y)}=x or y component of E field
E_{(x or y)o}=initial value of E_{(x or y)}
w=time constant
t=time
k=spatial constant
z=location in space
p=phase difference between the two components

Homework Equations

The Attempt at a Solution

I want to clarify what "at any instant" in the problem statement means before delving into a possibly long and tedious trig identity. Does this mean that either the time component or the spatial component of the E field equations can be ignored when solving/plugging equations in?

For that matter, would simplifying the terms through trig identities be the best approach here, or could this be a more scientifically-based question that depends on assumptions and reasoning to solve?

 

[KataKoniK]

I would approach this problem by first clarifying the meaning of "at any instant" in the context of the problem. This could potentially change the approach and assumptions used to solve the problem.

Assuming that "at any instant" means that both the time and spatial components of the E field equations are considered, I would then proceed to simplify the terms using trig identities. This would be a more mathematical approach and would provide a clear and concise solution to the problem.

However, if "at any instant" means that either the time or spatial component can be ignored, then the solution may require more scientific reasoning and assumptions. In this case, I would carefully consider the physical properties of light waves and how they behave in terms of time and space to explain the relationship between the Ex and Ey components and how they satisfy the ellipse equation.

In summary, as a scientist, I would first clarify the meaning of "at any instant" in the problem statement and then use a combination of mathematical and scientific reasoning to arrive at a solution that is both accurate and scientifically sound.