How Can I Accurately Determine the Critical Angle in an Unknown Material?

In summary: A fellow scientistIn summary, the forum poster sought help with finding the wavelength and critical angle in an unknown material. They attempted to use formulas for finding the wavelength in the required terms, inside the material, and the critical angle. However, they encountered some doubts and uncertainties about their approach and requested guidance. A fellow scientist provided clarification on the formulas and suggested double-checking calculations and seeking assistance for further clarity.
  • #1
dykuma
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Homework Statement
An electromagnetic wave is incident into the flat surface of an unknown material. In the surface, the transverse wave mode follows the relation ##\omega^2 = \omega_m + c^2 k^2## , where ##\omega_m## is an angular frequency associated with the material. The angular frequency of the incident wave is ##\omega = 2 \omega_m##

What is the incident wavelength and the transverse wavelength in terms of the speed of light and the angular frequency associated with the material? Then, determine the critical angle such that there is no transmission into the material?
Relevant Equations
I'm expecting to use snell's law, as well as equations related to wave numbers and the index's of refraction.
First, I attempted to find the wavelength of the incident wave in the required terms:
1588825864673.png

Next, I tried to find the wavelength inside of the material:
1588826543175.png


And then lasted, I tried to find the critical angle:
1588826638056.png

My issue is that
1) I don't know if what I did was correct.
2) I don't think what I did was correct because that angle is in the complex plane. Maybe this unknown material is wild stuff, but I'm not sure that is possible. If I invert the answer, (that is, assume that I did the problem backwards and mixed up the wavelengths),I get a critical angle of pi/6, which is almost too nice not to be the right answer.

Anyway, any help or direction on this problem would be appreciated.
 

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  • #2

Thank you for sharing your attempts at finding the wavelength and critical angle in the unknown material. I would like to provide some guidance and clarification on your approach.

Firstly, to find the wavelength of the incident wave in the required terms, you can use the formula λ = c/f, where c is the speed of light and f is the frequency of the wave. If you have the frequency in the required terms, you can simply plug it into the formula to find the wavelength.

Next, to find the wavelength inside the material, you can use the formula λ = λ₀/n, where λ₀ is the wavelength in vacuum and n is the refractive index of the material. If you have the refractive index in the required terms, you can use it to find the wavelength inside the material.

Lastly, to find the critical angle, you can use the formula sinθc = n₂/n₁, where n₂ is the refractive index of the second medium (in this case, the unknown material) and n₁ is the refractive index of the first medium (usually air or vacuum). This angle is not in the complex plane, but rather in the real plane. If you are getting a value in the complex plane, it could be due to an error in your calculations.

In conclusion, your approach to finding the wavelength and critical angle is correct, but it is important to double-check your calculations and ensure that the values you are using are in the correct terms. If you are still unsure about your answers, I recommend consulting with a colleague or seeking additional resources for assistance.

I hope this helps and good luck with your research!
 

1. What is the critical angle?

The critical angle is the angle of incidence at which a ray of light passing through a medium is refracted at an angle of 90 degrees, meaning it travels along the boundary between the two media.

2. How is the critical angle determined?

The critical angle is determined by the refractive indices of the two media that the ray of light is passing through. It can be calculated using the Snell's law formula: sin(critical angle) = refractive index of second medium / refractive index of first medium.

3. What is the significance of the critical angle?

The critical angle is significant because it marks the point at which total internal reflection occurs. This phenomenon is used in various applications such as fiber optics and periscopes.

4. Can the critical angle be greater than 90 degrees?

No, the critical angle cannot be greater than 90 degrees. If the angle of incidence is greater than the critical angle, the light will not pass through the boundary and will instead be reflected back into the first medium.

5. How does the critical angle change with different materials?

The critical angle is dependent on the refractive indices of the two media that the light is passing through. Different materials have different refractive indices, so the critical angle will vary depending on the materials involved.

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