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A thin film of MgF₂ (n = 1.38) coats a piece of glass. Constructive interference is observed for the reflection of light with wavelengths of 500 nm and 625 nm. What is the thinnest film for which this can occur?

Respuesta :

Answer:

t = 905.8 nm

Explanation:

Given:

- Wavelength λ_1 = 500 nm

- Wavelength λ_2 = 625 nm

- MgF₂ refractive index n = 1.38

Find:

What is the thinnest film for which this can occur?

Solution:

- We have two different wavelength of light that constructively interfere at the surface of the film. We need the minimum thickness of film that would satisfy the condition of constructive interference for both!

-Since the refractive index of glass is greater than that of MgF_2, the expression of constructive interference would be as follows:

                                2*t = m*λ / n

- Since, the orders m are unknown for each wavelength, also different. We will try to determine the first for each wavelength of light.

- Construct two equation:

                                t = m_1*(500 nm ) / (2*1.38 )

                                t = 181.1594203*m_1  nm

                                t = m_2*(625 nm ) / (2*1.38 )

                                t = 226.4492754*m_2  nm

- Now equate the two thicknesses which should be equal:

                               226.4492754*m_2 = 181.1594203*m_1

                               m_2 = 0.8*m_1

- Now we know that m can only take integer values, and m is proportional to thickness t. So for thinnest thickness m's must take the least integer values. Hence, we have:                    

                               m_2 = (4 / 5) * m_1

So,                           m_1 = 5 , m_2 = 4   ..... Least integer values.

- Now that we have m's we can compute the thickness t as follows:

                                t = 181.1594203*m_1  nm

- Substitute m_1 = 5, we have:

                                t = 181.1594203*5  nm

                               t = 905.8 nm

- Substitute m_2 = 4 in:

                                 t = 226.4492754*m_2

                                 t = 226.4492754*4

                                t = 905.8 nm

- Our values of t = 905.8 nm matches for both wavelengths.

                                   

jayilych4real

The thinnest film for which this can occur is 905.8 nm

Calculations and Parameters:

Given:

  • Wavelength λ_1 = 500 nm
  • - Wavelength λ_2 = 625 nm
  • - MgF₂ refractive index n = 1.38

Based on the fact that the refractive index of glass is greater than that of MgF_2, this would be expressed as: 2*t = m*λ / n

We would create equations to find the unknown m for the wavelengths

                               

  • t = m_1*(500 nm ) / (2*1.38 )
  • t = 181.1594203*m_1  nm.......(eq i)

  • t = m_2*(625 nm ) / (2*1.38 )
  • t = 226.4492754*m_2  nm........(eq ii)

We would equate them

226.4492754*m_2 = 181.1594203*m_1

m_2 = 0.8*m_1

m_2 = (4 / 5) * m_1

So, m_1 = 5 , m_2 = 4   ..... Least integer values.

We would then substitute the values and we would get:

  • t = 226.4492754*m_2
  • t = 226.4492754*4
  • t = 905.8 nm

Hence, the values of t = 905.8 nm match both wavelengths.

Read more about wavelengths here:

https://brainly.com/question/10728818

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