The phenomenon of constructive interference in thin films occurs when light waves reflected from the top and bottom surfaces of the film interfere constructively, which leads to brighter reflections at certain wavelengths. For constructive interference to occur, the path difference between the two reflected rays must be an integer multiple of the wavelength.

In this case, you are dealing with a thin film of MgF₂ (with refractive index $n = 1.38$) coating a piece of glass. The condition for constructive interference for reflected light is given by the equation:

$2nt = m\lambda$

where:

• $n$ is the refractive index of the thin film,
• $t$ is the thickness of the film,
• $m$ is an integer representing the order of interference (which can be 1, 2, 3, etc.),
• $\lambda$ is the wavelength of light in a vacuum (or air).

Given that constructive interference occurs at two different wavelengths, $\lambda_1 = 470 \, \text{nm}$ and $\lambda_2 = 627 \, \text{nm}$, you can use these wavelengths to find the thickness of the film by solving for $t$.

Step-by-step Calculation

1. The equation $2nt = m\lambda$ can be rewritten for the thickness $t$: $t = \frac{m\lambda}{2n}$ For two wavelengths, we have two separate equations: $t = \frac{m_1 \lambda_1}{2n} \quad \text{and} \quad t = \frac{m_2 \lambda_2}{2n}$

2. Now, using the refractive index $n = 1.38$, you can plug in the values of $\lambda_1$ and $\lambda_2$ to calculate possible thicknesses, based on different orders of interference (values of $m_1$ and $m_2$).

Would you like me to calculate the thickness based on specific values of $m_1$ and $m_2$, or should I explore possible combinations for you?

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Here are 5 related questions you might find interesting:

1. How does destructive interference affect thin-film interference?
2. What are the conditions for destructive interference in thin films?
3. How does the refractive index of the film affect the observed wavelengths?
4. What is the effect of thickness variation on interference patterns in thin films?
5. How do you calculate the phase shift in thin-film interference?

Tip: For multiple wavelengths to produce constructive interference, the film's thickness must be such that it satisfies the interference condition for both wavelengths simultaneously.