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The problem involves a modified Newton's rings setup, which involves interference by reflection. We're given three indices of refraction \(n_1 = 1.6\), \(n_2 = 1.5\), and \(n_3 = 1.4\). The task is to find the three minimum values of the air gap thickness \(d_0\) that result in a maximum intensity at the center when illuminated with light of wavelength \(\lambda = 550 \, \text{nm}\).

### **Step-by-step Solution:**

When light reflects off a thin film between two different media, constructive interference (a maximum) occurs when the optical path difference is an integer multiple of the wavelength:

\[
2 n_2 d_0 = m \lambda
\]

where:
- \(n_2 = 1.5\) (refractive index of the medium where the interference happens),
- \(d_0\) is the air gap or thickness,
- \(m\) is the order of the fringe (an integer \(m = 0, 1, 2, \ldots\)),
- \(\lambda = 550 \, \text{nm}\) is the wavelength of the light.

Since the problem asks for the first three minimum values of \(d_0\), we will calculate for \(m = 0, 1, 2\).

Let's calculate the values for \(d_0\):

\[
d_0 = \frac{m \lambda}{2 n_2}
\]

Now, let's compute the three values.The three minimum values of the air gap thickness \(d_0\) that result in a maximum at the center are:

1. For \(m = 0\): \(d_0 = 0 \, \text{m}\) (which corresponds to the surfaces being in contact)
2. For \(m = 1\): \(d_0 = 1.83 \times 10^{-7} \, \text{m} = 183 \, \text{nm}\)
3. For \(m = 2\): \(d_0 = 3.67 \times 10^{-7} \, \text{m} = 367 \, \text{nm}\)

Would you like a more detailed explanation, or do you have any questions?

### Related Questions
1. How does the refractive index affect the formation of Newton's rings?
2. What would happen if the wavelength of light was changed to a different value?
3. How do destructive interference patterns differ from constructive interference in this setup?
4. What is the significance of the phase change on reflection in Newton's rings?
5. How would the shape of the rings change if \(R_1\) and \(R_2\) were different?

**Tip:** Remember that in optics, the phase change upon reflection depends on whether light reflects off a medium with a higher or lower refractive index.

Considere el dispositivo de anillos de Newton modificado que se muestra en la figura. Se observan anillos por reflexión. ¿Cuáles son los 3 mínimos valores de d0 para que el centro de los anillos corresponda a un máximo cuando se lo ilumina con luz de longitud de onda λ = 550 nm?

Learn how to calculate the minimum values of the air gap thickness d0 in a modified Newton's rings experiment. This problem involves thin-film interference and the calculation of the constructive interference condition for a wavelength of λ = 550 nm, using refractive indices n1 = 1.6, n2 = 1.5, and n3 = 1.4. Suitable for university-level physics students.

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