Question

A mica sheet \(1.00 \mu \mathrm{m}\) thick is suspended in air. In reflected light, there are gaps in the visible spectrum at \(450,525,\) and $630 \mathrm{nm} .$ Calculate the index of refraction of the mica sheet.

Short answer

Answer: The index of refraction of the mica sheet is approximately 1.125.

Step-by-step solution

Recall the formula for destructive interference in thin films

For a thin film suspended in air, the path difference between the reflected rays from top and bottom surfaces will result in destructive interference when the path difference is an odd multiple of half-wavelength, i.e., \((2n-1) \frac{\lambda}{2}\). Here, n is the order of interference, and λ is the wavelength of light. The path difference in the case of thin films can be expressed as \(2 \times t \times \mu\), where t is the thickness of the film and μ is the index of refraction of the material. So, the formula for the destructive interference in thin films is given by: \(2 \times t \times \mu = (2n-1) \frac{\lambda}{2}\)

Solving for the index of refraction using the given wavelengths

Since we are given 3 wavelengths with gaps in the visible spectrum, we can use any one of them to calculate the index of refraction. Let's start with the first wavelength, λ₁ = 450 nm. We will also convert the thickness to meters: t = 1.00 μm = 1.00 x 10⁻⁶ m. Substitute the values into the formula and solve for μ: \(2 \times 1.00 \times 10^{-6} \times \mu = (2n-1) \frac{450 \times 10^{-9}}{2}\) We assume this is for the first order (n=1) and solve for μ: \(\mu = \frac{(2*1-1) \frac{450 \times 10^{-9}}{2}}{2 \times 1.00 \times 10^{-6}} = 1.125\)

Check if the other wavelengths also result in the same index of refraction

We need to check if the other two wavelengths (λ₂ = 525 nm and λ₃ = 630 nm) also result in the same index of refraction. If so, we can be confident in our solution. Let's first check for λ₂ = 525 nm: \(2 \times 1.00 \times 10^{-6} \times \mu = (2n-1) \frac{525 \times 10^{-9}}{2}\) To satisfy the equation, n must be an integer value. Solving for n: \(n = \frac{2 \times 1.00 \times 10^{-6} \times 1.125 + \frac{525 \times 10^{-9}}{2}}{525 \times 10^{-9}} \approx 4.29\) Since we don't get an integer value for n, this means that the destructive interference occurs at a different order for λ₂. We can check for λ₃ = 630 nm similarly: \(2 \times 1.00 \times 10^{-6} \times \mu = (2n-1) \frac{630 \times 10^{-9}}{2}\) Solving for n: \(n = \frac{2 \times 1.00 \times 10^{-6} \times 1.125 + \frac{630 \times 10^{-9}}{2}}{630 \times 10^{-9}} \approx 3.58\) Again, we don't get an integer value for n, which means that the destructive interference occurs at a different order for λ₃ as well. We can conclude that the index of refraction, μ ≈ 1.125, is consistent with the given wavelengths having gaps in the visible spectrum due to destructive interference at different orders.