Question

White light shines on a sheet of mica that has a uniform thickness of \(1.30 \mu \mathrm{m} .\) When the reflected light is viewed using a spectrometer, it is noted that light with wavelengths of \(433.3 \mathrm{nm}, 487.5 \mathrm{nm}, 557.1 \mathrm{nm}, 650.0 \mathrm{nm}\), and \(780.0 \mathrm{nm}\) is not present in the reflected light. What is the index of refraction of the mica?

Short answer

Answer: To find the index of refraction, we will follow the steps provided in the solution. Step 1: Identify the destructive interference condition. We have the equation for destructive interference: $$2tn = m\lambda$$ Step 2: Calculate the order of interference for each wavelength. For the 666 nm (0.666 µm) wavelength: $$m_1 = \frac{2(1.30)}{0.666} = 3.90$$ Round to the nearest whole number: $$m_1 = 4$$ For the 532 nm (0.532 µm) wavelength: $$m_2 = \frac{2(1.30)}{0.532} = 4.88$$ Round to the nearest whole number: $$m_2 = 5$$ Step 3: Determine the index of refraction. For the 666 nm (0.666 µm) wavelength: $$n_1 = \frac{4(0.666)}{2(1.30)} = 1.026$$ For the 532 nm (0.532 µm) wavelength: $$n_2 = \frac{5(0.532)}{2(1.30)} = 1.024$$ Step 4: Calculate the index of refraction and round the result. Find the average index of refraction: $$n = \frac{1.026 + 1.024}{2} = 1.025$$ The index of refraction of the mica is 1.025.

Step-by-step solution

Identify the destructive interference condition

The condition for destructive interference is given by: $$2 t n \cos(\theta)= m \lambda$$ Since \(\theta\) is small, we can approximate \(\cos(\theta) \approx 1\). Therefore, the equation becomes: $$2tn = m\lambda$$ Remember that \(t = 1.30 \mu m\), and we are given the wavelengths of the light that are not present in the reflected light.

Calculate the order of interference for each wavelength

To find the index of refraction, we will first need to find the order of interference for each wavelength. We will use the given wavelengths and the equation for destructive interference: $$m = \frac{2tn}{\lambda}$$ Calculate the order of interference for each wavelength, rounding to the nearest whole number.

Determine the index of refraction

Once we have the values of \(m\) for each wavelength, we can determine the index of refraction using the equation: $$n = \frac{m\lambda}{2t}$$ Calculate the value of \(n\) for each wavelength using their respective values of \(m\) and take the average of these values to find the overall index of refraction of the mica.

Calculate the index of refraction and round the result

Calculate the index of refraction using the average value of \(n\) from Step 3 and round the result to an appropriate number of significant figures. This will be the final answer for the index of refraction of the mica.