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A very thin oil film (n=1.25) floats on water (n=1.33). What is the minimum width of the oil film required to produce a strong reflection for green light with 500nm wavelength.

This question aims to find the width of the oil film required for a strong reflection of green light with 500nm of wavelength.

The basic concepts required for this question are reflection, refraction, and wavelength of different light colors. Refraction is the phenomenon in physics in which light changes its direction when it passes from one surface to another surface with a different refractive index. Depending upon the refractive indices of two mediums, the light bends towards the normal vector or away from it.

Reflection is the phenomenon of light where a ray of light bounces back completely after hitting the surface of a medium which do not absorb light. Each color in the light spectrum has a different wavelength. The two extreme wavelengths on the color spectrum are given as:

\[ Wavelength\ of\ Violet\ Color\ \lambda_v\ =\ 380\ nm \]

\[ Wavelength\ of\ Red\ Color\ \lambda_r\ =\ 700\ nm \]

Expert Answer

We need to find the thickness of the oil film where the green light will hit to have a strong reflection of the light.

The information we have for this problem is given as:

\[ Wavelength\ of\ Green\ Light\ \lambda_g\ =\ 500\ nm \]

\[ Refractive\ Index\ of\ Oil\ n_1\ =\ 1.25 \]

\[ Refractive\ Index\ of\ Water\ n_2\ =\ 1.33 \]

The formula to calculate the thickness of the oil film is given as:

\[ \lambda = \dfrac{2 n_1 d} {m} \]

Rearranging the formula for thickness, we get:

\[ d = \dfrac{m \lambda}{2 n_1} \]

Here, $m$ is a constant, for strong reflection, its value is $1$. Substituting the values in the above equation, we get:

\[ d = \dfrac{1 \times 500 \times 10^{-9}}{2 \times 1.25} \]

\[ d = \dfrac{500 \times 10^{-9}}{2.5} \]

\[ d = 200 \times 10^{-9} \]

\[ d = 200 nm \]

This means that an oil film with a refractive index of $1.25$ needs to have at least $200nm$ thickness to completely reflect the green light with a $500nm$ wavelength.

Numerical Result

The minimum thickness required for the oil to have a strong reflection of green light with $500nm$ is calculated to be:

\[ d = 200 nm \]

Example

An oil film with a refractive index of $1.15$ is required to have a strong reflection of red light with a wavelength of $650nm$. Find the minimum thickness of the oil film.

The given information about this problem is given as:

\[ Wavelength\ of\ Red\ Light\ \lambda_g\ =\ 650\ nm \]

\[ Refractive\ Index\ of\ Oil\ n_1\ =\ 1.15 \]

\[ Refractive\ Index\ of\ Water\ n_2\ =\ 1.33 \]

The formula to calculate the thickness of the surface to have a strong reflection of the red light is given as:

\[ d = \dfrac{m \lambda}{2 n_1} \]

Substituting the values, we get:

\[ d = \dfrac{1 \times 650\times 10^{-9}}{2 \times 1.15} \]

\[ d = \dfrac{650 \times 10^{-9}}{2.3} \]

\[ d = 282.6 \times 10^{-9} \]

\[ d = 282.6 nm \]

The minimum thickness required to have a strong reflection of the red light with a wavelength of $650 nm$ is calculated to be $282.6 nm$.

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