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 $282.6 nm$.