A glass surface is coated by an oil film of uniform thickness `1.00xx10^-4`cm. The index of refraction of the oil is 1.25 and that of the glass is 1.50. Find the wavelengths of light in the visible region `(400nm-750nm) which are completely transmitted by the oil film under normal incidence.

Explanation

To solve the problem, we need to determine the wavelengths of light in the visible region (400 nm to 750 nm) that are completely transmitted by an oil film of uniform thickness coated on a glass surface. The given parameters are:

- Thickness of the oil film, \( t = 1.00 \times 10^{-4} \) cm = \( 1.00 \times 10^{-6} \) m
- Index of refraction of the oil, \( \mu = 1.25 \)
- Index of refraction of the glass, \( n_g = 1.50 \)

### Step-by-Step Solution:

1. **Understand the Condition for Strong Transmission:**
For constructive interference (which leads to strong transmission) in thin films, the condition is given by:
\[ 2 \mu t = (2n + 1) \frac{\lambda}{2} \]
where \( n \) is an integer (0, 1, 2, ...).

2. **Rearranging the Formula:**
We can rearrange the formula to solve for the wavelength \( \lambda \):
\[ \lambda = \frac{4 \mu t}{2n + 1} \]

3. **Substituting the Known Values:**
Substitute \( \mu = 1.25 \) and \( t = 1.00 \times 10^{-6} \) m into the equation:
\[ \lambda = \frac{4 \times 1.25 \times 1.00 \times 10^{-6}}{2n + 1} \]
Simplifying this gives:
\[ \lambda = \frac{5.00 \times 10^{-6}}{2n + 1} \text{ m} \]
Converting to nanometers (1 m = \( 10^9 \) nm):
\[ \lambda = \frac{5000 \text{ nm}}{2n + 1} \]

4. **Finding Values of \( n \) and Corresponding Wavelengths:**
Now we will calculate \( \lambda \) for different integer values of \( n \) and check if they fall within the visible range (400 nm to 750 nm).

- For \( n = 0 \):
\[ \lambda = \frac{5000}{1} = 5000 \text{ nm} \quad (\text{not in visible range}) \]
- For \( n = 1 \):
\[ \lambda = \frac{5000}{3} \approx 1666.67 \text{ nm} \quad (\text{not in visible range}) \]
- For \( n = 2 \):
\[ \lambda = \frac{5000}{5} = 1000 \text{ nm} \quad (\text{not in visible range}) \]
- For \( n = 3 \):
\[ \lambda = \frac{5000}{7} \approx 714.29 \text{ nm} \quad (\text{in visible range}) \]
- For \( n = 4 \):
\[ \lambda = \frac{5000}{9} \approx 555.56 \text{ nm} \quad (\text{in visible range}) \]
- For \( n = 5 \):
\[ \lambda = \frac{5000}{11} \approx 454.55 \text{ nm} \quad (\text{in visible range}) \]

5. **Final Result:**
The wavelengths of light in the visible region that are completely transmitted by the oil film are:
- \( 714.29 \text{ nm} \)
- \( 555.56 \text{ nm} \)
- \( 454.55 \text{ nm} \)