To determine the thickness $t$ of the oil film at point B, we need to consider the phenomenon of thin-film interference, specifically the constructive interference of light waves reflecting within the film. Given the refractive indices and the wavelength of yellow light, we can use the conditions for constructive interference.

Given Information

• Refractive index of oil, $n_o = 1.50$
• Refractive index of water, $n_w = 1.33$
• Wavelength of yellow light in air, $\lambda_{\text{air}} = 580 \, \text{nm}$

Since yellow light is visible at point B, this corresponds to constructive interference at that thickness.

Step-by-Step Solution

1. Wavelength in the Oil Film: First, calculate the wavelength of yellow light within the oil, since the wavelength changes when light enters a medium with a different refractive index. $\lambda_{\text{oil}} = \frac{\lambda_{\text{air}}}{n_o} = \frac{580 \, \text{nm}}{1.50} = 386.67 \, \text{nm}$

2. Condition for Constructive Interference in Thin Film: For constructive interference in a thin film with a higher refractive index (oil) surrounded by media with lower refractive indices (air and water), the condition is: $2t = m \lambda_{\text{oil}}$ where $m$ is an integer representing the order of interference, and $t$ is the thickness of the film.

   Since we are looking for the minimum thickness that produces constructive interference, we assume the first order of interference, $m = 1$.

3. Solve for Thickness $t$: $2t = 1 \times 386.67 \, \text{nm}$ $t = \frac{386.67 \, \text{nm}}{2} = 193.33 \, \text{nm}$

Final Answer

The thickness of the oil film at point B is: $t = 193.33 \, \text{nm}$

Would you like more details on this calculation or have any questions?

Here are some follow-up questions that could deepen your understanding:

1. What is the effect of changing the refractive index of the oil on the observed colors?
2. How would the interference pattern change if the light source were monochromatic instead of white?
3. What would be the required thickness for constructive interference of blue light?
4. How does the thickness requirement differ for destructive interference in this setup?
5. What are the practical applications of thin-film interference in everyday life?

Tip: When analyzing thin-film interference problems, always adjust the wavelength according to the medium’s refractive index. This change affects the interference conditions directly.