To find the smallest thickness of a soap bubble that produces constructive interference, we can use the formula for thin-film interference.

Constructive Interference in Thin Films

For constructive interference in a thin film, the condition is:

$2nt = m\lambda$

where:

• $n$ is the refractive index of the soap bubble,
• $t$ is the thickness of the soap film,
• $m$ is the interference order (an integer, typically starting at $m = 0$),
• $\lambda$ is the wavelength of light in a vacuum.

However, when the light reflects off a higher-index medium (in this case, from air $n = 1$ to soap film $n = 1.36$), a phase shift of $\lambda / 2$ occurs. This shifts the condition for constructive interference to:

$2nt = (m + \frac{1}{2})\lambda$

For the smallest thickness (first order of constructive interference, $m = 0$):

$2nt = \frac{\lambda}{2}$

Solving for $t$:

$t = \frac{\lambda}{4n}$

Plugging in the values:

• $\lambda = 627 \, \text{nm}$ (wavelength of light),
• $n = 1.36$ (index of refraction of soap).

$t = \frac{627 \, \text{nm}}{4 \times 1.36} = \frac{627 \, \text{nm}}{5.44} \approx 115.29 \, \text{nm}$

Therefore, the smallest thickness $t$ is approximately 115.3 nm.

Would you like to dive into the interference mechanisms or explore further related problems?

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Here are 5 related questions to explore:

1. How would the thickness change for a different wavelength of light?
2. What happens if the index of refraction of the soap changes?
3. How does the interference pattern differ for destructive interference?
4. What is the effect of having multiple layers of films?
5. Can interference still occur if the bubble's thickness is not uniform?

Tip: Thin-film interference is responsible for the iridescent colors seen in soap bubbles due to variations in thickness across the bubble!