When light reflects off a film, we can use the principles of constructive interference to determine the conditions for a bright reflection. The film in this case is surrounded by air, which has a lower index of refraction than the soap film. Thus, when light reflects off the top and bottom surfaces of the soap film, a phase shift occurs at the top surface.
For constructive interference in a thin film where light reflects from a medium of lower refractive index (air) to one of higher refractive index (soap film), the additional path difference that contributes to constructive interference is given by:
\[ 2nt = m\lambda \]
where:
- \( n \) is the index of refraction of the film,
- \( t \) is the thickness of the film,
- \( m \) is the order of interference (an integer),
- \( \lambda \) is the wavelength of light in vacuum or air.
Since the light in air is 588 nm, we need to use this wavelength in the equation.
Given:
- \( n = 1.33 \) (index of refraction of the soap film),
- \( \lambda = 588 , \text{nm} \) (wavelength in air).
Substituting these values into the equation we get:
\[ 2(1.33)t = m(588 , \text{nm}) \] \[ 2.66t = m(588) \] \[ t = \frac{m(588)}{2.66} \]
Now we need to find the thickness for different values of \( m \).
Let's calculate this for \( m = 1 \) and \( m = 2 \):
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For \( m = 1 \): \[ t_1 = \frac{1 \times 588 , \text{nm}}{2.66} \approx \frac{588}{2.66} \approx 220.3 , \text{nm} \]
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For \( m = 2 \): \[ t_2 = \frac{2 \times 588 , \text{nm}}{2.66} \approx \frac{1176}{2.66} \approx 442.2 , \text{nm} \]
The second smallest possible value for the thickness of the film corresponds to \( m = 2 \):
\[ \boxed{442.2 , \text{nm}} \]