Question

A soap bubble has a thickness of $100\;nm\;$ and a refractive index $1.35$. Given that the wavelength of red light is about $700\;nm\;$ and that of blue light is$400\;nm$ , what colour does the bubble appear to be at the point on its surface closest to an observer when it is illuminated by white light?
A White
B Black
C Red
D Green

Answer 1

In thin-film interference light waves reflected by the upper and lower boundaries either enhancing or reducing the reflected light. When a white light incident on the film, certain wavelengths are intensified and others are attenuated.
The light through the soap bubble will be transmitted for the same condition as minimum reflection and the transmitted light will be in phase with the incident light.The phase is never inverted on transmission.
The internal surface reflection of light will also be in phase with the incident light.

Formula used:
When the thickness of the soap bubble is a multiple of a quarter-wavelength of the light the two reflected waves cancel each other. The wave was completely transmitted.
The film thickness must be $\dfrac{\lambda }{4}$
As we know, refractive index $\mu = \dfrac{{{\lambda _{air}}}}{{{\lambda _{soap{\text{ }}bubble}}}}$

Complete step by step answer:
As per the given vale,
Let the thickness of a soap bubble is ${t_{film}} = 100nm$, the refractive index of the bubble is ${\mu _{film}} = 1.35$.
The film thickness is $\dfrac{\lambda }{4}$
$\therefore$The wavelength in the soap bubble is ${\lambda _{film}} = 4{t_{film}}$
$\Rightarrow {\lambda _{film}} = 4 \times 100nm$
$\Rightarrow {\lambda _{film}} = 400nm$
Now, as we know ${\mu _{film}} = \dfrac{{{\lambda _{air}}}}{{{\lambda _{film}}}}$
$\Rightarrow {\lambda _{air}} = 1.35 \times 400nm$
$\Rightarrow {\lambda _{air}} = 540nm$
$\therefore$ The light having a wavelength $540nm$ will be transmitted through. And the green color has the wavelength of $540nm$.
Hence, The correct answer is option D.

Note: The true thickness of the film (here the soap bubble) depends on the refractive index and angle of incident of light. Higher the index medium slower the speed of light. In the normal angle of incidence, the thickness of the film will be a quarter or half multiple of the certain wavelength. But in the oblique angle of incidence, the thickness of the film will be the cosine of the angle at the quarter or half wavelength position.