Final answer:
To find the wavelengths of light with maximum transmittance through a thin film, one must consider the conditions for constructive interference caused by thin film interference. The number of different wavelengths can be found by dividing the thickness by integer multiples and considering only the visible spectrum.
Step-by-step explanation:
Thin Film Interference and Maximum Transmittance -
The question involves understanding thin film interference in the context of a glass plate coated with a thin film. To determine the number of light beams with different wavelengths that have maximum transmittance through the film, we analyze the condition for constructive interference. This occurs when the phase shift caused by the light waves reflecting off the top and bottom surfaces of the film results in them combining to reinforce each other.
For maximum transmittance and constructive interference, the path length difference must be an integer multiple of the wavelength in the medium (mλfilm), where m is an integer (order of interference) and λfilm is the wavelength of light in the film. The refractive index (n) of the film reduces the wavelength of light within the film according to λfilm = λair / n.
The phase shift upon reflection depends on whether the reflection occurs at a boundary with higher or lower refractive index. In this case, light reflects from a medium with lower refractive index (air) to higher (film), and no phase shift occurs at the first surface. However, at the second surface (film to glass), there is a phase shift as the light reflects from a medium of higher refractive index (film) to one of lower refractive index (glass). The effective path length difference for maximum transmittance is thus given by 2t = mλair / n, where t is the thickness of the film.
To find the wavelengths that correspond to maximum transmittance, divide the thickness by integer multiples. However, we only consider wavelengths in the visible spectrum (approximately 400 nm to 700 nm). Each integer m that results in a wavelength within this range corresponds to a maximum transmittance peak.