Scientists are testing a transparent material whose index of refraction for visible light varies with wavelength as n = 30.0 nm^1/2/λ^1/2 , where λ is in nm. If a 295-nm-thick coating is placed on glass (n=1.50), for what visible wavelengths will the reflected light have maximum constructive interference?

Verified step by step guidance

1. Step 1: Understand the concept of constructive interference. Constructive interference occurs when the path difference between two reflected waves is an integer multiple of the wavelength. For thin-film interference, this condition is given by: 2t = mλ/n, where t is the thickness of the film, n is the refractive index of the film, λ is the wavelength in vacuum, and m is an integer (order of interference).
2. Step 2: Substitute the given refractive index formula into the equation. The refractive index of the material is given as n = 30.0 nm^(1/2)/λ^(1/2). Replace n in the thin-film interference equation with this expression to account for the wavelength dependence of the refractive index.
3. Step 3: Rearrange the equation to solve for λ. After substituting n, the equation becomes 2t = mλ/(30.0 nm^(1/2)/λ^(1/2)). Simplify this to express λ in terms of t, m, and the constants provided.
4. Step 4: Plug in the given values. The thickness of the coating is t = 295 nm, and the refractive index of the glass is n = 1.50. Use these values to calculate the wavelengths λ for different integer values of m (e.g., m = 1, 2, 3, etc.) that fall within the visible spectrum (approximately 400 nm to 700 nm).
5. Step 5: Identify the visible wavelengths. After calculating λ for different values of m, filter out the wavelengths that lie within the visible range. These are the wavelengths for which the reflected light will have maximum constructive interference.

Key Concepts

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. In this case, the index of refraction varies with wavelength, which affects how light interacts with the material, particularly in terms of reflection and refraction.

Constructive Interference

Constructive interference occurs when two or more waves combine to produce a wave of greater amplitude. For light waves, this happens when the path difference between the waves is an integer multiple of the wavelength. In the context of thin films, such as the coating on glass, specific wavelengths will reinforce each other, leading to bright reflections.

Thin Film Interference

Thin film interference is a phenomenon that occurs when light waves reflect off the boundaries of a thin layer of material. The thickness of the film and the wavelength of light determine the conditions for constructive or destructive interference. In this scenario, the thickness of the coating and the varying index of refraction with wavelength will dictate which visible wavelengths experience maximum constructive interference.