The wings of a certain beetle have a series of parallel lines across them. When normally incident 520-nm light is reflected from the wing, the wing appears bright when viewed at an angle of 56°. How far apart are the lines?

Verified step by step guidance

1. Identify the problem as involving diffraction grating, where light of a known wavelength is diffracted by parallel lines and observed at a specific angle.
2. Recognize that the formula to use is the diffraction grating equation: $d \sin(\theta) = m \lambda$, where $d$ is the distance between the lines (grating spacing), $\theta$ is the angle of the observed bright fringe, $m$ is the order of the maximum (which is 1 for the first order maximum), and $\lambda$ is the wavelength of the light.
3. Substitute the given values into the diffraction grating equation. Here, $\theta = 56^\circ$, $\lambda = 520$ nm, and $m = 1$.
4. Solve the equation for $d$ to find the spacing between the lines: $d = \frac{m \lambda}{\sin(\theta)}$.
5. Calculate $d$ using the values of $m$, $\lambda$, and $\sin(\theta)$ to find the distance between the lines on the beetle's wing.

Key Concepts

Interference of Light

Interference occurs when two or more light waves overlap, resulting in a new wave pattern. In the context of the beetle's wings, the parallel lines create conditions for constructive and destructive interference, which affects the color and brightness observed at different angles. This phenomenon is crucial for understanding how the beetle's wings reflect light and appear bright at specific angles.

Wavelength of Light

The wavelength of light is the distance between successive peaks of a light wave, measured in nanometers (nm). In this question, the 520-nm wavelength is significant because it determines the color of the light and influences how it interacts with the beetle's wing structure. The spacing of the lines on the wing must be comparable to this wavelength to produce noticeable interference effects.

Angle of Reflection

The angle of reflection is the angle at which light reflects off a surface, measured from the normal (a line perpendicular to the surface). In this scenario, the angle of 56° is critical as it indicates the specific conditions under which the reflected light is most intense. This angle helps in determining the spacing of the lines on the beetle's wings, as it relates to the interference pattern created by the reflected light.