How many complete orders of the visible spectrum \((400 \mathrm{nm}<\lambda<700 \mathrm{nm})\) can be produced with a diffraction grating that contains 5000 lines per centimeter?

The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. This range includes wavelengths from approximately 400 nanometers (nm) to 700 nanometers. It's important to understand that this is the range where visible light exists, and each color we see falls into a specific wavelength within this range. For example, the color violet has the shortest wavelength, around 400 nm, and red has the longest, near 700 nm. In the context of diffraction gratings, this wavelength range is crucial because it determines which light we can analyze through diffraction patterns.

When we work with diffraction gratings, we're often interested in understanding how the grating will affect light within the visible spectrum. Since different colors will be diffracted by different amounts, depending on their wavelengths, it's essential to know this range to predict and calculate the behavior of light as it interacts with the grating.

The diffraction grating equation is foundational for understanding how light behaves when it encounters a surface with multiple slits. The equation itself is given by:\[\begin{equation} m\theta = d\theta (\theta) \end{equation}\] Here, \( m \) is the order of the spectrum, which can be thought of as the 'version' of the diffracted light you're observing—a higher order means a repetition of the spectrum. The variable \( \theta \) is the diffraction angle, which is how much the light bends when passing through the grating. The letter \( d \) is the distance between adjacent lines on the diffraction grating, also known as the grating line spacing. Wavelength (\theta) is measured in meters and often expressed in nanometers for convenience in the visible spectrum.

To visualize it, imagine light hitting the grating: some of it passes through the slits and gets bent at different angles, creating a spectrum. This equation helps us determine exactly at what angles these spectra can be observed. It is fundamental when calculating the specific details of how light of various wavelengths will be diffracted by the grating, which is especially useful in various fields like spectroscopy.

Grating line spacing is essentially the distance between adjacent lines in a diffraction grating, and it is paramount when examining the diffraction patterns produced by different wavelengths of light. To calculate the grating line spacing (\( d \)), we use the formula:\[\begin{equation} d = \frac{1}{\text{number of lines per unit length}} \end{equation}\] The unit length is typically centimeters (cm), so if we know the number of lines per centimeter, we can find the spacing by taking the reciprocal. For instance, if a grating has 5000 lines per cm, then the spacing would be \frac{1}{5000} cm between each line. Converting this to meters, which is a more standard unit in physics, we multiply by \frac{1}{100} to get the spacing in meters (m).

Understanding and calculating the grating line spacing are critical for predicting the diffraction pattern of light passing through the grating. With this knowledge, one can determine the angles at which different wavelengths of light will diffract and hence analyze the light's properties.