The analysis shown below also applies to diffraction gratings with lines separated by a distance \(d .\) What is the distance between fringes produced by a diffraction grating having 125 lines per centimeter for 600 -nm light, if the screen is \(1.50 \mathrm{m}\) away? (Hint: The distance between adjacent fringes is \(\Delta y=x \lambda / d, \quad\) assuming the slit separation \(d\) is comparable to \(\lambda_{-}\) )

When dealing with diffraction and interference of light, it's essential to comprehend the nature of the wavelength involved. In physics exercises, such as the diffraction grating fringe calculation, you might come across the wavelength of light mentioned in nanometers (nm). To fit into the standard metric system used in most scientific calculations, this unit needs to be converted into meters (m), which is the SI unit for length.

To convert nanometers to meters, we divide the quantity in nanometers by 1 billion, or to put it numerically, by \( 10^9 \). This is because 1 meter equals 1 billion nanometers. For example, in the given problem, the conversion of a 600 nm wavelength of light into meters is calculated by dividing 600 by \( 10^9 \), resulting in a wavelength \( \lambda \) which is \( 6 \times 10^{-7} \) meters. Converting wavelengths correctly is crucial to ensure that all units align, allowing for proper fringe distance calculation using the diffraction grating equation.

In a diffraction grating, the term 'slit separation' refers to the distance between adjacent slits or lines and is denoted as \( d \). It plays a vital role in determining the pattern produced by the grating when light of a specific wavelength falls upon it. To find this separation, you need to know the number of lines or slits per unit length of the grating, which is often given per centimeter or per meter.

For instance, if a grating has 125 lines per centimeter, you first convert this into lines per meter by multiplying by 100, since 1 meter equals 100 centimeters. In our problem, this results in 12,500 lines per meter. The slit separation \( d \) is then found by taking the reciprocal of this value, essentially dividing 1 meter by the number of lines per meter. This yields \( d = 8 \times 10^{-5} \) meters. Understanding how to calculate this separation is crucial because it directly affects the interference pattern's fringe spacing observed on a screen.

Once the slit separation and wavelength are determined, analyzing the diffraction pattern becomes the next step. The pattern on a screen due to a diffraction grating is made of bright and dark fringes, caused by constructive and destructive interference, respectively. The distances between these fringes is a measure of the diffraction pattern's detail.

The distance between adjacent fringes \( \Delta y \) can be calculated using the formula \( \Delta y = x \frac{\lambda}{d} \) where \( x \) is the distance from the grating to the screen, \( \lambda \) is the wavelength of light, and \( d \) is the slit separation. By plugging in the values into the formula, one can calculate the space between the fringes that appear on the screen. In the exercise, by substituting \( x = 1.50 \) meters, \( \lambda = 6 \times 10^{-7} \) meters, and \( d = 8 \times 10^{-5} \) meters, we can calculate the fringe spacing \( \Delta y \) to be approximately 11.25 millimeters.

Analyzing the diffraction pattern helps in the understanding of light behavior and the properties of the material composing the grating. This also aids in various applications such as the measurement of light wavelengths and the testing of optical components.