91Ó°ÊÓ

Grating spacing is a crucial concept when dealing with diffraction gratings. It refers to the separation between adjacent rulings on a grating and is often denoted by the symbol \( d \). For the given problem, a diffraction grating has a width of 20.0 mm and features 6000 rulings. The formula to calculate grating spacing is:

• \( d = \frac{W}{N} \)

Here, \( W \) is the total width of the grating, and \( N \) is the number of rulings. Substituting the given values, \( d = \frac{20.0 \, \text{mm}}{6000} = 3.33 \, \times \, 10^{-6} \, \text{m} \). This result is also expressed as \( 3333 \, \text{nm} \) for consistency with wavelength units. Having accurate grating spacing is essential for calculations involving diffraction patterns.

In optics, diffraction maxima occur when waves overlap constructively, leading to bright bands on a viewing screen. This phenomenon is mathematically expressed by the formula:

• \( d \sin \theta = m \lambda \)

Here, \( d \) is the grating spacing, \( \lambda \) is the wavelength of the light, and \( m \) denotes the order of the maximal intensity points, starting with \( m = 0, 1, 2, \) and so on. The condition \( \sin \theta \leq 1 \) restricts the maximum order \( m \) and thus determines the possible angles \( \theta \). In the example provided, the integer values of \( m \) are found by dividing \( d \) by \( \lambda \) and rounding down to the closest integer, which in this case allows up to \( m = 5 \). This allows us to compute the angles \( \theta \) for each consecutive maxima.

Wavelength calculation is integral to understanding diffraction patterns because it helps determine the angles at which maxima occur. Using the relationship from the diffraction grating formula:

• \( \sin(\theta) = \frac{m \lambda}{d} \)

Here, \( m \) is the order of the maxima, \( \lambda \) represents the wavelength of the incident light, and \( d \) is the grating spacing. In the problem, the wavelength \( 589 \, \text{nm} \) is given. Calculating the diffraction angle involves substituting values for \( m \) and solving for \( \theta \). For instance, for the highest order \( m = 5 \), \( \theta \) is approximately \( 62.3^\circ \). Similarly, for \( m = 4 \) and \( m = 3 \), the respective angles are \( 45.0^\circ \) and \( 32.0^\circ \). These calculations reveal how light of a specific wavelength interacts with a grating, manifesting as bright lines at distinct angles.