A diffraction grating with 2000 lines per centimeter is used to measure the wavelengths emitted by a hydrogen gas discharge tube. (a) At what angles will you find the maxima of the two first-order blue lines of wavelengths 410 and \(434 \mathrm{nm} ?\) (b) The maxima of two other first-order lines are found at \(\theta_{1}=0.097 \mathrm{rad}\) and \(\theta_{2}=0.132 \mathrm{rad} .\) What are the wavelengths of these lines?

Understanding the concept of grating spacing is crucial when working with a diffraction grating. The grating spacing, denoted as (d), refers to the distance between adjacent lines or grooves on the diffraction grating. It's vital because it directly influences the diffraction pattern produced by the grating. To calculate the grating spacing, the formula

\[ d = \frac{1}{N} \]

is used, where (N) represents the number of lines per unit length on the grating. In the given exercise, to convert lines per centimeter to lines per meter, the value is multiplied by 100, which then allows us to find the spacing in meters. Here's how it works: Let's say we're given a grating with 2000 lines per centimeter. We convert this to 200,000 lines per meter because there are 100 centimeters in a meter. The spacing (d) is therefore 1 divided by 200,000, ending up with a very small distance indicative of the precision of these optical devices.

The diffraction grating equation is the core tool used to understand how light diffracts through a grating. Demonstrating how this formula works can significantly enhance a student's comprehension of diffraction phenomena. The equation

\[ n\lambda = d\sin(\theta) \]

is fundamental in this context. It relates the wavelength of the light (\( \lambda \)), the grating spacing (\( d \)), and the angle of diffraction (\( \theta \)) to the order of the maximum (\( n \)). To clarify, \( n \) represents the order of the maximum, which refers to the sequence of bright bands produced, with 1 being the first order. Using this equation, one can predict the angles (\( \theta \)) where light at specific wavelengths will form bright fringes. In the exercise, we apply this equation to find the angles for blue light wavelengths of 410 nm and 434 nm, revealing the intricate dance between light and the precision-engineered grating lines.

When it comes to identifying spectral lines, the wavelengths provided by the diffraction grating hold the key. Spectral lines are the unique fingerprints of elements—which emit or absorb specific wavelengths of light. Determining these wavelengths is a foundational aspect of studying atomic and molecular properties. In the given exercise, the problem provides the angles at which the maxima occur for two spectral lines and asks to find their respective wavelengths. By rearranging the diffraction grating equation to solve for wavelength (\( \lambda \)), we can plug in the known values of \( d \) and \( \theta \) to calculate the wavelength associated with those spectral lines. This process uses the wonderful precision of diffraction gratings to unlock the mysteries of atomic emission and absorption, which has profound implications in fields like astrophysics, quantum mechanics, and even art conservation.