A \(5.000-\mathrm{cm}\) -wide diffraction grating with 200 grooves is used to resolve two closely spaced lines (a doublet) in a spectrum. The doublet consists of two wavelengths, \(\lambda_{\mathrm{a}}=\) \(629.8 \mathrm{nm}\) and \(\lambda_{\mathrm{b}}=630.2 \mathrm{nm} .\) The light illuminates the entire grating at normal incidence. Calculate to four significant digits the angles \(\theta_{1 \mathrm{a}}\) and \(\theta_{1 \mathrm{~b}}\) with respect to the normal at which the first-order diffracted beams for the two wavelengths, \(\lambda_{\mathrm{a}}\) and \(\lambda_{\mathrm{b}}\), respectively, will be reflected from the grating. Note that this is not \(0^{\circ}\) What order of diffraction is required to resolve these two lines using this grating?

The grating equation is given by: \[d(\sin{\theta_m} + \sin{\theta_i}) = m\lambda\] In our case, the light is illuminating the grating at normal incidence, so \(\theta_i = 0\). Therefore, the grating equation simplifies to: \[d\sin{\theta_m} = m\lambda\]

We are given that there are 200 grooves in a \(5.000\,\text{cm}\)-wide grating. To find the grating spacing \(d\), we will divide the width of the grating by the number of grooves. \[d = \frac{5.000\,\text{cm}}{200} = \frac{5.000 \times 10^{-2}\,\text{m}}{200} = 2.500 \times 10^{-4} \,\text{m}\]

Now we will use the grating equation to find the angles \(\theta_{1a}\) and \(\theta_{1b}\) for the first-order diffraction (\(m = 1\)) and the given wavelengths \(\lambda_a = 629.8\,\text{nm}\) and \(\lambda_b = 630.2\,\text{nm}\). For \(\theta_{1a}\): \[2.500 \times 10^{-4} \,\text{m} \sin{\theta_{1a}} = (1) (629.8 \times 10^{-9}\,\text{m})\] \[\sin{\theta_{1a}} = \frac{(1) (629.8 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}}\] \[\theta_{1a} = \arcsin{(\frac{(1) (629.8 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}})} = 0.14601 \,\text{rad}\] Now converting radians to degrees, we get: \[\theta_{1a} = 0.14601 \,\text{rad} \times \frac{180^\circ}{\pi} = 8.3715 ^\circ\] For \(\theta_{1b}\): \[2.500 \times 10^{-4} \,\text{m} \sin{\theta_{1b}} = (1) (630.2 \times 10^{-9}\,\text{m})\] \[\sin{\theta_{1b}} = \frac{(1) (630.2 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}}\] \[\theta_{1b} = \arcsin{(\frac{(1) (630.2 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}})} = 0.14624 \,\text{rad}\] Now converting radians to degrees, we get: \[\theta_{1b} = 0.14624 \,\text{rad} \times \frac{180^\circ}{\pi} = 8.3797 ^\circ\] So, to four significant digits, the angles are \(\theta_{1a} = 8.372^\circ\) and \(\theta_{1b} = 8.380^\circ\).

Resolving power of a grating is given by: \[R = mN\] where \(R\) is the resolving power, \(m\) is the order of diffraction, and \(N\) is the total number of grooves. We are given that there are 200 grooves in the grating. To find the minimum order of diffraction \(m\) that can resolve the two lines, we can use the shortest wavelength \(\lambda_a\) and the wavelength difference \(\Delta\lambda = \lambda_b - \lambda_a\) as follows: \[R = \frac{\lambda_a}{\Delta\lambda}\] \[mN = \frac{\lambda_a}{\Delta\lambda}\] \[m = \frac{\lambda_a}{\Delta\lambda \times N}\] \[m = \frac{629.8 \times 10^{-9}\,\text{m}}{(630.2 - 629.8) \times 10^{-9}\,\text{m} \times 200}\] \[m \approx 0.998\] Since the order of diffraction must be an integer, we will round \(m\) up to the next integer. Therefore, the minimum order of diffraction required to resolve the two lines is \(m = 1\).