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(i) A diffraction grating has 560 lines per mm. Calculate the grating spacing. If the total length of the grating is 2 cm, how many lines are there?

(ii) Light of wavelength 500 nm is incident on the grating with this spacing. Calculate the deviation angle for 1st order diffraction.

I am having difficulty with the (ii) part. For my spacing I got 0.0018 mm. So, then I used

\begin{align} n \lambda &= d \sin{\Theta} \\ \frac{n \lambda}{d} &= \sin{\Theta} \\ 0.27 &= \sin{\Theta} \\ \Theta &= \sin^{-1}{(0.27)} = 15.7^\circ \end{align}

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The grating equation you've used is correct, and in order to find the first order diffraction angle ($n = 1$) you need to use arcsine trigonometric function and carefully place data with respective units:

\begin{align} n \lambda &= d \sin{\Theta} \tag{1}\\ d &= \frac{\pu{1 mm}}{N}~[\pu{mm}] \tag{2} \end{align}

where $N$ is number of lines per unit length. Combining (1) and (2) and converting the wavelength from nm to mm:

$$\Theta = \arcsin{\left(\frac{n \lambda}{d}\right)} = \arcsin{\left(\frac{n \lambda N}{\pu{1 mm}}\right)} = \arcsin{\left(\frac{1 \cdot \pu{5e-4 mm} \cdot 560}{\pu{1 mm}}\right)} = 16.26^\circ \tag{3}$$

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