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Question

# The light shining on a diffraction grating has a wavelength of 495 nm (in vacuum). The grating produces a second-order bright fringe whose position is defined by an angle of$9.34^\circ.$How many lines per centimeter does the grating have?

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Constructive interference creates the principal fringes. In a diffraction grating set up, the principal maxima can be determined using Equation 27.7:

\begin{align*} \sin \theta = m \frac{\lambda}{d} \quad \quad \text{m = 0, 1, 2, 3, ...} \end{align*}

where $d$ is the separation between the slits, $\lambda$ is the wavelength of the light and $m$ is the order of the maxima. We solve for $d$ and we supply the known values. We note that the second-order maximum of a diffraction grating corresponds to $m =2$.

\begin{align*} d &= \frac{m\lambda}{\sin \theta} \\ &= \frac{(2)(495\times 10^{-9}\;\text{m})}{\sin 9.34^{\;\circ}} \\ &= 6.10\times 10^{-6}\;\text{m} \end{align*}

The grating number of lines ($N$) per centimeter is determined by getting the reciprocal of the separation distance between the slits, $d$. We convert $d$ in cm.

\begin{align*} N = \frac{1}{d} = \frac{1}{6.10\times 10^{-4}\;\text{cm}} = \boxed{1640\;\text{lines per cm}} \end{align*}

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