Question

a grating that has 3600 slits per cm produces a third - order fringe at a 22.0 ° angle.
part a
what wavelength of light is being used?
express your answer to two significant figures and include the appropriate units.
λ = value units
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Explanation:

Step1: Find the grating spacing \( d \)

The grating has 3600 slits per cm. So, \( d=\frac{1\ cm}{3600}=\frac{1\times10^{-2}\ m}{3600}\approx2.778\times10^{-6}\ m \)

Step2: Use the grating equation \( d\sin\theta = m\lambda \)

We know \( m = 3 \) (third - order), \( \theta=22.0^{\circ} \), and \( d = 2.778\times10^{-6}\ m \). Rearranging for \( \lambda \), we get \( \lambda=\frac{d\sin\theta}{m} \)
Substitute the values: \( \sin(22.0^{\circ})\approx0.3746 \), \( d = 2.778\times10^{-6}\ m \), \( m = 3 \)
\( \lambda=\frac{2.778\times10^{-6}\ m\times0.3746}{3}\approx3.4\times10^{-7}\ m = 340\ nm \) (or \( 3.4\times10^{2}\ nm \), or \( 3.4\times10^{-7}\ m \))

Answer:

\( \lambda=\boxed{340\ nm} \) (or \( \lambda=\boxed{3.4\times10^{-7}\ m} \))