Answer to Question #214918 in Optics for lala

Question

A diffraction grating produces a third-order bright spot for wavelength 550 nm at 50^o from the central maximum. What is/are the other angle(s) where the bright spot can occur? [angle is expressed in 2 significant figures]

A. 8^o

B. 18^o

C. 27^o

D. 50^o

Answer 1

$\text{For a diffraction grating}\newline d\sin\theta_m = m\lambda$

$\lambda= 550 nm$

$\theta_3 = 50\degree$

$d= \frac{m\lambda}{\sin\theta_m}=\frac{3*550}{\sin50\degree}\approx2154\ nm$

$m= \frac{d\sin\theta_m}{\lambda}$

$A.\text{if }\theta_m= 8\degree\newline m= \frac{d\sin8\degree}{\lambda}= \frac{2154\sin8\degree}{550}=0.54\newline m\notin Z$

$m= \frac{d\sin8\degree}{\lambda}= \frac{2154\sin8\degree}{550}=0.54$

$B.\text{if }\theta_m=18\degree\newline m= \frac{d\sin18\degree}{\lambda}= \frac{2154\sin18\degree}{550}=1.21\newline m\notin Z$

$C.\text{if }\theta_m= 27\degree\newline m= \frac{d\sin27\degree}{\lambda}= \frac{2154\sin27\degree}{550}=1.77\newline m\notin Z$

$D.\text{if }\theta_m= 50\degree\newline m= \frac{d\sin50\degree}{\lambda}= \frac{2154\sin50\degree}{550}=3\newline m\in Z$

$\text{Answer: D}.50\degree$