Answer to Question #102516 in Optics for DD

Question

A diffraction grating has 2000 lines. Two sources of monochromatic lights emanating wavelengths λ1 and λ2 are just resolved by the grating in the first order spectrum. If the mean of λ1 and λ2 is 600 nm, calculate the difference between the wavelengths

Answer 1

$R=mN=\dfrac{\lambda_1}{\lambda_2-\lambda_1} ​$

$\frac{\lambda_1+\lambda_2}{2}=600$ $\to\lambda_1=1200-\lambda_2$

$m=1$

So, we have

$1200N-\lambda_1N-\lambda_1N=\lambda_1$

$1200N=\lambda_1(2N+1)$

from these two equations we get

$\lambda_1=\dfrac{1200N}{2N+1}=\dfrac{1200\cdot 2000}{2\cdot 2000+1}=599.85 nm$

$\lambda_2=1200-599.85=600.15 nm$

Hence the difference is

$\lambda_2-\lambda_1=600.15-599.85=0.3 nm$