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

$R=mN=\frac{\lambda_1}{\lambda_2-\lambda_1}$

$\frac{\lambda_1+\lambda_2}{2}=600$

$m=1$

$N\cdot \lambda_2-N\cdot \lambda_1=\lambda_1$

$\lambda_1=1200-\lambda_2$

So, we have

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

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

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

$\lambda_2=1200-599.85=600.15 nm$

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