Question

A diffraction grating with 161 slits per centimeter is used to measure the wavelengths emitted by hydrogen gas. At what angles in the first-order spectrum would you expect to find the two violet lines of wavelength 434 nm and of wavelength 410 nm

Answer 1

Answer:

$\theta_1 = 0.400^o$

$\theta_2 =0.378^o$

Explanation:

From the question we are told that

    The  number of slits per cm is  k =  $161\ slits\ per\ cm = 161 \ slits\ per\ 0.01 m$

    The order of the maxima is  n =  1

    The wavelength are  $\lambda_1 = 434 nm = 434 *10^{-9} \ m \ \ \ , \lambda_2 = 410nm = 410 *10^{-9} \ m$

The  spacing between the slit is mathematically represented as

           $d = \frac{ 0.01}{k}$

=>       $d = \frac{ 0.01}{161}$

=>         $d = 6.211 *10^{-5} \ m$

Generally the condition for constructive interference is  

        $n\lambda = d \ sin \theta$

At  $\lambda_1$

      $\theta _1 = sin^{-1} [ \frac{1 * 434 *10^{-9}}{6.211 *10^{-5}} ]$

      $\theta_1 = 0.400^o$

At  $\lambda_2$

       $\theta _2 = sin^{-1} [ \frac{1 * 410 *10^{-9}}{6.211 *10^{-5}} ]$

       $\theta_2 =0.378^o$