Question

A diffraction grating 3.00 cm wide produces the second order at 33.0° with light of wavelength 600 nm. What is the total number of lines on the grating?

Short answer

The number of diffraction grating is 13617 lines.

Step-by-step solution

Identification of given data

The given data can be listed below,

• The width of the grating is,$D=0.03\text{m}$
• The order of diffraction is,$m=2$
• The wavelength of light is,$\lambda =600\text{nm}$

Concept/Significance of Bragg’s law

According to the Bragg’s law, the angle of the x-ray when it strikes a crystal surface,, will reflect back with the same angle of scattering,. Additionally, a constructive interference will take place when the path difference, d, equals a wavelength number, n.

Determination of the total number of lines on the grating

The brags law is given by,

$$\begin{gathered} d\sin \theta =m\lambda \\ d=\frac{m\lambda }{\sin \theta } \end{gathered}$$

Here, is the wavelength of the light, dis the slit width, m is the order of diffraction minima.

Substitute all the values in the above, the slit width of the grating,

$$\begin{gathered} d=\frac{2\left(600\times {10}^{-9}\text{m}\right)}{\sin 33°} \\ =2.203\times {10}^{-6}\text{m} \end{gathered}$$

The number of gratings is given by,

$N=\frac{D}{d}$

Here, D is the width of the grating and dis the slit width.

Substitute all the values in the above,

$$\begin{gathered} N=\frac{0.030\text{m}}{2.203\times {10}^{-6}\text{m}} \\ =13617\text{lines} \end{gathered}$$

Thus, the number of diffraction grating is 13617 lines.