Chapter 24, Problem 35P

To determine

The maximum number of lines per cm for the grating.

Answer to Problem 35P

  $6.67\times {10}^3$ lines $/cm$

Explanation of Solution

Given info:

Number of grating per unit length,N=6000 lines $/cm$

Wavelength, $\lambda =750nm=750\times {10}^{-9}m$

Formula used:

It is known that $\sin \theta =\frac{m\lambda }{d}$

Where,

   m is an integer

  λ is wavelength

  d is distance between slits

Calculation:

In constructive interference, path difference is an integer multiple of wavelength.

Rearrange the given formula$\sin \theta =\frac{m\lambda }{d};So,d\sin \theta =m\lambda$

Because the angle increases with wavelength, to have a full order, use the largest wavelength.

So, the maximum Angle, $\theta ={90}^{°}$

Now, calculate the minimum separation from by using d $\sin \theta =m\lambda ,$ with $m=2$

  $d_{\min }\sin {90}^{°}=(2)\left(750\times {10}^{-9}m\right)$

  $d_{mim}=1.50\times {10}^{-6}m=1.50\times {10}^{-4}cm$

Now, calculate the maximum number of lines per cm:

  $\frac{1}{d_{mim}}=\frac{1}{\left(1.50\times {10}^{-4}cm\right)}=6.67\times {10}^{3}lines/cm$

Conclusion:

The required value is $6.67\times {10}^{3}lines/cm$ .