91影视

The equation 36.28 from the textbook is

The expression to calculate the phase between the two adjacent slit is given as follows.

$\mathbf{鈭�}\mathit{蠒}\mathbf{=}\mathbf{2}\mathbf{蟺尘}\mathbf{+}\frac{\mathbf{2}\mathbf{蟺}}{\mathbf{N}}$

Here, $\mathit{N}$is the number of the slits and localid="1663142714174" $\mathit{m}$is the order of diffraction.

The expression to calculate the path difference between the two adjacent slit is given as follows.

鈥︹�� (i)

Here, $\mathit{位}$is the wavelength.

The condition for the diffraction maxima is given by,

$d\sin 胃=m位$

Calculate the path difference,

Substitute$\left(2\mathrm{蟺尘}+\frac{2\mathrm{蟺}}{\mathrm{N}}\right)$for $鈭�蠒$in to equation (i).

鈥︹�� (ii)

$$\begin{gathered} 鈭�L=\left(2\mathrm{蟺尘}+\frac{2\mathrm{蟺}}{\mathrm{N}}\right)\frac{位}{2\mathrm{蟺}} \\ 鈭�L=2\mathrm{蟺}\left(m+\frac{1}{\mathrm{N}}\right)\frac{位}{2\mathrm{蟺}} \\ 鈭�\mathrm{L}=\left(\mathrm{m}+\frac{1}{\mathrm{N}}\right)\mathrm{位} \\ 鈭�\mathrm{L}=\mathrm{尘位}+\frac{\mathrm{位}}{\mathrm{N}} \end{gathered}$$

Let assume the$胃$is the angular position of the $m^{th}$order maxima, and $d$is the slit separation, therefore the path difference is also given by,

鈥︹�� (iii)

$鈭�L=sdin(胃+鈭�胃]$

Equate the equation (ii) and (iii).

$\begin{array}{ll}m位+\frac{位}{N}=d\sin \left(胃+鈭�胃\right)& \\ m位+\frac{位}{N}=d\left[\sin \left(胃\right)\cos \left(鈭�胃\right)+\sin \left(鈭�胃\right)\cos \left(胃\right)\right]& \end{array}$

Since the is very small, therefore $\sin \left(鈭�胃\right)=鈭�胃$and$\cos \left(鈭�胃\right)=1$ .

$\begin{array}{l}m位+\frac{位}{N}=d(sin胃脳1+鈭�胃\cos 胃)\\ m位+\frac{位}{N}=dsin胃+d鈭�胃cos胃\end{array}$

Substitute$m位$ for $d\sin \mathrm{胃}$into above equation.

role="math" localid="1663142681080" $$\begin{gathered} m位+\frac{位}{n}=m位+d鈭�胃\cos 胃 \\ \frac{位}{n}=d鈭�胃\cos 胃 \\ 鈭�胃=\frac{位}{Nd\cos 胃} \end{gathered}$$

Hence the required equation for the half width of the lines is diffraction grating pattern is$鈭�胃=\frac{位}{Nd\cos 胃}$ .