91影视

The wavelength consists by light beam from $460nm$to $640nm$.

The diffraction grading, $\frac{W}{N}=\frac{1}{160lines/mm}$

The expression for the maxima of the diffraction is given as follows.

$\mathrm{诲蝉颈苍胃}=\mathrm{尘位}鈥�鈥�\left(\mathrm{i}\right)$
The expression to calculate the angle is given as follows.

$\mathrm{胃}={\sin }^{-1}\left(\frac{\mathrm{尘位}}{\mathrm{d}}\right)鈥�鈥�.\left(\mathrm{ii}\right)$

(a)

The width can be calculated as,

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

$d=\frac{1}{160lines/mm}$

$d=\frac{1}{160脳{10}^{-3}lines/m}$

$d=6.25脳{10}^{-6}m$

Calculate the angle for the wavelength$位=460\text{nm}$and$m=2$.

Substitute for $460nm$for $位$, $2$for $m$ and $6.25脳{10}^{-6}\text{m}$and for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{2脳460脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.1472\right) \\ 胃=8.46掳 \end{gathered}$$

Calculate the angle for the wavelength $位=640\text{nm}$and $m=2$.

Substitute $640nm$for $位$,$2$for $m$and $6.25脳{10}^{-6}\text{m}$for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{2脳640脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.2048\right) \\ 胃=11.82掳 \end{gathered}$$

Calculate the angle for the wavelength$位=460\text{nm}$ and $m=3$.

Substitute $460nm$for $位$,$3$for $m$and$6.25脳{10}^{-6}\text{m}$for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{3脳460脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.2208\right) \\ 胃=12.75掳 \end{gathered}$$

Calculate the angle for the wavelength$位=640nm$ and $m=3$.

Substitute$640nm$for $位$, $3$for $m$and$6.25脳{10}^{-6}\text{m}$for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{3脳640脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.3072\right) \\ 胃=17.89掳 \end{gathered}$$

Calculate the angle for the wavelength$位=460nm$and $m=4$.

Substitute $460nm$for $位$,$4$for $m$and $6.25脳{10}^{-6}\text{m}$for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{4脳460脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.2944\right) \\ 胃=17.12掳 \end{gathered}$$

Calculate the angle for the wavelength$位=640nm$ and $m=4$.

Substitute $640nm$for $位$, $4$for $m$and $6.25脳{10}^{-6}\text{m}$for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{4脳640脳0^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.4096\right) \\ 胃=24.17掳 \end{gathered}$$

Since from the above calculation, it can be seen that the value of $胃$the for $位=460\text{nm}$,$m=4$ and $位=640nm$,$m=3$ is nearly equals and overlaps. Therefore, the lowest order is $m=3$.

Hence the lowest order that is overlaps by the another is $3$.

(b)

To present the full spectral, the angle with the axis,$胃=90掳$

Substitute$90掳$for $胃$, $6.25脳{10}^{-6}\text{m}$for $d$and $640nm$for$位$into equation (i).

$$\begin{gathered} 6.25脳{10}^{-6}脳\sin 90掳=m脳640脳{10}^{-9} \\ m=\frac{6.25脳{10}^{-6}脳\sin 90}{640脳{10}^{-9}} \\ m=\frac{6.25脳{10}^3}{640} \\ m=9.76 \end{gathered}$$

Hence the highest order at which full spectral can be seen is$9$ .

(c)

Calculate the value of angle,

Substitute $460nm$ for $位$,$9$for $m$and $6.25脳{10}^{-6}\text{m}$ for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{9脳460脳{10}^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.6624\right) \\ 胃=41.48掳 \end{gathered}$$

Hence the angle at the highest order for wavelength$460nm$ is$41.48掳$ .

(d)

Calculate the value of angle,

Substitute $640nm$for $位$,$9$for$m$and $6.25脳{10}^{-6}\text{m}$ for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{9脳640脳{10}^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.9216\right) \\ 胃=67.16掳 \end{gathered}$$

Hence the angle at the highest order for wavelength$640nm$ is $67.16掳$.

(e)

Calculate the highest order for which full spectral can be seen.

Substitute $90掳$for $胃$, $6.25脳{10}^{-6}\text{m}$ for $d$and $460nm$for $位$ into equation (i).

$$\begin{gathered} 6.25脳{10}^{-6}脳\sin 90掳=m脳460脳{10}^{-9} \\ m=\frac{6.25脳{10}^{-6}脳\sin 90}{460脳{10}^{-9}} \\ m=\frac{6.25脳{10}^3}{460} \\ m=13.58 \end{gathered}$$

Calculate the value of angle,

Substitute $460nm$for $位$, $13$for $m$and $6.25脳{10}^{-6}\text{m}$
for $d$into equation (ii).

$$\begin{gathered} 胃={\sin }^{-1}\left(\frac{13脳460脳{10}^{-9}}{6.25脳{10}^{-6}}\right) \\ 胃={\sin }^{-1}\left(0.9568\right) \\ 胃=73.09掳 \end{gathered}$$

Hence the greatest angle which the light at wavelength $460nm$appears is$73.09掳$ .