91影视

Write the expression forelectric field.

$E=\frac{1}{4蟺{蔚}_0}\frac{qd}{a^3}$ 鈥︹�� (1)

Here, ${蔚}_0$is the permittivity of free space, q is the charge, d is the distance and a is the radius of an atom.

Write the expression for binding force using equation (1) as,

$\begin{array}{c}{F}_{binding}=-qE\\ =-\left(\frac{1}{4蟺{蔚}_0}\frac{q^2}{a^3}\right)x\\ =-k_{spring}x\end{array}$ 鈥︹�� (2)

Here,$k_{\text{spring}}$ is the spring constant.

Substitute $F_{\text{binding}}=\frac{1}{4蟺{蔚}_0}\frac{q^2}{a^3}$and $k_{\text{spring}}=m{蝇}_0^2$in equation (2).

$\begin{array}{c}\left(\frac{1}{4蟺{蔚}_0}\frac{q^2}{a^3}\right)x=鈭�m{蝇}_0^{2}x\\ {蝇}_0=\sqrt{\frac{q^2}{4蟺{蔚}_{0}m{a}^3}}\end{array}$ 鈥︹�� (3)

Write the expression for the natural frequency.

${谓}_0=\frac{{蝇}_0}{2蟺}$

Substitute${蝇}_0=\sqrt{\frac{q^2}{4蟺{蔚}_{0}m{a}^3}}$ in the above expression.

${谓}_0=\frac{1}{2蟺}\sqrt{\frac{q^2}{4蟺{蔚}_{0}m{a}^3}}$ 鈥︹�� (4)

Here, $q$is charge of electron,$m$ is mass of electron, ${蔚}_0$is permittivity of space, and$a$ is distance.

Substitute ,$q=1.6脳{10}^{鈭�19}\text{鈥塁}$ ,${蔚}_0=8.85脳{10}^{鈭�12}{\text{鈥塁}}^2/\text{N}鈰�{\text{m}}^2$,$m=9.11脳{10}^{鈭�31}\text{鈥塳驳}$and $a=0.5\text{鈥�}\stackrel{\text{0}}{\text{A}}$in equation (4).

$\begin{array}{c}{谓}_0=\frac{1}{2蟺}\sqrt{\frac{{(1.6脳{10}^{鈭�19}\text{鈥塁})}^2}{4蟺(8.85脳{10}^{鈭�12}{\text{鈥塁}}^2/\text{N}鈰�{\text{m}}^2)(9.11脳{10}^{鈭�31}\text{鈥塳驳}){(0.5\text{鈥�}\stackrel{0}{\text{A}}脳\frac{{10}^{鈭�10}\text{鈥尘}}{1\text{鈥�}\stackrel{0}{\text{A}}})}^3}}\\ {谓}_0=7.16脳{10}^{15}\text{鈥塇锄}\end{array}$

Therefore, the natural frequency is $7.16脳{10}^{15}\text{鈥塇锄}$, and it lies in ultraviolet range.

Write the formula for the index of refraction.

$\begin{array}{c}n=1+\left(\frac{N{q}^2}{2m{蔚}_0}\underset{j}{鈭�}\frac{f_j}{{蝇}_j^2}\right)+{蝇}^2\left(\frac{N{q}^2}{2m{蔚}_0}\underset{j}{鈭�}\frac{f_j}{{蝇}_j^4}\right)\\ n=1+A(1+\frac{B}{{位}^2})\end{array}$

Here,$位$ is the wavelength.

Here,$位=\frac{2蟺c}{{蝇}_0}$

Write the formula for coefficient of refraction (A).

$A=\frac{N{q}^2}{2m{蔚}_0}\frac{f}{{蝇}_0^2}$ 鈥︹�� (5)

Here, $N$is number of molecules per unit volume, and $f$number of electrons per molecules.

Here,$f=2\left({\text{蹿辞谤鈥塇}}_{\text{2}}\right)$ .

Write the formula for coefficient of dispersion (B).

$B={\left(\frac{2蟺c}{{蝇}_0}\right)}^2$ 鈥︹�� (6)

Here, c is the speed of light.

Substitute$q=1.6脳{10}^{鈭�19}\text{鈥塁}$,${蔚}_0=8.85脳{10}^{鈭�12}{\text{鈥塁}}^2/\text{N}鈰�{\text{m}}^2$, $m=9.11脳{10}^{鈭�31}\text{鈥塳驳}$and $a=0.5\text{鈥�}\stackrel{\text{0}}{\text{A}}$in equation (3).

$\begin{array}{c}{蝇}_0=\sqrt{\frac{q^2}{4蟺{蔚}_{0}m{a}^3}}\\ {蝇}_0=\sqrt{\frac{{(1.6脳{10}^{鈭�19}\text{鈥塁})}^2}{4蟺(8.85脳{10}^{鈭�12})(9.11脳{10}^{鈭�31}\text{鈥塳驳}){(0.5\text{鈥�}\stackrel{0}{\text{A}}脳\frac{{10}^{鈭�10}\text{鈥尘}}{1\text{鈥�}\stackrel{0}{\text{A}}})}^3}}\\ {蝇}_0=4.5脳{10}^{16}\text{鈥塇锄}\end{array}$

Calculate the value of $N$as follows.

$\begin{array}{c}N=\frac{\text{础惫辞驳补诲谤辞鈥檚鈥塶耻尘产别谤}}{22.4\text{鈥塋}}\\ N=\frac{6.02脳{10}^{23}}{22.4脳{10}^{鈭�3}}\\ N=2.69脳{10}^{25}\end{array}$

Substitute$N=2.69脳{10}^{25}$, $q=1.6脳{10}^{鈭�19}\text{鈥塁}$,$f=2$,$m=9.11脳{10}^{鈭�31}\text{鈥塳驳}$, ${蔚}_0=8.85脳{10}^{鈭�12}{\text{鈥塁}}^2/\text{N}鈰�{\text{m}}^2$and${蝇}_0=4.5脳{10}^{16}\text{鈥塇锄}$in equation (5).

$\begin{array}{c}A=\frac{(2.69脳{10}^{25}){(1.6脳{10}^{鈭�19}\text{鈥塁})}^2\left(2\right)}{2(9.11脳{10}^{鈭�31}\text{鈥塳驳})(8.85脳{10}^{鈭�12}{\text{鈥塁}}^2/\text{N}鈰�{\text{m}}^2){({蝇}_0=4.5脳{10}^{16}\text{鈥塇锄})}^2}\\ A=4.2脳{10}^{鈭�5}\end{array}$

This value is about$\frac{1}{3}$of the actual value.

Substitute$c=3脳{10}^8\text{鈥尘}/\text{s}$and${蝇}_0=4.5脳{10}^{16}\text{鈥塇锄}$in equation (6).

$\begin{array}{c}B={\left(\frac{2蟺脳3脳{10}^8}{4.5脳{10}^{16}}\right)}^2\\ B=1.8脳{10}^{鈭�15}{\text{鈥尘}}^2\end{array}$

This value is about $\frac{1}{4}$of the actual value.

Therefore, the coefficient of refraction is $4.2脳{10}^{鈭�5}$and it is about $\frac{1}{3}$of the actual value. The coefficient of dispersionis $1.8脳{10}^{鈭�15}{\text{鈥尘}}^2$, and it is about$\frac{1}{4}$ of the actual value.