91影视

Write the expressions forthe index of refraction.

$n=1+\frac{N{q}^2}{2m{蔚}_0}\frac{({蝇}_0^2-{蝇}^2)}{[{({蝇}_0^2-{蝇}^2)}^2+{纬}^2{蝇}^2]}$ 鈥︹�� (1)

Here, N is the number of molecules per unit volume, q is the charge, m is the mass,${蔚}_0$ is the permittivity of free space,${蝇}_0$ is the resonance frequency and$纬$ is the Lorentz contraction.

Let the denominator part of equation (1) is equal to $D$.

Differentiate the equation (1) with respect to $蝇$.

$\frac{dn}{d蝇}=\frac{N{q}^2}{2m{蔚}_0}\{\frac{鈭�2蝇}{D}鈭�\frac{({蝇}_0^2鈭�{蝇}^2)}{D^2}[2({蝇}_0^2鈭�{蝇}^2)(鈭�2蝇)+2蝇{纬}^2]\}$

Substitute $\frac{dn}{d蝇}=0$in the equation.

$\begin{array}{l}\frac{N{q}^2}{2m{蔚}_0}\{\frac{鈭�2蝇}{D}鈭�\frac{({蝇}_0^2鈭�{蝇}^2)}{D^2}[2({蝇}_0^2鈭�{蝇}^2)(鈭�2蝇)+2蝇{纬}^2]\}=0\\ \frac{鈭�2蝇}{D}鈭�\frac{({蝇}_0^2鈭�{蝇}^2)}{D^2}[2({蝇}_0^2鈭�{蝇}^2)(鈭�2蝇)+2蝇{纬}^2]=0\\ 2蝇D=2蝇({蝇}_0^2鈭�{蝇}^2)[2({蝇}_0^2鈭�{蝇}^2)鈭�{纬}^2]\\ {({蝇}_0^2鈭�{蝇}^2)}^2+{纬}^2{蝇}^2=2{({蝇}_0^2鈭�{蝇}^2)}^2鈭�{纬}^2({蝇}_0^2鈭�{蝇}^2)\end{array}$

On further solving, the above equation becomes,

$\begin{array}{l}{({蝇}_0^2鈭�{蝇}^2)}^2={纬}^2({蝇}^2+{蝇}_0^2鈭�{蝇}^2)\\ {({蝇}_0^2鈭�{蝇}^2)}^2={纬}^2{蝇}_0^2\\ {蝇}_0^2鈭�{蝇}^2=卤纬{蝇}_0\\ {蝇}^2={蝇}_0^2鈭�纬{蝇}_0\end{array}$

Again on further solving, the above equation becomes,

$\begin{array}{c}蝇={蝇}_0\sqrt{1鈭�\frac{纬}{{蝇}_0}}\\ ={蝇}_0(1鈭�\frac{纬}{2{蝇}_0})\\ ={蝇}_0鈭�\frac{纬}{2}\end{array}$

Hence, the initial and final width of an anomalous region will be,

$\begin{array}{c}{蝇}_1={蝇}_0鈭�\frac{纬}{2}\\ {蝇}_2={蝇}_0+\frac{纬}{2}\end{array}$

Calculate the width of the anomalous region.

$\begin{array}{c}螖蝇={蝇}_2鈭�{蝇}_1\\ 螖蝇={蝇}_0+\frac{纬}{2}鈭�{蝇}_0+\frac{纬}{2}\\ 螖蝇=纬\end{array}$

Write the equation for the absorption coefficient.

$伪=\frac{N{q}^2{蝇}^2}{m{蔚}_{0}c}\frac{纬}{{({蝇}_0^2鈭�{蝇}^2)}^2+{纬}^2{蝇}^2}$ 鈥︹��. (2)

Here $蝇={蝇}_0$,

Substitute $蝇={蝇}_0$in equation (2).

$\begin{array}{c}{伪}_{\max }=\frac{N{q}^2{蝇}_0^2}{m{蔚}_{0}c}\frac{纬}{{({蝇}_0^2鈭�{蝇}_0^2)}^2+{纬}^2{蝇}_0^2}\\ {伪}_{\max }=\frac{N{q}^2}{m{蔚}_{0}c纬}\end{array}$

At${蝇}_1$and ${蝇}_2$,${蝇}^2={蝇}_0^2鈭�纬{蝇}_0$, so, the equation (2) becomes,

$\begin{array}{c}伪=\frac{N{q}^2{蝇}^2}{m{蔚}_{0}c}\frac{纬}{{纬}^2{蝇}_0^2+{纬}^2{蝇}^2}\\ 伪={伪}_{\max }\left(\frac{{蝇}^2}{{蝇}_0^2+{蝇}^2}\right)\end{array}$ 鈥︹�� (3)

Here, ${蝇}^2={蝇}_0^2鈭�纬{蝇}_0$.

Calculatethe value of $\left(\frac{{蝇}^2}{{蝇}_0^2+{蝇}^2}\right)$from equation (3).

$\begin{array}{c}\left(\frac{{蝇}^2}{{蝇}_0^2+{蝇}^2}\right)=\frac{{蝇}_0^2鈭�纬{蝇}_0}{2{蝇}_0^2鈭�纬{蝇}_0}\\ =\frac{1}{2}\frac{(1鈭�纬/{蝇}_0)}{(1鈭�纬/2{蝇}_0)}\\ 鈮�\frac{1}{2}(1鈭�\frac{纬}{{蝇}_0})(1卤\frac{纬}{2{蝇}_0})\\ 鈮�\frac{1}{2}(1鈭�\frac{纬}{2{蝇}_0})\end{array}$

Simplify further.

$\left(\frac{{蝇}^2}{{蝇}_0^2+{蝇}^2}\right)鈮�\frac{1}{2}$

Substitute$\left(\frac{{蝇}^2}{{蝇}_0^2+{蝇}^2}\right)鈮�\frac{1}{2}$ in equation (3).

$伪=\frac{1}{2}{伪}_{\max }$

Therefore, the width of the anomalous region is .it is proved that the index of refraction assumes its maximum and minimum values at points where the absorption coefficient is at half-maximum.