91影视

The radius of the electron cloud from the hydrogen atom is $R$.

The total charge is $e$.

The magnitude of the applied electrical field is $E$.

The polarization of the atomic hydrogen is $伪$.

The effective stiffness string for atomic hydrogen is $k_s$.

The expression to calculate the applied electric field is given as follows.

$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{蟺蔚}}_{\mathbf{0}}}\frac{\mathbf{e}\mathbf{s}}{{\mathbf{R}}^{\mathbf{3}}}.........\left(\mathrm{i}\right)$

Here, s is the displacement distance of the proton from the centre of the electron cloud.

The expression to calculate the shift of the proton is given as follows.

$\mathbf{s}\mathbf{=}\frac{{\mathbf{伪}}_{\mathbf{measured}}\mathbf{E}}{\mathbf{e}}..\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{ii}\mathbf{\right)}$

Here, ${\mathbf{伪}}_{\mathbf{measured}}$is the measured polarization.

The expression to calculate the force acting electron cloud due to nucleus is given as follows.

$\mathbf{F}\mathbf{=}\mathbf{eE}\mathbf{}\mathbf{}\mathbf{}......\left(\mathrm{iii}\right)$

The expression to calculate the force due to the stiffness of the string is given as follows.

(a)

Calculate the expression for the polarization of atomic hydrogen.

Substitute $伪E$for into equation (i).

$$\begin{gathered} E=\frac{1}{4{\mathrm{蟺蔚}}_0}\frac{伪E}{R^3} \\ 1=\frac{1}{4{\mathrm{蟺蔚}}_0}\frac{伪}{R^3} \\ 伪=4{\mathrm{蟺蔚}}_0{R}^3.......\left(v\right) \end{gathered}$$

Hence the expression for the polarization of atomic hydrogen in terms of$R$ is $4{\mathrm{蟺蔚}}_0{R}^3$.

(b)

The radius of the electron cloud from the hydrogen atom, $R=1脳{10}^{-10}\mathrm{m}$

The measured value of the polarization for hydrogen atom, ${伪}_{measured}=7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right)$

Calculate the polarization for the atomic hydrogen.

Substitute $1脳{10}^{-10}\mathrm{m}$for $R$and $8.85脳{10}^{-12}\mathrm{F}/\mathrm{m}$for ${蔚}_0$into equation (v).

$$\begin{gathered} 伪=4\mathrm{蟺}脳8.85脳{10}^{-12}脳{\left(1脳{10}^{-10}\right)}^3 \\ \mathrm{伪}=111.21脳{10}^{-12}脳{10}^{-30} \\ \mathrm{伪}=111.21脳{10}^{-42} \\ \mathrm{伪}=111.21脳{10}^{-41}\mathrm{C}路\mathrm{m}/\mathrm{N}/\mathrm{C} \end{gathered}$$

Hence the value of the polarization for the atomic hydrogen is$111.21脳{10}^{-41}\mathrm{C}路\mathrm{m}/\mathrm{N}/\mathrm{C}$ and it is very close to measured polarization.

(c)

The applied electric field,$E=1脳{10}^6\mathrm{N}/\mathrm{C}$

The measured value of the polarization for hydrogen atom, ${伪}_{measured}=7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right)$

Calculate the shift of the proton.

Substitute $1脳{10}^6\mathrm{N}/\mathrm{C}$for $E$, $1.6脳{10}^{-19}\mathrm{C}$for $e$and $7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right)$for ${伪}_{measured}$into equation (ii).

$$\begin{gathered} s=\frac{7.4脳{10}^{-41}脳1脳{10}^6}{1.6脳{10}^{-19}} \\ s=\frac{7.4脳{10}^{-35}}{1.6脳{10}^{-19}} \\ s=4.625脳{10}^{-16}\mathrm{m} \end{gathered}$$

Hence the shift of the proton is $4.625脳{10}^{-16}\mathrm{m}$.

(d)

Derive the expression for the effective stiffness by equate the equation (iii) and (iv).

$$\begin{gathered} eE=k_{s}s \\ eE=k_s\left(\frac{es}{e}\right) \\ {k}_s=\frac{e^{2}E}{es} \end{gathered}$$

Substitute ${伪}_{measured}E$for $es$into above equation.

$$\begin{gathered} k_s=\frac{e^{2}E}{{伪}_{measured}E} \\ {k}_s=\frac{e^2}{{伪}_{measured}}...\mathbf{.}\mathbf{.}\left(\mathrm{vi}\right) \end{gathered}$$

Substitute $1.6脳{10}^{-19}C$for $e$and $7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right)$ for ${伪}_{measured}$into equation (vi).

$$\begin{gathered} k_s=\frac{{\left(1.6脳{10}^{-19}\right)}^2}{7.4脳{10}^{-41}} \\ {k}_s=\frac{2.56脳{10}^{-38}}{7.4脳{10}^{-41}} \\ {k}_s=0.346脳{10}^3 \\ {k}_s=346\mathrm{N}/\mathrm{m} \end{gathered}$$

Hence the value of the effective string stiffness is $346\mathrm{N}/\mathrm{m}$.

(e)

Since the value of the polarization is twice as large therefore measured value of the polarization for hydrogen atom,

$$\begin{gathered} {伪}_{measured}=2脳7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right) \\ {\mathrm{伪}}_{\mathrm{m}\mathrm{easured}}=14.8脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right) \end{gathered}$$

Calculate the value of the effective string stiffness.

Substitute $1.6脳{10}^{-19}\mathrm{C}$for $e$and $7.4脳{10}^{-41}\mathrm{C}路\mathrm{m}/\left(\mathrm{N}/\mathrm{c}\right)$for ${伪}_{measured}$into equation (vi).

$$\begin{gathered} k_s=\frac{{\left(1.6脳{10}^{-19}\right)}^2}{14.8脳{10}^{-41}} \\ {k}_s=\frac{2.56脳{10}^{-38}}{14.8脳{10}^{-41}} \\ {k}_s=0.173脳{10}^3 \\ {k}_s=173\mathrm{N}/\mathrm{m} \end{gathered}$$

Hence the value of the effective string stiffness for twice measured value of polarization is 173 N/m.