91影视

Initial energy of the photon,

The expression to calculate the initial energy is given as follows.

鈥�(颈)

Here,h is the plank鈥檚 constant and c is the speed of light.

The expression to calculate the change in the wavelength is given as follows.

$\mathit{螖}\mathit{位}\mathbf{=}\frac{\mathbf{h}}{\mathbf{m}\mathbf{c}}\mathbf{(}\mathbf{1}\mathbf{鈭�}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{胃}\mathbf{)}$ 鈥�(颈颈)

Here,m is the mass of the electron.

The expression to calculate the kinetic energy of the electron is given as follows.

$\mathit{K}\mathbf{=}\mathit{E}\mathbf{-}{\mathit{E}}^{\mathbf{\text{'}}}$ 鈥�(颈颈颈)

Here, $E^{\text{'}}$is the energy of backscattered photon.

The value of plank鈥檚 constant is, $6.62脳{10}^{鈭�34}\text{鈥尘办驳}/\text{s}$.

The value of mass of electron is,$9.11脳{10}^{鈭�31}\text{鈥塳驳}$.

Since the photon is backscattered and electron is knocked forward, therefore the angle, $胃=180掳$.

Substitute$180掳$for$胃$into equation (ii).

$\begin{array}{l}螖位=\frac{h}{mc}(1鈭�\cos 180掳)\\ 螖位=\frac{h}{mc}(1鈭�(鈭�1))\\ 螖位=\frac{2h}{mc}\end{array}$

The new wavelength is given by,

role="math" localid="1663071812908" $\begin{array}{l}{位}^{\text{'}}=螖位+位\\ {位}^{\text{'}}=位\left(\frac{螖位}{位}+1\right)\end{array}$

Calculate the energy of photon when it is backscattered.

$E^{\text{'}}=\frac{hc}{{位}^{\text{'}}}$

Substitute $位\left(\frac{螖位}{位}+1\right)$for${位}^{\text{'}}$into above equation.

$\begin{array}{l}{E}^{\text{'}}=\frac{hc}{位\left(\frac{螖位}{位}+1\right)}\\ E^{\text{'}}=\frac{\frac{hc}{位}}{\left(\frac{螖位}{位}+1\right)}\end{array}$

Substitute E for$\frac{hc}{位}$into above equation.

$E^{\text{'}}=\frac{E}{\left(\frac{螖位}{位}+1\right)}$

Substitute$2h/\text{mc}$for$螖位$and $hc/E$for$位$into above equation.

$\begin{array}{l}{E}^{\text{'}}=\frac{E}{\left(\frac{\frac{2h}{mc}}{\frac{hc}{E}}+1\right)}\\ E^{\text{'}}=\frac{E}{\left(\frac{2hE}{mh{c}^2}+1\right)}\\ E^{\text{'}}=\frac{E}{\left(\frac{2E}{m{c}^2}+1\right)}\end{array}$

And,

$\begin{array}{c}m{c}^2=9.11脳{10}^{鈭�31}\text{鈥塳驳}脳{(3脳{10}^8\text{鈥尘}/\text{s})}^2\\ =8.199脳{10}^{鈭�14}\text{鈥塉}\\ =8.199脳{10}^{鈭�14}脳6.242脳{10}^{15}\text{鈥塳别痴}\\ =511.78\text{鈥塳别痴}\\ m{c}^2鈮�511\text{鈥塳别痴}\end{array}$

Substitute$50\text{鈥塳别痴}$for E and $511\text{鈥塳别痴}$for $m{c}^2$into above equation

$\begin{array}{c}{E}^{\text{'}}=\frac{50\text{鈥塳别痴}}{\frac{2脳50\text{鈥塳别痴}}{511\text{鈥塳别痴}}+1}\\ E^{\text{'}}=\frac{50\text{鈥塳别痴}}{\frac{100}{511}+1}\\ E^{\text{'}}=\frac{50\text{鈥塳别痴}}{1.1956}\\ E^{\text{'}}=41.8\text{鈥塳别痴}\end{array}$

Hence the energy of the backscattered photon is .$41.8\text{鈥塳别痴}$

Calculate the kinetic energy of electron.

Substitute$41.8\text{鈥塳别痴}$for $E^{\text{'}}$and $50\text{鈥塳别痴}$for E into equation (iii).

$\begin{array}{l}K=50\text{鈥塳别痴}鈭�41.8\text{鈥塳别痴}\\ K=8.2\text{鈥塳别痴}\end{array}$

Hence the kinetic energy of the electron is$8.2\text{鈥塳别痴}$.