Answer :
Sure! Let's break down this problem step-by-step.
### Step 1: The Problem's Parameters
We are given the following values:
- Scattering angle, [tex]\(\theta = 180.0^\circ\)[/tex]
- Speed of light, [tex]\(c = 3.00 \times 10^8 \, m/s\)[/tex]
- Mass of the electron, [tex]\(m = 9.10938356 \times 10^{-31} \, kg\)[/tex]
- The speed of the electron after scattering, [tex]\(v = 4.30 \times 10^6 \, m/s\)[/tex]
### Step 2: Convert Scattering Angle to Radians
[tex]\[ \theta_{rad} = \theta \times \frac{\pi}{180} = 180.0^\circ \times \frac{\pi}{180} = \pi \, \text{radians} \][/tex]
### Step 3: Calculate the Relativistic Momentum of the Electron After Scattering
The momentum ([tex]\(p\)[/tex]) of the electron after scattering is given by:
[tex]\[ p = m \times v \][/tex]
Substituting the given values:
[tex]\[ p = (9.10938356 \times 10^{-31} \, kg) \times (4.30 \times 10^6 \, m/s) = 3.9170349307999995 \times 10^{-24} \, kg \cdot m/s \][/tex]
### Step 4: Compton Wavelength Shift
The Compton wavelength shift formula is:
[tex]\[ \Delta \lambda = \frac{h}{m \cdot c} \left( 1 - \cos \theta \right) \][/tex]
Where [tex]\(h\)[/tex] is Planck's constant:
[tex]\[ h = 6.62607015 \times 10^{-34} \, J \cdot s \][/tex]
Since [tex]\(\cos(180^\circ) = -1\)[/tex]:
[tex]\[ \Delta \lambda = \frac{6.62607015 \times 10^{-34}}{9.10938356 \times 10^{-31} \times 3.00 \times 10^8} \left( 1 - (-1) \right) \][/tex]
[tex]\[ \Delta \lambda = \frac{6.62607015 \times 10^{-34}}{9.10938356 \times 10^{-31} \times 3.00 \times 10^8} \times 2 \][/tex]
[tex]\[ \Delta \lambda = 4.849263477494848 \times 10^{-12} \, m \][/tex]
### Step 5: Calculate the Initial Photon Energy
To determine the initial wavelength, we utilize the energy relation for the photon and the momentum of the electron after scattering.
Energy of the photon right after scattering:
[tex]\[ E_{photon} = p \times c \][/tex]
[tex]\[ E_{photon} = 3.9170349307999995 \times 10^{-24} \, kg \cdot m/s \times 3.00 \times 10^8 \, m/s \][/tex]
[tex]\[ E_{photon} = 1.17511047924 \times 10^{-15} \, J \][/tex]
### Step 6: Calculate the Initial Photon Wavelength
The wavelength [tex]\(\lambda\)[/tex] is obtained by:
[tex]\[ \lambda = \frac{h}{p} \][/tex]
[tex]\[ \lambda_{initial} = \frac{6.62607015 \times 10^{-34}}{1.17511047924 \times 10^{-15}} \][/tex]
[tex]\[ \lambda_{initial} = 1.6916035386609936 \times 10^{-10} \, m \][/tex]
### Step 7: Find the Incident Wavelength
Finally, the incident wavelength [tex]\(\lambda_{initial}\)[/tex] can be calculated by adjusting for the Compton shift:
[tex]\[ \lambda_{incident} = \lambda_{initial} - \Delta \lambda \][/tex]
[tex]\[ \lambda_{incident} = 1.6916035386609936 \times 10^{-10} - 4.849263477494848 \times 10^{-12} \][/tex]
[tex]\[ \lambda_{incident} = 1.643110903886045 \times 10^{-10} \, m \][/tex]
So, the wavelength of the incident X-ray photon is:
[tex]\[ \lambda_{incident} = 1.643110903886045 \times 10^{-10} \, meters \][/tex]
### Step 1: The Problem's Parameters
We are given the following values:
- Scattering angle, [tex]\(\theta = 180.0^\circ\)[/tex]
- Speed of light, [tex]\(c = 3.00 \times 10^8 \, m/s\)[/tex]
- Mass of the electron, [tex]\(m = 9.10938356 \times 10^{-31} \, kg\)[/tex]
- The speed of the electron after scattering, [tex]\(v = 4.30 \times 10^6 \, m/s\)[/tex]
### Step 2: Convert Scattering Angle to Radians
[tex]\[ \theta_{rad} = \theta \times \frac{\pi}{180} = 180.0^\circ \times \frac{\pi}{180} = \pi \, \text{radians} \][/tex]
### Step 3: Calculate the Relativistic Momentum of the Electron After Scattering
The momentum ([tex]\(p\)[/tex]) of the electron after scattering is given by:
[tex]\[ p = m \times v \][/tex]
Substituting the given values:
[tex]\[ p = (9.10938356 \times 10^{-31} \, kg) \times (4.30 \times 10^6 \, m/s) = 3.9170349307999995 \times 10^{-24} \, kg \cdot m/s \][/tex]
### Step 4: Compton Wavelength Shift
The Compton wavelength shift formula is:
[tex]\[ \Delta \lambda = \frac{h}{m \cdot c} \left( 1 - \cos \theta \right) \][/tex]
Where [tex]\(h\)[/tex] is Planck's constant:
[tex]\[ h = 6.62607015 \times 10^{-34} \, J \cdot s \][/tex]
Since [tex]\(\cos(180^\circ) = -1\)[/tex]:
[tex]\[ \Delta \lambda = \frac{6.62607015 \times 10^{-34}}{9.10938356 \times 10^{-31} \times 3.00 \times 10^8} \left( 1 - (-1) \right) \][/tex]
[tex]\[ \Delta \lambda = \frac{6.62607015 \times 10^{-34}}{9.10938356 \times 10^{-31} \times 3.00 \times 10^8} \times 2 \][/tex]
[tex]\[ \Delta \lambda = 4.849263477494848 \times 10^{-12} \, m \][/tex]
### Step 5: Calculate the Initial Photon Energy
To determine the initial wavelength, we utilize the energy relation for the photon and the momentum of the electron after scattering.
Energy of the photon right after scattering:
[tex]\[ E_{photon} = p \times c \][/tex]
[tex]\[ E_{photon} = 3.9170349307999995 \times 10^{-24} \, kg \cdot m/s \times 3.00 \times 10^8 \, m/s \][/tex]
[tex]\[ E_{photon} = 1.17511047924 \times 10^{-15} \, J \][/tex]
### Step 6: Calculate the Initial Photon Wavelength
The wavelength [tex]\(\lambda\)[/tex] is obtained by:
[tex]\[ \lambda = \frac{h}{p} \][/tex]
[tex]\[ \lambda_{initial} = \frac{6.62607015 \times 10^{-34}}{1.17511047924 \times 10^{-15}} \][/tex]
[tex]\[ \lambda_{initial} = 1.6916035386609936 \times 10^{-10} \, m \][/tex]
### Step 7: Find the Incident Wavelength
Finally, the incident wavelength [tex]\(\lambda_{initial}\)[/tex] can be calculated by adjusting for the Compton shift:
[tex]\[ \lambda_{incident} = \lambda_{initial} - \Delta \lambda \][/tex]
[tex]\[ \lambda_{incident} = 1.6916035386609936 \times 10^{-10} - 4.849263477494848 \times 10^{-12} \][/tex]
[tex]\[ \lambda_{incident} = 1.643110903886045 \times 10^{-10} \, m \][/tex]
So, the wavelength of the incident X-ray photon is:
[tex]\[ \lambda_{incident} = 1.643110903886045 \times 10^{-10} \, meters \][/tex]
