A photon of wavelength 200 nm is scattered by an electron that is initially at rest. Which one of the following statements concerning the wavelength of the scattered photon is true?

Options

A. The wavelength is zero nm.
B. The wavelength is greater than 200 nm.
C. The wavelength is 200 nm.
D. The wavelength is 100 nm.

Answer

To solve the problem of a photon of wavelength 200 nm being scattered by an electron initially at rest, we can use the principles of the Compton effect. The Compton effect describes how the wavelength of a photon changes when it scatters off a charged particle, such as an electron.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- We have a photon with an initial wavelength \( \lambda_i = 200 \, \text{nm} \).
- The electron is initially at rest.

2. **Energy of the Incident Photon**:
- The energy of the photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
- Where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)), \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)), and \( \lambda \) is the wavelength in meters.
- Convert \( 200 \, \text{nm} \) to meters:
\[ \lambda_i = 200 \times 10^{-9} \, \text{m} \]

3. **Calculate the Initial Energy**:
- Substitute the values into the energy formula:
\[ E_i = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{200 \times 10^{-9} \, \text{m}} \]
- Calculate \( E_i \).

4. **Compton Scattering**:
- When the photon scatters off the electron, it transfers some of its energy to the electron. The energy of the scattered photon will be less than the energy of the incident photon.
- This means the wavelength of the scattered photon \( \lambda_f \) will be greater than the initial wavelength \( \lambda_i \).

5. **Conclusion**:
- Since the energy of the photon decreases after scattering, the wavelength of the scattered photon must increase. Therefore, the wavelength of the scattered photon \( \lambda_f \) will be greater than \( 200 \, \text{nm} \).

### Final Answer:
The wavelength of the scattered photon is greater than \( 200 \, \text{nm} \).