The minimum wavelength `lambda_min` in the continuous spectrum of X-rays is

Options

A. Proportional to the potential difference V between the cathode and anode.
B. Inversely proportional to potential differece V between the cathode and anode
C. Proportional to the square root of the potential difference V between the cathode and the anode.
D. Inversely proportional to the square root of the potential difference V between the cathode and the anode.

Answer

To find the minimum wavelength \( \lambda_{\text{min}} \) in the continuous spectrum of X-rays, we can use the relationship between the minimum wavelength and the kinetic energy of the electrons that produce the X-rays.

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy**:
The kinetic energy (\( KE \)) of the electrons when they are accelerated through a potential difference \( V \) is given by:
\[ KE = eV \]
where \( e \) is the charge of the electron.

2. **Relating Kinetic Energy to Wavelength**:
The energy of a photon can also be expressed in terms of its wavelength \( \lambda \) using the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

3. **Setting the Energies Equal**:
For the minimum wavelength of X-rays, the kinetic energy of the electrons is converted into the energy of the X-ray photon. Therefore, we can set the two expressions for energy equal:
\[ eV = \frac{hc}{\lambda_{\text{min}}} \]

4. **Rearranging for Wavelength**:
Rearranging the equation to solve for \( \lambda_{\text{min}} \):
\[ \lambda_{\text{min}} = \frac{hc}{eV} \]

5. **Analyzing the Relationship**:
From the equation \( \lambda_{\text{min}} = \frac{hc}{eV} \), we can see that \( \lambda_{\text{min}} \) is inversely proportional to the potential difference \( V \). This means that as the potential difference increases, the minimum wavelength decreases.

6. **Conclusion**:
Therefore, the minimum wavelength in the continuous spectrum of X-rays is inversely proportional to the potential difference \( V \).

### Final Answer:
The minimum wavelength \( \lambda_{\text{min}} \) in the continuous spectrum of X-rays is **inversely proportional to the potential difference \( V \)**.