The minimum wavelength of the `X` - rays produced by electrons accelerated through a potential difference of `V "volts"` is directly proportional to

Options

A. `sqrt(V)`
B. `V^(2)`
C. `1//sqrt(V)`
D. `1//V`

Explanation

To determine the minimum wavelength (\(\lambda_{\text{min}}\)) of X-rays produced by electrons accelerated through a potential difference \(V\) volts, we can follow these steps:

1. **Understand the energy conversion:**
- When an electron is accelerated through a potential difference \(V\), it gains kinetic energy equal to \(eV\), where \(e\) is the charge of the electron.
- This kinetic energy is then converted into the energy of the X-ray photon when the electron strikes the target.

2. **Relate kinetic energy to photon energy:**
- The energy of the X-ray photon (\(E\)) is given by \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light.
- Since the entire kinetic energy of the electron is converted into the energy of the X-ray photon, we have \(eV = \frac{hc}{\lambda_{\text{min}}}\).

3. **Solve for the minimum wavelength:**
- Rearrange the equation to solve for \(\lambda_{\text{min}}\):
\[ \lambda_{\text{min}} = \frac{hc}{eV} \]

4. **Determine the proportionality:**
- From the equation \(\lambda_{\text{min}} = \frac{hc}{eV}\), it is clear that \(\lambda_{\text{min}}\) is inversely proportional to the potential difference \(V\).

Therefore, the minimum wavelength of the X-rays produced is inversely proportional to the potential difference \(V\).

### Step-by-Step Solution:

1. **Energy Conversion:**
- An electron accelerated through a potential difference \(V\) gains kinetic energy:
\[ \text{Kinetic Energy} = eV \]

2. **Photon Energy Relation:**
- The energy of the X-ray photon is:
\[ E = \frac{hc}{\lambda} \]
- For minimum wavelength (\(\lambda_{\text{min}}\)), the entire kinetic energy is converted into photon energy:
\[ eV = \frac{hc}{\lambda_{\text{min}}} \]

3. **Solve for \(\lambda_{\text{min}}\):**
- Rearrange the equation to solve for \(\lambda_{\text{min}}\):
\[ \lambda_{\text{min}} = \frac{hc}{eV} \]

4. **Proportionality:**
- From the equation, we see that:
\[ \lambda_{\text{min}} \propto \frac{1}{V} \]
- Hence, \(\lambda_{\text{min}}\) is inversely proportional to \(V\).