Calculate the minimum wavelength of X-rays produced by an X-ray tube operating at 30 kV.
`h=6.6 times 10^(-27) erg-sec]`

Options

A. 42.4 nm
B. 4.2 `overset(@)(A)`
C. 0.4125 `overset(@)(A)`
D. `4.8 mum`

Explanation

To calculate the minimum wavelength of X-rays produced by an X-ray tube operating at 30 kV, we can use the relationship between energy and wavelength. The energy (E) of the X-rays can be calculated using the formula:

\[ E = eV \]

Where:
- \( E \) is the energy in ergs,
- \( e \) is the charge of an electron (approximately \( 4.8 \times 10^{-10} \) ergs/V),
- \( V \) is the voltage in volts (30 kV = 30,000 V).

1. **Calculate the energy of the X-rays:**
\[ E = eV = (4.8 \times 10^{-10} \text{ ergs/V}) \times (30,000 \text{ V}) \]
\[ E = 1.44 \times 10^{-5} \text{ ergs} \]

2. **Use the energy-wavelength relationship:**
The energy of a photon is also related to its wavelength (\( \lambda \)) by the equation:
\[ E = \frac{hc}{\lambda} \]
Where:
- \( h \) is Planck's constant (\( 6.6 \times 10^{-27} \text{ erg-sec} \)),
- \( c \) is the speed of light (\( 3 \times 10^{10} \text{ cm/sec} \)).

3. **Rearranging the equation to find the wavelength:**
\[ \lambda = \frac{hc}{E} \]

4. **Substituting the values:**
\[ \lambda = \frac{(6.6 \times 10^{-27} \text{ erg-sec}) \times (3 \times 10^{10} \text{ cm/sec})}{1.44 \times 10^{-5} \text{ ergs}} \]
\[ \lambda = \frac{1.98 \times 10^{-16} \text{ erg-cm}}{1.44 \times 10^{-5} \text{ ergs}} \]
\[ \lambda = 1.375 \times 10^{-11} \text{ cm} \]

5. **Convert to Angstroms:**
Since \( 1 \text{ Angstrom} = 10^{-8} \text{ cm} \):
\[ \lambda = 1.375 \times 10^{-11} \text{ cm} \times \frac{1 \text{ Angstrom}}{10^{-8} \text{ cm}} = 0.1375 \text{ Angstroms} \]

Thus, the minimum wavelength of X-rays produced by the X-ray tube operating at 30 kV is approximately **0.1375 Angstroms**.