Hard X–rays for the study of fractures in bones should have a minimum wavelength of `10^(–11)` m. The accelerating voltage for electrons in X–rays machine should be –

Options

A. lt 124·2 kV
B. gt 124·2 kV
C. between 60 kV and 70 kV
D. = 100 kV

Answer

To find the accelerating voltage for electrons in an X-ray machine that produces hard X-rays with a minimum wavelength of \(10^{-11}\) m, we can follow these steps:

### Step-by-Step Solution

1. **Understand the relationship between energy and wavelength**:
The energy \(E\) of a photon is related to its wavelength \(\lambda\) by the equation:
\[ E = \frac{hc}{\lambda} \]
where \(h\) is Planck's constant and \(c\) is the speed of light.

2. **Relate the energy to voltage**:
The energy of an electron accelerated through a potential difference \(V\) is given by:
\[ E = eV \]
where \(e\) is the charge of an electron.

3. **Set the two expressions for energy equal**:
Since both expressions represent the energy of the photons produced, we can set them equal to each other:
\[ eV = \frac{hc}{\lambda} \]

4. **Rearrange to find voltage \(V\)**:
Rearranging the equation gives:
\[ V = \frac{hc}{e\lambda} \]

5. **Substitute known values**:
- Planck's constant \(h = 6.626 \times 10^{-34} \, \text{J s}\)
- Speed of light \(c = 3.00 \times 10^{8} \, \text{m/s}\)
- Charge of an electron \(e = 1.602 \times 10^{-19} \, \text{C}\)
- Wavelength \(\lambda = 10^{-11} \, \text{m}\)

Substituting these values into the equation:
\[ V = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^{8} \, \text{m/s})}{(1.602 \times 10^{-19} \, \text{C})(10^{-11} \, \text{m})} \]

6. **Calculate the numerator**:
\[ hc = 6.626 \times 10^{-34} \times 3.00 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]

7. **Calculate the denominator**:
\[ e\lambda = 1.602 \times 10^{-19} \times 10^{-11} = 1.602 \times 10^{-30} \, \text{C m} \]

8. **Calculate the voltage**:
\[ V = \frac{1.9878 \times 10^{-25}}{1.602 \times 10^{-30}} \approx 1.24 \times 10^{5} \, \text{V} = 124 \, \text{kV} \]

9. **Conclusion**:
The accelerating voltage for the X-ray machine should be approximately \(124 \, \text{kV}\) or less.

### Final Answer
The accelerating voltage for electrons in the X-ray machine should be less than or equal to \(124 \, \text{kV}\).