What is the operating voltage of a medical x-ray machine that has a cut-off wavelength of `2.20xx10^(-11)m`?

Options

A. 83 800 V
B. 56 500 V
C. 10 900 V
D. 44 900 V

Answer

To find the operating voltage of a medical x-ray machine with a given cut-off wavelength, we can use the formula that relates the wavelength of the emitted x-rays to the voltage. The formula is given by:

\[ V = \frac{hc}{\lambda} \]

Where:
- \( V \) is the operating voltage in volts,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the cut-off wavelength in meters.

### Step-by-step Solution:

1. **Identify the given values:**
- Cut-off wavelength, \( \lambda = 2.20 \times 10^{-11} \, \text{m} \)
- Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- Speed of light, \( c = 3.00 \times 10^8 \, \text{m/s} \)

2. **Substitute the values into the formula:**
\[ V = \frac{(6.626 \times 10^{-34} \, \text{Js}) \times (3.00 \times 10^8 \, \text{m/s})}{2.20 \times 10^{-11} \, \text{m}} \]

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

4. **Calculate the voltage:**
\[ V = \frac{1.9878 \times 10^{-25}}{2.20 \times 10^{-11}} \]
\[ V = 9.025 \times 10^{5} \, \text{V} \]

5. **Final result:**
The operating voltage of the medical x-ray machine is approximately \( 9.03 \times 10^{5} \, \text{V} \) or \( 903,000 \, \text{V} \).