An X-rays tube has a short wavelength end at `0.45A`. The voltage of tube is

Options

A. 450000 V
B. 9600 V
C. 27500 V
D. 60600 V

Answer

To find the voltage of the X-ray tube given the short wavelength of `0.45 Å`, we can follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength is in Angstroms (Å). We need to convert it to meters for our calculations.

\[ \text{Wavelength} (\lambda) = 0.45 \, \text{Å} = 0.45 \times 10^{-10} \, \text{m} \]

### Step 2: Use the Energy-Wavelength Relation
We can calculate the energy (E) of the X-rays using the formula:

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

Where:
- \( h \) (Planck's constant) = \( 6.63 \times 10^{-34} \, \text{J s} \)
- \( c \) (speed of light) = \( 3 \times 10^{8} \, \text{m/s} \)

Substituting the values:

\[ E = \frac{(6.63 \times 10^{-34} \, \text{J s})(3 \times 10^{8} \, \text{m/s})}{0.45 \times 10^{-10} \, \text{m}} \]

### Step 3: Calculate Energy
Now, we will calculate the energy:

\[ E = \frac{(6.63 \times 10^{-34})(3 \times 10^{8})}{0.45 \times 10^{-10}} \]

Calculating the numerator:

\[ 6.63 \times 3 = 19.89 \quad \text{(approximately 20)} \]

Now, substituting back:

\[ E \approx \frac{20 \times 10^{-34}}{0.45 \times 10^{-10}} = \frac{20}{0.45} \times 10^{-24} = 44.44 \times 10^{-24} \, \text{J} \]

### Step 4: Relate Energy to Voltage
The energy of the X-rays can also be related to the voltage (V) of the tube using the formula:

\[ E = eV \]

Where \( e \) (the charge of an electron) = \( 1.6 \times 10^{-19} \, \text{C} \).

Rearranging for voltage:

\[ V = \frac{E}{e} \]

### Step 5: Substitute Energy and Charge
Now substituting the values we found:

\[ V = \frac{44.44 \times 10^{-24}}{1.6 \times 10^{-19}} \]

### Step 6: Calculate Voltage
Calculating the voltage:

\[ V \approx \frac{44.44}{1.6} \times 10^{-5} = 27.75 \times 10^{5} \, \text{V} \approx 27500 \, \text{V} \]

### Conclusion
Thus, the voltage of the tube is approximately **27,500 V**.