The energy of the electromagetic wave is of the order of `15` keV. To which part of the spectrum dose it belong?

Options

A. `X-rays`
B. Infrared rays
C. Ultraviolet rays
D. `gamma-rays`

Answer

To determine the part of the electromagnetic spectrum that corresponds to an energy of 15 keV, we can follow these steps:

### Step-by-Step Solution:

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

2. **Convert the energy from keV to joules**: The given energy is 15 keV. To convert this to joules, we use the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).
\[ E = 15 \text{ keV} = 15 \times 10^3 \text{ eV} = 15 \times 10^3 \times 1.6 \times 10^{-19} \text{ J} \]
\[ E = 2.4 \times 10^{-15} \text{ J} \]

3. **Calculate the wavelength**: We can now use the formula \( \lambda = \frac{hc}{E} \) to find the wavelength.
- Planck's constant \( h = 6.626 \times 10^{-34} \text{ J s} \)
- Speed of light \( c = 3 \times 10^8 \text{ m/s} \)
\[ \lambda = \frac{(6.626 \times 10^{-34} \text{ J s})(3 \times 10^8 \text{ m/s})}{2.4 \times 10^{-15} \text{ J}} \]

4. **Perform the calculation**:
\[ \lambda = \frac{1.9878 \times 10^{-25} \text{ J m}}{2.4 \times 10^{-15} \text{ J}} \approx 8.282 \times 10^{-11} \text{ m} \]

5. **Identify the part of the spectrum**: The wavelength \( 8.282 \times 10^{-11} \text{ m} \) or \( 82.82 \text{ pm} \) (picometers) falls within the range of X-rays, which is approximately from \( 10^{-13} \text{ m} \) to \( 3 \times 10^{-8} \text{ m} \).

### Conclusion:
The energy of the electromagnetic wave of the order of 15 keV corresponds to the X-ray part of the electromagnetic spectrum.

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