Electromagnetic Wave Wavelength Calculation

An electromagnetic (EM) wave is a type of wave that can travel through the vacuum of outer space. It is characterized by its frequency and wavelength, and these two properties are related by the speed of light.

The question provides the frequency of an electromagnetic (EM) wave and asks for its wavelength and the region of the electromagnetic spectrum it falls into. To solve this, we use the fundamental relationship between the speed of light, frequency, and wavelength of an electromagnetic wave.

Electromagnetic Wave Formula

The speed of an electromagnetic wave in a vacuum, denoted by \(c\), is constant and approximately \(3 \times 10^8\) meters per second. The relationship between the speed of light \(c\), frequency \(\nu\), and wavelength \(\lambda\) is given by the formula:

\[c = \nu \lambda\]

Where:

• \(c\) is the speed of light (\(3 \times 10^8\) m/s)
• \(\nu\) is the frequency of the EM wave (in Hz)
• \(\lambda\) is the wavelength of the EM wave (in meters)

Wavelength Calculation Steps

Given the frequency of the electromagnetic (EM) wave as \(\nu = 6 \times 10^{14}\) Hz, we can rearrange the formula to find the wavelength \(\lambda\):

\[\lambda = \frac{c}{\nu}\]

Substitute the given values into the formula:

\[\lambda = \frac{3 \times 10^8 \text{ m/s}}{6 \times 10^{14} \text{ Hz}}\]

Perform the division:

\[\lambda = \left(\frac{3}{6}\right) \times \left(\frac{10^8}{10^{14}}\right) \text{ m}\]

\[\lambda = 0.5 \times 10^{(8 - 14)} \text{ m}\]

\[\lambda = 0.5 \times 10^{-6} \text{ m}\]

To express this wavelength in nanometers (nm), we recall that 1 nanometer (nm) is \(10^{-9}\) meters. We can convert meters to nanometers by multiplying by \(10^9\):

\[\lambda = 0.5 \times 10^{-6} \text{ m} \times \left(\frac{10^9 \text{ nm}}{1 \text{ m}}\right)\]

\[\lambda = 0.5 \times 10^{(-6 + 9)} \text{ nm}\]

\[\lambda = 0.5 \times 10^3 \text{ nm}\]

\[\lambda = 500 \text{ nm}\]

So, the wavelength of the electromagnetic (EM) wave is 500 nm.

Electromagnetic Spectrum Range

Now, let's determine where this 500 nm wavelength falls in the electromagnetic spectrum. The electromagnetic spectrum is a range of all types of electromagnetic radiation, ordered by wavelength and frequency. Different regions of the spectrum have different applications and properties.

Here's a general overview of some key regions of the electromagnetic spectrum:

Comparing our calculated wavelength of 500 nm with the typical ranges:

• 500 nm falls within the 400 nm to 700 nm range.
• This range corresponds to the visible light part of the electromagnetic (EM) spectrum.
• Specifically, 500 nm is in the green-blue part of the visible light spectrum.

Conclusion

Based on our calculation, the wavelength of the electromagnetic (EM) wave is 500 nm, and it falls in the range of visible rays.

Therefore, the correct option is the one stating "500 nm, visible rays".