Question

(a) What is the wavelength (in nanometers) of light having a frequency of \(8.6 \times 10^{13} \mathrm{~Hz} ?\) (b) What is the frequency (in Hz) of light having a wavelength of \(566 \mathrm{nm} ?\)

Short answer

The wavelength of light having a frequency of \(8.6 × 10^{13} Hz\) is \(3.5 nm\), and the frequency of light having a wavelength of \(566 nm\) is \(5.3 × 10^{14} Hz\).

Step-by-step solution

Calculate Wavelength

Given a frequency of \(8.6 × 10^{13} Hz\), use the formula \(λ = c/f\). Replace \(c\) with \(3.0 × 10^8 m/s\) and \(f\) with the given frequency, \(8.6 × 10^{13} Hz\). Perform the calculation to find the wavelength in meters. Then, convert meters to nanometers by multiplying by \(1 × 10^{9}\), since 1 meter is equal to \(1 × 10^{9}\) nanometers.

Calculate Frequency

Given a wavelength of \(566 nm\), use the formula \(f = c/λ\). Replace \(c\) with \(3.0 × 10^8 m/s\) and convert the wavelength from nanometers to meters by multiplying by \(1 × 10^{-9}\), as \(1 nm = 1 × 10^{-9} m\). Perform the calculation to find the frequency in Hz.

Key concepts

Speed of Light

The speed of light is a fundamental constant, vital for understanding electromagnetic radiation. It measures how fast light travels in a vacuum and is approximately \(3.0 \times 10^8\) meters per second (m/s). This constant speed is pivotal in connecting the concepts of frequency and wavelength because any electromagnetic radiation – from radio waves to visible light – travels at this speed.
In the formula \(c = \lambda \times f\), \(c\) represents the speed of light. Here, \(\lambda\) is the wavelength, and \(f\) is the frequency. This relationship implies that as the frequency of a light wave increases, its wavelength decreases, maintaining a constant product equal to the speed of light.
For example:

• An increase in frequency means the light wave oscillates more times per second.
• A decrease in wavelength means each oscillation occupies a shorter space.

By understanding this consistent speed, you gain insight into the fundamental behavior of light across different media.

Conversion of Units

Converting units is essential when performing calculations in physics, especially with electromagnetic radiation. It’s common to switch between meters and nanometers, or Hertz (Hz) and megahertz (MHz), to suit the scale of the problem.
For instance, in the exercise, we often need to convert between meters (m) and nanometers (nm). Here is how:

• 1 meter equals \(1 \times 10^9\) nanometers.
• 1 nanometer equals \(1 \times 10^{-9}\) meters.

Similarly, converting frequency units is important:

• 1 Hz equals \(1 \times 10^0\) cycles per second.
• 1 MHz is \(1 \times 10^6\) Hz.

Unit conversion helps align calculations to a common metric, making equations easier to solve while interpreting results in a tangible form. Proper conversion assures that all components of the formula are in compatible units, ensuring accurate computations.

Electromagnetic Radiation

Electromagnetic radiation comprises waves of electric and magnetic fields travelling through space at the speed of light. This spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
Each type of electromagnetic radiation is defined by its wavelength and frequency - critical parameters that influence how we perceive and use these waves. For instance:

• Visible light has wavelengths ranging roughly from 400 to 700 nanometers – this is the light we can see.
• Infrared radiation has longer wavelengths, not visible but felt as heat.
• On the shorter end, we have ultraviolet light, which can cause sunburn.

The relationship between wavelength and frequency applies across the electromagnetic spectrum, dictating how we utilize each type of radiation in technology and daily life. From communicating via radio waves to powering solar panels with visible light, understanding electromagnetic radiation opens the door to many applications.