Question

What is the wavelength of light with a frequency of \(5.77 \times 10^{14} \mathrm{Hz}\) ?

Short answer

The wavelength of light with a frequency of \(5.77 \times 10^{14} Hz\) is approximately 520 nanometers.

Step-by-step solution

Write down the given information and formula

We are given the frequency of light, \(f = 5.77 \times 10^{14} Hz\), and need to find the wavelength, \(\lambda\). The formula relating the speed of light, frequency, and wavelength is: \(c = \lambda f\)

Use the speed of light constant

The speed of light in a vacuum is approximately \(3 \times 10^8 m/s\). We plug this value into the equation: \(3 \times 10^8 m/s = \lambda (5.77 \times 10^{14} Hz)\)

Solve for the wavelength

To solve for \(\lambda\), we will divide both sides of the equation by the given frequency: \(\lambda = \frac{3 \times 10^8 m/s}{5.77 \times 10^{14} Hz}\)

Calculate the wavelength

Now, we will carry out the division: \(\lambda = \frac{(3 \times 10^{8}) m/s}{(5.77 \times 10^{14}) Hz} = \frac{3}{5.77} \times \frac{10^8}{10^{14}} m\) \(\lambda \approx 0.52 \times 10^{-6} m\)

Express the result in nanometers

To express the result in nanometers, we need to convert meters to nanometers by multiplying by \(10^{9}\): \(\lambda \approx 0.52 \times 10^{-6} m \times 10^{9} nm/m \approx 520 nm\) The wavelength of light with a frequency of \(5.77 \times 10^{14} Hz\) is approximately 520 nanometers.

Key concepts

Frequency of Light

Frequency refers to the number of oscillations or cycles that occur in a wave per unit of time. In the case of light waves, it indicates how many wave peaks pass a given point in one second. The frequency of light is measured in Hertz (Hz), which means cycles per second.

The frequency is crucial when calculating other properties of light waves, such as their wavelength. The relationship is expressed by the formula: \(c = \lambda f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency.

• High frequency implies shorter wavelength.
• Low frequency results in a longer wavelength.
• Frequency doesn't change even if light travels through different mediums.

Understanding the frequency of light helps in a variety of fields such as physics, optics, and even telecommunications as it determines the color of visible light and the type of electromagnetic waves (radio, microwave, etc.).

Speed of Light

The speed of light is one of the most fundamental constants in physics. In a vacuum, it is usually represented as \(c\) and is approximately equal to \(3 \times 10^8\) meters per second (m/s). This constant speed makes light unique among all types of waves.

The speed of light is a cornerstone concept in equations that relate frequency and wavelength as seen in the formula: \(c = \lambda f\). This relationship is indispensable in physics when analyzing how light behaves, how it is refracted or reflected, and how it travels across various distances.

• The speed of light remains constant in a vacuum.
• It is slower in other mediums like water or glass.
• Understanding it is essential for calculations in both classical and modern physics.

Thus, the speed of light not only unifies different domains of physics but also remains critical in applications like astrophysics, time calculations, and even global positioning systems.

Nanometer Conversion

Nanometer conversion is an important concept in science, particularly in optics and materials science, because wavelengths in light are generally expressed in nanometers. A nanometer is one-billionth of a meter or \(10^{-9}\) meters. This unit is especially useful when dealing with tiny scales that meters cannot adequately express.

When calculating the wavelength of light from a given frequency using the formula \(\lambda = \frac{c}{f}\), the result is typically in meters. To convert this to nanometers, which is a more convenient unit for expressing wavelengths of light, you multiply by \(10^9\) to translate meters into nanometers:

\[ \lambda (nm) = \lambda (m) \times 10^9 \]

• Nanometers provide a more comprehensible scale for light wavelengths.
• Conversions simplify comparisons with known values.
• Commonly used in industries like optics or semiconductor technology.

Using nanometer conversion ensures that calculated wavelengths have practical relevance and allows efficient communication and understanding of scales in scientific contexts.