Question

A laser emits a beam of green light that has a wavelength of \(5.00 \times 10^{-5} \mathrm{~cm} .\) What is the wavelength of this light in nanometers?

Short answer

The wavelength of the green light in nanometers is 500 nm.

Step-by-step solution

Write down the given wavelength in centimeters

The wavelength of the green light is given as \(5.00 \times 10^{-5} \mathrm{~cm}\).

Set up the conversion factor

We know that 1 cm = 10000000 nm (1 x 10^7 nm). So our conversion factor is: \[ \frac{1 \mathrm{~cm}}{10^7 \mathrm{~nm}} \]

Convert the wavelength from centimeters to nanometers

We will now multiply the wavelength given in centimeters by the conversion factor: \[ \lambda_{\mathrm{nm}} = (5.00 \times 10^{-5} \mathrm{~cm}) \times \frac{1 \mathrm{~cm}}{10^7 \mathrm{~nm}} \]

Complete the calculation

Perform the multiplication and simplify the result: \[ \lambda_{\mathrm{nm}} = (5.00 \times 10^{-5}) \times (10^7 \mathrm{~nm}) \] \[ \lambda_{\mathrm{nm}} = 500 \mathrm{~nm} \]

State the final answer

The wavelength of the green light in nanometers is 500 nm.

Key concepts

Wavelength of Light

The wavelength of light is a fundamental property that determines its color and is part of what makes, for instance, a laser beam appear green or red. Wavelength is the distance between successive peaks (or troughs) in a wave and is commonly measured in meters or parts of it, such as centimeters or nanometers. In the visible spectrum, wavelengths range from about 380 nm (violet light) to 700 nm (red light). Shorter wavelengths correspond to higher energies and bluer colors, while longer wavelengths correspond to lower energies and redder colors.

Light behaves both as a wave and a particle, which is a concept known as wave-particle duality. When considering its wave-like properties, understanding wavelength helps in analyzing various phenomena, such as diffraction, interference, and the determination of light's interaction with different materials.

Nanometers

A nanometer (nm) is a unit of length in the metric system that equals one billionth of a meter (\(10^{-9}\text{ meters}\)). It is a common unit for measuring the wavelength of light, particularly in the field of optics, because it allows for a more convenient and tangible number when dealing with the extremely small wavelengths of visible light and other parts of the electromagnetic spectrum.

Example of Scale

The thickness of a sheet of paper is around 100,000 nanometers, highlighting how incredibly small a nanometer is. The use of nanometers is essential in areas like nanotechnology, semiconductor fabrication, and the study of biological structures at the molecular level.

Conversion Factors

Conversion factors are used to convert quantities from one unit to another. They are essentially a form of multiplicative identity (like the number 1) but expressed as a ratio of two different units, such that when multiplied by the original quantity, the desired unit conversion is achieved while the actual value remains unchanged.

It's crucial in physics and engineering to be precise with unit conversions to ensure accurate calculations and measurements. Using conversion factors helps maintain this accuracy and allows scientists and engineers to work with units that are most convenient for their specific field or the particular aspect of a phenomenon they are studying.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is based on powers of ten. In this notation, a number is written as the product of a coefficient (usually between 1 and 10) and a power of 10. This method is particularly helpful in fields like physics and astronomy where very large or very small values are common.

For example, the number 500 can be written as \(5.00 \times 10^2\) in scientific notation, which illustrates how this method can create a more compact and manageable representation of numerical information. It also simplifies arithmetic operations such as multiplication and division, especially with numbers that have many zeros.