Question

Electromagnetic radiation (light) consists of waves. More than a century ago, scientists thought that light, like other waves, required a medium (called the ether) to support its transmission. Glass, having a typical mass density of \(\rho=2500 \mathrm{~kg} / \mathrm{m}^{3},\) also supports the transmission of light. What would the elastic modulus of glass have to be to support the transmission of light waves at a speed of \(v=2.0 \cdot 10^{8} \mathrm{~m} / \mathrm{s} ?\) Compare this to the actual elastic modulus of window glass, which is \(5 \cdot 10^{10} \mathrm{~N} / \mathrm{m}^{2}\).

Short answer

Answer: The calculated elastic modulus required to support the given light wave speed in glass is \(1 \cdot 10^{11} \mathrm{N/m^2}\). When compared to the actual elastic modulus of window glass, which is \(5 \times 10^{10}\mathrm{\,N/m^2}\), the calculated value is approximately twice the actual value.

Step-by-step solution

Write down the speed of light and the mass density of glass.

The provided speed of the light waves is \(v = 2.0 \cdot 10^8 \mathrm{\,m/s}\), and the mass density of glass \(\rho = 2500 \mathrm{\,kg/m^3}\).

Recall the formula for wave speed in a solid material.

In a solid material, the wave speed can be calculated using the formula: \(v = \sqrt{\frac{E}{\rho}}\), where \(v\) is the wave speed, \(E\) is the elastic modulus, and \(\rho\) is the mass density.

Rearrange the formula to find the elastic modulus.

We need to find the value of \(E\) to support the given wave speed. Rearrange the formula to get: \(E = \rho v^2\).

Input the values of mass density and wave speed into the formula.

Plug in the known values of \(\rho\) and \(v\) to get: \(E = (2500 \mathrm{\,kg/m^3}) \times (2.0 \cdot 10^8 \mathrm{\,m/s})^2\).

Calculate the required elastic modulus.

Perform the calculation: \(E = 2500 \times (2.0 \cdot 10^8)^2 \mathrm{\,N/m^2} = 1 \cdot 10^{11} \mathrm{\,N/m^2}\).

Compare the calculated elastic modulus with the actual value.

The calculated elastic modulus of glass to support the given light wave speed would be \(1 \cdot 10^{11} \mathrm{~N/m^2}\). The actual elastic modulus of window glass is \(5 \times 10^{10}\mathrm{\,N/m^2}\). Comparing the two values, the calculated elastic modulus is approximately twice the actual value of window glass.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that is all around us and takes the form of waves. These waves are composed of oscillating electric and magnetic fields that can travel through the vacuum of space, which was a groundbreaking realization that debunked the need for the hypothetical 'ether' medium that was once thought necessary for light propagation.

Common examples of electromagnetic radiation include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type of electromagnetic wave is distinguished by its wavelength—or frequency—in the electromagnetic spectrum. Visible light, a tiny part of the spectrum, is the type of electromagnetic wave that is most familiar to us, as it is what is perceived by the human eye.

In the context of the given problem, light waves traveling through glass are considered. The speed at which light travels through any material depends on the material's optical properties, affecting its refractive index and consequently its wave speed within the medium.

Wave Speed in Solids

The speed of wave propagation in solids is a fundamental aspect of understanding how sound, seismic activities, and even light waves travel through different materials. For solids, the wave speed can be expressed as \( v = \sqrt{\frac{E}{\rho}} \), where \( v \) represents the wave speed, \( E \) is the elastic modulus, and \( \rho \) is the mass density of the material.

This relationship indicates that the wave speed is directly proportional to the square root of the elastic modulus of the material and inversely proportional to the square root of the mass density. The elastic modulus is a measure of a material's stiffness or resistance to deformation, while the mass density is a measure of its compactness. A higher elastic modulus—which suggests a more rigid structure—will generally result in a higher wave speed, all other factors being equal.

Mass Density of Materials

Mass density, denoted by \( \rho \), is a property that measures how much mass is contained in a given volume. It is typically expressed in units of \( \mathrm{kg/m^3} \). Mass density plays a crucial role in not just mechanical and structural engineering fields, but also in understanding the propagation of waves through different media.

Materials with higher mass densities tend to be heavier and may reflect or absorb waves more than less dense materials, which may allow waves to pass through more readily. In the exercise improvement advice, understanding mass density is particularly important as it influences wave speed in solids. As seen in the step by step solution, the formula \( E = \rho v^2 \) establishes the relationship between the mass density and the elastic modulus required for a given wave speed.

When considering the exercise solution, the mass density of glass is crucial for calculating the necessary elastic modulus to support the transmission of light waves at a specified speed. In the real world, this knowledge is applied to design materials with particular properties, such as the glass in fiber optic cables, where light needs to be transmitted efficiently over long distances.