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 bulk 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 bulk modulus of window glass, which is \(5.0 \cdot 10^{10} \mathrm{~N} / \mathrm{m}^{2}\).

Short answer

Answer: The required bulk modulus of glass to support the transmission of light waves at a speed of \(2.0\cdot10^8\; m/s\) is \(1.0\cdot10^{14}\; N/m^2\). This is significantly larger than the actual bulk modulus of window glass, which is \(5.0\cdot10^{10}\; N/m^2\).

Step-by-step solution

Extract given information from the problem

We are given: - Speed of light in glass (v): \(2.0\cdot10^8 \; m/s\) - Mass density of glass (ρ): \(2500 \; kg/m^3\) - Actual bulk modulus of window glass: \(5.0\cdot10^{10} \; N/m^2\)

Calculate the required bulk modulus of glass

Using the equation for the speed of a mechanical wave in a solid: \(v=\sqrt{\frac{B}{\rho}}\) We can rearrange the formula to solve for the bulk modulus (B): \(B = \rho v^2\) Now plug in the given values of ρ and v: \(B = 2500 \; kg/m^3 \cdot (2.0\cdot10^8 \; m/s)^2\) Calculate the result: \(B = 1.0\cdot10^{14} \; N/m^2\)

Compare the calculated bulk modulus with the actual bulk modulus

The calculated bulk modulus for glass to support the transmission of light waves at the given speed is \(1.0\cdot10^{14} \; N/m^2\). Comparing this to the actual bulk modulus of window glass, which is \(5.0\cdot10^{10} \; N/m^2\), we can see that the calculated bulk modulus is much larger than the actual bulk modulus. This means that the actual bulk modulus of window glass would not be sufficient to support the transmission of light waves at the speed given. In conclusion, the required bulk modulus of glass to support the transmission of light waves at a speed of \(2.0\cdot10^8\; m/s\) is \(1.0\cdot10^{14}\; N/m^2\). This is significantly larger than the actual bulk modulus of window glass, which is \(5.0\cdot10^{10}\; N/m^2\).

Key concepts

Speed of Light in Materials

The speed of light in materials is an important concept in physics, particularly in optics and materials science, which refers to the velocity at which light waves travel through different mediums. Unlike the vacuum of space where light travels at approximately 299,792,458 meters per second (the speed of light in a vacuum, denoted as 'c'), light slows down when it passes through materials such as glass, water, or plastic. This slower speed is due to the material's refractive index, a dimensionless number that describes how much light bends when entering the material.

The speed of light in a material, denoted as 'v', can be calculated using the equation: \(v = \frac{c}{n}\),where 'c' is the speed of light in vacuum and 'n' is the refractive index of the material. A higher refractive index indicates a slower speed of light within that material. This is crucial in designing lenses, glasses, and other optical devices as it affects the way light is refracted and reflected within these materials.

The exercise provided asks for the calculation of the bulk modulus based on the speed at which light waves should propagate within glass, offering a wonderful opportunity to explore the relationship between these two properties.

Mass Density

Mass density, denoted by \(\rho\),is a critically important concept in various scientific disciplines like physics, material science, and engineering. It is defined as the mass per unit volume of a substance and is typically measured in kilograms per cubic meter (kg/m³) in the International System of Units (SI). The mass density of a material influences many of its physical properties, including strength, buoyancy, and thermal conductivity.

In the context of mechanical waves, the mass density plays a role in determining how these waves propagate through different materials. For example, when discussing the movement of sound through air versus through water, the denser water transmits sound faster due to its higher mass density. The exercise we are looking at uses the concept of mass density of glass to explore how light waves would propagate through the material.

Mechanical Wave Propagation

Mechanical wave propagation refers to the process by which waves travel through a medium. Unlike electromagnetic waves, mechanical waves like sound require a material medium to travel through. They can be transverse or longitudinal, depending on the direction of oscillation relative to the wave's direction of travel.

Mechanical waves in solids are governed by the material's elasticity and density. The speed at which these waves travel is crucial for understanding phenomena such as earthquakes, acoustics, and the physical properties of materials. The formula used in the textbook exercise, \(v=\sqrt{\frac{B}{\rho}}\),demonstrates this by relating the speed of a wave 'v' to the material's bulk modulus 'B' and its mass density 'ρ'.

The exercise asks to calculate the bulk modulus needed for glass to transmit light at a certain speed. While light waves are not mechanical and don't require a medium, they still slow down in denser materials, a concept which parallels mechanical wave behavior in materials such as glass.