Compute the velocity of light in diamond, which has a dielectric constant \(\epsilon_{r}\) of 5.5 (at frequencies within the visible range) and a magnetic susceptibility of \(-2.17 \times 10^{-5}\).

Understanding the dielectric constant is crucial when studying how light propagates through different materials. In essence, the dielectric constant, often represented as \(\epsilon_r\), is a measure of a material’s ability to store electrical energy within its structure. It's also known as the relative permittivity of the material and indicates how much the electric field is reduced compared to a vacuum.

A vacuum has a dielectric constant of 1, and when light enters a medium with a higher dielectric constant, such as diamond with \(\epsilon_r = 5.5\), the electric field component of the electromagnetic wave inducing the light is affected. The higher the dielectric constant, the slower the light will travel through the material. This is because the material's atoms or molecules polarize in response to the light’s electric field, creating internal fields that oppose the light's propagation, effectively causing a reduction in the velocity of light.

This concept is key for various applications, including the design of lenses, optical fibers, and even in understanding natural phenomena such as rainbows, where refractive indices play a pivotal role.

Magnetic susceptibility, represented by \(\chi_m\), is a measure of how much a material will become magnetized in an applied magnetic field. This magnetization is directly proportional to the applied field, such that \(M = \chi_m H\), where \(M\) is the magnetization of the material and \(H\) is the strength of the magnetic field.

In the context of light propagation, magnetic susceptibility affects the magnetic component of the electromagnetic wave. The value of \(\chi_m\) can be positive or negative, indicating whether the material is paramagnetic (+) or diamagnetic (−), respectively. Diamond's negative magnetic susceptibility, \(\chi_m = -2.17 \times 10^{-5}\), means it's a diamagnetic material and repels magnetic fields slightly, which may contribute to changes in the velocity of light within the diamond.

It's important for students to comprehend that the magnetic susceptibility of most materials is very small, and therefore the effect on light speed is typically negligible. However, for precise scientific applications, even this small effect can be significant.

The relative permeability, denoted as \(\mu_r\), is a ratio that describes the permeability of a material compared to the vacuum permeability. Permeability is a characteristic that indicates how easy it is for a magnetic field to pass through a material. For a vacuum, \(\mu_r\) is set to 1.

Relative permeability is closely related to magnetic susceptibility through the relationship \(\mu_r = 1 + \chi_m\). Since magnetic fields are part of electromagnetic waves, such as light, \(\mu_r\) indirectly influences the propagation speed of light in a material. For a material like diamond with a \(\mu_r\) that is slightly less than 1 due to its negative \(\chi_m\), we can infer that it is diamagnetic and opposes the magnetic field of the light slightly.

This relationship determines how light waves interact with the medium's magnetic field and affects the index of refraction, which in turn influences the actual speed of light in that material. In diamonds, the \(\mu_r\) is very close to 1, indicating that magnetic effects on light speed are minimal and primarily dominated by the dielectric constant when considering the velocity of light within the material.