Wave propagation in dielectrics: If the free-space wavelength is λ0 and a uniform, lossless insulator has relative permeability μr and relative permittivity εr, what is the wavelength λ inside that insulator?

Introduction / Context:
When an electromagnetic wave travels through a uniform, lossless dielectric, both its phase velocity and wavelength change compared with free space. Understanding how the wavelength scales with the material properties μr (relative permeability) and εr (relative permittivity) is fundamental in microwave engineering, antenna design, and transmission-line calculations.

Given Data / Assumptions:

• Free-space wavelength: λ0.
• Material parameters: relative permeability μr and relative permittivity εr.
• Medium is uniform, isotropic, and lossless; dispersion is ignored over the band of interest.
• We seek the wavelength λ inside the material.

Concept / Approach:

The phase velocity in a material is v = c / sqrt(εr * μr), where c is the speed of light in free space. Wavelength is related to phase velocity by λ = v / f. Since free-space wavelength is λ0 = c / f, substitution yields λ = (c / f) / sqrt(εr * μr) = λ0 / sqrt(εr * μr). This compact relation links material constants directly to wavelength reduction.

Step-by-Step Solution:

Verification / Alternative check:

Special cases confirm the formula. For nonmagnetic dielectrics (μr ≈ 1), λ ≈ λ0 / sqrt(εr). In air (εr ≈ μr ≈ 1), λ = λ0. For magnetic media (μr > 1), the wavelength shortens further, consistent with slower phase velocity.

Why Other Options Are Wrong:

• λ = λ0 * sqrt(εr * μr): predicts wavelength increases in denser media, which contradicts physics.
• λ = λ0 / εr or λ = λ0 * μr: ignore the square-root dependence and the joint role of both μr and εr.
• λ independent of material: false; materials clearly alter phase velocity and wavelength.

Common Pitfalls:

Mixing up velocity and wavelength scaling, forgetting the square root, or assuming μr = 1 without checking if the medium is nonmagnetic. Always verify whether μr departs from unity in ferrites or metamaterials.

Final Answer:

λ = λ0 / sqrt(εr * μr)