Electromagnetic wave velocity in a medium: If permittivity = ε and permeability = μ, what is the wave speed?

Introduction / Context:
The speed of an electromagnetic (EM) wave in a material medium depends on that medium’s electric permittivity and magnetic permeability. This relationship is a cornerstone of classical electromagnetics and directly connects material properties to propagation speed and refractive index.

Given Data / Assumptions:

• Material characterized by absolute permittivity ε and absolute permeability μ.
• Losses are negligible (non-conducting, non-dispersive medium) for the basic relation.
• We refer to phase velocity of a plane EM wave.

Concept / Approach:
Maxwell’s equations yield the wave equation with wave speed v satisfying v^2 = 1 / (ε * μ). In free space, ε = ε0 and μ = μ0, giving c = 1 / √(ε0 * μ0). In materials, ε and μ alter the speed relative to c, often summarized by refractive index n where n = c / v = √(ε_r * μ_r).

Step-by-Step Solution:
Start with Maxwell curl equations to derive the homogenous wave equation.Identify that the wave equation constant gives v^2 = 1/(εμ).Take square root for positive speed: v = 1 / √(ε * μ).Relate to refractive index: n = √(ε_r * μ_r) = c / v, as a consistency check.

Verification / Alternative check:
Plugging ε0 ≈ 8.854×10^-12 F/m and μ0 = 4π×10^-7 H/m yields c ≈ 3×10^8 m/s, matching measured light speed, validating the formula.

Why Other Options Are Wrong:
εμ or √(ε/μ): Dimensional mismatch for speed.1/(ε*μ): Missing square root; wrong magnitude by many orders.None of these: Incorrect because a standard closed-form exists.

Common Pitfalls:
Forgetting the square root; confusing relative (ε_r, μ_r) with absolute (ε, μ) values; ignoring frequency-dependent dispersion in real materials (the ideal relation is still foundational).

Final Answer:
v = 1 / √(ε * μ).