Plane Wave in a Dielectric: A plane electromagnetic wave travels in a medium with relative permittivity εr = 9 (μr ≈ 1). How does its speed compare to free space?

Introduction / Context:Propagation velocity of electromagnetic waves in a lossless dielectric depends on the medium’s permittivity and permeability. This question reinforces the relationship between speed, refractive index, and relative permittivity.

Given Data / Assumptions:

• Relative permittivity εr = 9.
• Relative permeability μr ≈ 1 (non-magnetic).
• Free-space speed c ≈ 3 × 10^8 m/s.

Concept / Approach:

The wave speed in a medium is v = c / sqrt(εr μr). For μr ≈ 1, v = c / sqrt(εr). The refractive index n = sqrt(εr μr) = sqrt(εr) when μr = 1.

Step-by-Step Solution:

Verification / Alternative check:

Equivalently, n = 3 implies v = c/n = c/3. This matches standard optics/electromagnetics relations.

Why Other Options Are Wrong:

• Increased by 9 or 3: contradicts v = c/√εr.
• Unchanged: only true for εr = 1.
• Reduced by 1/9: would require εr = 81.

Common Pitfalls:

• Confusing phase velocity scaling with εr instead of √εr.
• Assuming μr ≠ 1 without being given.

Final Answer:

reduced by a factor of 1/3