Electromagnetic waves in good conductors: what is the approximate propagation velocity compared with free space?

Introduction / Context:
When an electromagnetic (EM) wave enters a good conductor such as copper or aluminum, its behavior differs markedly from that in free space or dielectrics. This question checks core understanding of how conductivity affects propagation velocity, attenuation, and field penetration depth.

Given Data / Assumptions:

• The medium is a good conductor with high conductivity σ and approximately μ ≈ μ0, ε ≈ ε0.
• We compare wave velocity qualitatively to c ≈ 3 × 10^8 m/s in free space.
• We focus on physical trends rather than an exact numeric value.

Concept / Approach:

In a good conductor at radio or microwave frequencies, the propagation constant γ = α + jβ satisfies α ≈ β ≈ √(π f μ σ). The phase velocity v_p = ω/β and the group/energy transport (as seen via Poynting vector) are both drastically affected by high loss and very shallow penetration depth δ = 1/α. The wave attenuates quickly and advances slowly within the conducting medium.

Step-by-Step Solution:

Verification / Alternative check:

For copper at 1 MHz: σ ≈ 5.8×10^7 S/m, μ ≈ μ0. One finds α ~ β on the order of hundreds of Np/m, yielding v_p far below c. This qualitative outcome matches the 'very low' choice.

Why Other Options Are Wrong:

• '3 × 10^8 m/s' and 'more than 3 × 10^8 m/s': contradict the strong dispersion and loss in conductors; such speeds occur in non-conductive media.
• 'high' is vague and incorrect; the speed is not high, it is small.
• 'approximately 1 × 10^8 m/s': still too high for a good conductor’s interior propagation; the wave is quickly absorbed.

Common Pitfalls:

• Confusing surface/skin effects (fast currents along surfaces) with wave speed in the bulk.
• Assuming dielectric-like behavior inside metals; conductors are dominated by conduction currents and losses.

Final Answer:

very low