A proton moves at half the speed of light in a perfectly straight line at constant velocity. What electromagnetic radiation does it emit?
AIt emits radiation because the velocity fields it produces carry energy outward
BIt emits radiation because relativistic beaming concentrates its fields
CIt emits no radiation — constant-velocity motion produces only velocity fields, which carry no energy to infinity
DIt emits radiation because κ ≠ 1 for relativistic motion
A charge in uniform motion has only velocity fields (∝ 1/r²). Their energy flux falls off as 1/r⁴, which integrates to zero over any large sphere — no energy escapes to infinity. Radiation requires acceleration (a⃗ ≠ 0 at the retarded time). κ ≠ 1 for relativistic motion affects the field's angular distribution but does not create radiation; it is a feature of the velocity (bound) fields only.
Question 2 Multiple Choice
The Lienard-Wiechert fields of an accelerating charge contain two components. Which has the slower spatial decay, and what is its physical significance?
AThe velocity field (∝ 1/r²) — it is responsible for radiation because it persists at large distances
BThe acceleration field (∝ 1/r) — it carries energy to infinity and constitutes electromagnetic radiation
CBoth components decay at the same rate; the distinction is only in their angular dependence
DThe acceleration field (∝ 1/r) — it accelerates nearby charges but carries no energy
The acceleration field falls off as 1/r, so its energy flux (∝ E² ∝ 1/r²) integrates to a finite nonzero value over a large sphere — energy escapes to infinity. This is radiation. The velocity field falls off as 1/r², giving energy flux ∝ 1/r⁴, which integrates to zero: no energy is carried to infinity regardless of how fast the charge moves.
Question 3 True / False
A charge undergoing acceleration has an acceleration field proportional to the component of acceleration perpendicular to the observation direction.
TTrue
FFalse
Answer: True
The radiation (acceleration) field is proportional to the component of a⃗ transverse to the retarded direction R̂. Acceleration along the line of sight to the observer produces no radiation in that direction — this is why a linearly oscillating charge radiates most strongly at 90° to its axis and nothing along its axis, consistent with Larmor's formula and classical antenna theory.
Question 4 True / False
The κ = 1 − (v⃗·R̂)/c factor in the Lienard-Wiechert potentials primarily determines whether a charge radiates.
TTrue
FFalse
Answer: False
κ appears in the denominator of both the scalar and vector potentials and controls relativistic beaming — it amplifies the fields of a charge moving toward you (κ < 1) and suppresses fields from a charge moving away (κ > 1). This is a property of the velocity (bound) fields. Radiation — whether it occurs at all — is determined solely by whether the charge is accelerating (a⃗ ≠ 0 at the retarded time), not by the value of κ.
Question 5 Short Answer
Why does a uniformly moving charge not radiate, even though it produces electromagnetic fields that vary in time as it passes an observer?
Think about your answer, then reveal below.
Model answer: Radiation requires acceleration. A uniformly moving charge produces only velocity (bound) fields that fall off as 1/r². Their energy flux falls off as 1/r⁴, integrating to zero over a distant sphere, so no net energy escapes to infinity. Radiation fields (∝ 1/r, energy flux ∝ 1/r²) appear only when acceleration is nonzero at the retarded time. This is consistent with the equivalence principle: an observer in free fall cannot detect radiation from an inertially moving charge.
The key is the 1/r versus 1/r² distinction. Fields that fall off faster than 1/r cannot deliver finite power flux through a large sphere (power ~ (1/r^n)² × r² → 0 if n > 1). Radiation is defined precisely as the component of the field that carries energy to infinity, which requires exactly the 1/r falloff that only acceleration produces.