A proton moves in a straight line at constant velocity 0.99c. Does it radiate electromagnetic energy?
AYes — at relativistic speed its fields become highly concentrated, releasing radiation
BNo — only acceleration produces radiation; constant velocity, regardless of speed, produces no radiation field
CYes — the velocity field increases as the proton approaches an observer, so energy is radiated
DNo — but only because 0.99c is below the threshold at which radiation begins
Radiation is caused by acceleration, not velocity. A charge in uniform motion — even at 0.99c — has only a velocity field that falls as 1/r² and carries no net energy to infinity. The radiation field ∝ 1/r only appears when the charge accelerates. This is why synchrotron radiation requires bending magnets (which accelerate electrons centripetally) rather than just high speed alone. There is no velocity threshold — the dividing line is zero versus nonzero acceleration.
Question 2 Multiple Choice
A non-relativistic electron undergoes simple harmonic motion. At the moment of maximum displacement (momentarily at rest), does it radiate?
ANo — it is at rest, so it behaves like a stationary charge and produces no radiation
BYes — at maximum displacement the restoring force (and thus acceleration) is largest, so it radiates most strongly
CYes — but only because the radiation is a delayed effect of its prior motion
DNo — it only radiates when it is moving, so radiation is maximum at the equilibrium position
Radiation depends on acceleration, not velocity. At maximum displacement in SHM, velocity is zero but acceleration is maximum (the restoring force F = −kx is largest there). Since radiation power is proportional to acceleration squared, maximum displacement is the moment of peak radiation. At the equilibrium position, velocity is maximum but acceleration is zero — no radiation at that instant. This perfectly illustrates why acceleration, not speed, is the operative quantity.
Question 3 True / False
The near (velocity) field of a moving charge also decreases as 1/r and therefore carries a finite amount of energy to infinity.
TTrue
FFalse
Answer: False
The near field decreases as 1/r², not 1/r. The Poynting vector (energy flux) is proportional to E × B, so for the near field it goes as 1/r⁴. Integrating over a sphere of radius r gives power ∝ r² × (1/r⁴) = 1/r² → 0 as r → ∞. The near field carries zero net energy to infinity — it's reactive energy oscillating near the charge. Only the radiation field, with 1/r dependence, produces a Poynting vector ∝ 1/r² that integrates to a constant nonzero power over any sphere.
Question 4 True / False
At relativistic speeds, radiation from an accelerated charge is distributed uniformly in most directions, just as in the non-relativistic case.
TTrue
FFalse
Answer: False
In the non-relativistic case, radiation follows a sin²θ pattern (a donut around the acceleration axis). At relativistic speeds, the radiation is concentrated strongly in the forward direction — relativistic beaming. This is why synchrotron light sources produce tightly collimated beams: relativistic electrons emit radiation mostly forward along their direction of motion. The beaming effect intensifies with increasing speed and is a direct consequence of the relativistic Liénard-Wiechert fields.
Question 5 Short Answer
Explain why the 1/r dependence of the radiation field, as opposed to the 1/r² dependence of the near field, is physically significant for energy transport.
Think about your answer, then reveal below.
Model answer: Energy flux (Poynting vector) is proportional to E × B. For a field component going as 1/r, the Poynting vector goes as 1/r². Integrating over a spherical surface of radius r gives total power ∝ r² × (1/r²) = constant, independent of r. Energy flows outward at the same rate through any sphere, no matter how large — it escapes to infinity permanently. For the near field (1/r²), the Poynting vector goes as 1/r⁴, integrating to 1/r² → 0 as r → ∞. Near-field energy never escapes; it oscillates reactively near the charge.
This is the precise reason why acceleration is necessary for radiation: only the acceleration-dependent 1/r field produces an outward energy flux that doesn't vanish at large distances. The near field is energy that's 'borrowed' from and returned to the source — never permanently radiated away.