Questions: Radiation from Accelerating Charges

Question 1

An electron travels at half the speed of light in a perfectly straight line through vacuum. Does it radiate electromagnetic energy?

• A. Yes — any charged particle moving through vacuum radiates due to the Cherenkov effect
• B. Yes — high-velocity charges always radiate because of their large kinetic energy
• C. No — a charge in uniform motion (constant velocity, straight line) produces only bound Coulomb-like fields that carry no net energy to infinity
• D. No — radiation only occurs at speeds exceeding the speed of light

Answer: (C)

Radiation requires acceleration. A charge at constant velocity produces fields that are purely electrostatic in the instantaneous rest frame. These Coulomb-like fields fall off as 1/r² and carry zero net energy flux through any distant sphere — they are bound fields accompanying the charge. The Larmor formula P ∝ a² confirms this: zero acceleration gives zero radiated power. Note: the Cherenkov effect (option A) occurs when a charge travels faster than light *in a medium*, not in vacuum, and arises from a different mechanism entirely.

Question 2

Why does the 1/r falloff of radiation fields — rather than the 1/r² falloff of Coulomb fields — determine whether energy escapes to infinity?

• A. 1/r fields are stronger near the source, so they carry more total energy outward
• B. Energy flux (Poynting vector) scales as |E|², so a 1/r field gives flux ∝ 1/r², which integrates to a constant over a sphere of area 4πr², giving nonzero power at any distance
• C. 1/r fields have lower frequency than 1/r² fields and therefore propagate further
• D. The falloff rate is irrelevant; what matters is that radiation fields oscillate while Coulomb fields are static

Answer: (B)

The energy flux (Poynting vector magnitude) scales as |E|². A field that falls as 1/r gives flux ∝ 1/r². Integrated over a sphere of radius r (area 4πr²), this gives total power ∝ 4πr² · (1/r²) = constant — independent of r. Energy passes through every sphere at the same rate: it has truly escaped to infinity. For 1/r² Coulomb fields, flux ∝ 1/r⁴, which integrated over a sphere gives zero as r → ∞. The bound fields carry no net energy outward.

Question 3

A uniformly charged sphere moving at constant velocity radiates electromagnetic energy because the charges within it undergo centripetal acceleration to maintain their relative positions.

• T. True
• F. False

Answer: False

Charges within a rigidly moving sphere do not accelerate if the sphere moves at constant velocity in a straight line — every charge undergoes the same translational motion, and in the center-of-mass frame each is stationary. More fundamentally, uniform translational motion (constant velocity) produces no radiation regardless of the geometry. The Larmor formula P ∝ a² gives zero radiated power for zero acceleration. The misconception conflates 'rigid body motion' with 'individual charge acceleration.'

Question 4

The classical picture of an electron orbiting a nucleus fails because an orbiting electron is continuously accelerating and should therefore radiate energy, spiraling inward.

• T. True
• F. False

Answer: True

Centripetal acceleration is real acceleration — the electron's direction changes continuously even if its speed is constant. By the Larmor formula P = q²a²/(6πε₀c³), any acceleration produces radiated power. For a classical orbiting electron, this power loss would cause it to spiral inward in a tiny fraction of a second. This 'ultraviolet catastrophe' for atomic stability was one of the central crises classical physics could not resolve, ultimately requiring quantum mechanics and the quantization of orbital angular momentum.

Question 5

Explain why accelerating charges radiate but charges moving at constant velocity do not. What physically happens to the electric field lines when a charge is suddenly accelerated?

Model answer: When a charge moves at constant velocity, its field lines rearrange smoothly and continuously, always appearing as the field of a charge at its current position. When a charge is suddenly accelerated, the field lines cannot rearrange instantaneously — information propagates at the speed of light. The 'old' Coulomb-field configuration persists at large distances while the 'new' configuration propagates outward from the charge. The mismatch creates a kink in the field lines that propagates outward as an electromagnetic wave — the radiation field. This 1/r kink carries energy independently of the source and persists at arbitrary distance, unlike the 1/r² bound Coulomb field.

The key concept is retardation: field rearrangement takes time, and when a charge accelerates, the field structure at large distances cannot keep up. This kink is permanent — it propagates outward forever — while the bound fields of a uniformly moving charge carry no energy to infinity. Every radiating antenna, synchrotron, and emitting atom exploits exactly this mechanism.