An accelerating charge produces two field terms: one decaying as 1/r and one decaying as 1/r². As r → ∞, which carries energy to infinity and why?
AThe 1/r² term carries energy to infinity because it is stronger near the source and dominates the total energy
BThe 1/r term carries energy to infinity: the Poynting vector scales as E² ∝ 1/r², so power through a sphere (∝ r²) is constant in r
CThe 1/r term dominates at large r, but the total power through a sphere still falls to zero as r → ∞
DBoth terms contribute equally to the total radiated power at all distances
Energy flux (the Poynting vector) scales as S ∝ E². For the 1/r radiation field, S ∝ 1/r². Integrated over the surface of a sphere of radius r (area ∝ r²), total power P = ∮ S · dA ∝ (1/r²)(r²) = constant — independent of r. This constant power flow is what it means to 'radiate': energy genuinely escapes to infinity. For the 1/r² near-field term, S ∝ 1/r⁴, so P ∝ (1/r⁴)(r²) = 1/r² → 0. The near-field term stores and returns energy locally; only the radiation field carries energy irreversibly away.
Question 2 Multiple Choice
In the radiation zone (kr >> 1), the electric field E of an oscillating dipole and the direction of propagation k̂ satisfy:
AE is parallel to k̂ — the wave is longitudinal, like a sound wave
BE has no fixed relationship to k̂ — the polarization depends on the observation angle in an arbitrary way
CE is perpendicular to k̂ (transverse), with B also perpendicular to both E and k̂, and |B| = |E|/c
DE and B are both parallel to k̂, since they must point in the direction of energy propagation
Far-field radiation is always a transverse electromagnetic wave: E and B are both perpendicular to k̂ (the propagation direction) and to each other, with |B| = |E|/c. This follows from the double cross product in the radiation field formula E ∝ k̂ × (k̂ × p̈), which projects p̈ onto the plane perpendicular to k̂. Option A (longitudinal) would violate Maxwell's equations in free space. The transverse nature is a defining feature of the radiation zone and underlies antenna theory.
Question 3 True / False
The total power radiated by an oscillating dipole, calculated as the integral of the Poynting vector over a sphere, decreases as the sphere's radius increases.
TTrue
FFalse
Answer: False
This is the defining property of radiation: the Poynting vector S ∝ 1/r², and the surface area of the sphere grows as 4πr², so total power P = ∮ S · dA is constant in r. This r-independence is what it means for radiation to carry energy to infinity — every spherical shell, no matter how large, captures the same amount of power per unit time. If power fell with r, the source would not be truly radiating in the sense of irreversible energy loss.
Question 4 True / False
In the radiation zone, the electric field of a dipole source is proportional to the second time derivative (acceleration) of the dipole moment, not the moment itself or its first derivative.
TTrue
FFalse
Answer: True
The radiation field formula E ∝ (1/r)[k̂ × (k̂ × p̈)] shows that E depends on p̈ — the acceleration of the charge distribution, not its position or velocity. This is why a charge moving at constant velocity does not radiate (p̈ = 0 for uniform motion), while an accelerating charge does. The p̈ dependence is also the origin of the Larmor formula for radiated power (P ∝ p̈²), which underpins all classical radiation theory.
Question 5 Short Answer
Why does the 1/r² near-field term of an accelerating charge NOT contribute to radiation, while the 1/r term does? Use the concept of total power through a sphere in your answer.
Think about your answer, then reveal below.
Model answer: The Poynting vector (energy flux) is proportional to E². For the 1/r² near-field term, E² ∝ 1/r⁴, so the total power through a sphere of radius r scales as (1/r⁴)(4πr²) = 4π/r² → 0 as r → ∞. The near-field carries no net energy to infinity; it stores energy in the fields near the source and returns it on each cycle. For the 1/r radiation term, E² ∝ 1/r², so power through a sphere scales as (1/r²)(4πr²) = 4π = constant. This r-independent power flow means energy irreversibly leaves the source — the hallmark of radiation.
The distinction between near-field and radiation field is fundamentally about whether energy escapes permanently. Near-field terms are associated with the static and inductive fields of the source — they create a reactive energy 'halo' that oscillates in and out but doesn't propagate. Only the 1/r term, produced by acceleration, creates the self-sustaining electromagnetic wave that propagates to infinity. This is why antennas must have accelerating charges (oscillating currents) to radiate — uniform current flow produces only near-field.