Questions: Multipole Expansion and Far-Field Radiation

Question 1

A small antenna oscillates at frequency ω. Its charge distribution is perfectly symmetric, so the electric dipole moment p = Σqᵢrᵢ is identically zero at all times. What is the dominant radiation from this antenna?

• A. The antenna still radiates electric dipole radiation, because the individual charges still accelerate
• B. No radiation occurs, since the dipole moment is zero and dipole radiation is the only significant contribution for small antennas
• C. Magnetic dipole or electric quadrupole radiation dominates, suppressed by a factor of (ka)² relative to what dipole would have been
• D. The radiation pattern is identical to dipole radiation, but weaker by a factor of ka

Answer: (C)

When the dipole moment vanishes by symmetry, the dipole term in the multipole expansion is zero, and the next terms — magnetic dipole and electric quadrupole — dominate. These are suppressed by a factor (ka)² = (a/λ)² relative to electric dipole radiation. Dipole radiation requires a time-varying dipole moment; if that moment is zero, the leading-order radiation comes from the next multipole. This is exactly the situation for gravitational radiation: conservation laws forbid monopole and dipole gravitational radiation, forcing the dominant emission to be quadrupole.

Question 2

A small dipole antenna radiates at frequency f. If the frequency is doubled while the dipole moment amplitude is held constant, what happens to the radiated power?

• A. The power doubles, since power is proportional to frequency
• B. The power quadruples, since power is proportional to frequency squared
• C. The power increases by a factor of 16, since dipole radiated power scales as ω⁴
• D. The power is unchanged, since the dipole moment amplitude is the same

Answer: (C)

The electric dipole radiated power is P = p̈²/(6πε₀c³). For a dipole moment oscillating as p(t) = p₀cos(ωt), we have p̈ = −ω²p₀cos(ωt), so p̈² ∝ ω⁴. Doubling the frequency (ω → 2ω) increases the power by 2⁴ = 16. This strong frequency dependence is why higher-frequency radiation is so intense, and why efficient generation at low frequencies requires large antenna structures or high dipole moments to compensate.

Question 3

For a source much smaller than a wavelength (ka << 1), the multipole expansion converges rapidly because higher multipole contributions are suppressed by successive powers of (ka)².

• T. True
• F. False

Answer: True

This is the physical foundation of the multipole approach. When the source size a is small compared to the wavelength λ = 2π/k, the parameter ka = 2πa/λ << 1. Each successive multipole term carries an extra factor of (ka)², making the series converge rapidly. The electric dipole term then provides an excellent approximation by itself, without needing to track the detailed internal structure of the source.

Question 4

A system of two equal and opposite charges oscillating symmetrically about the origin has no net electric dipole moment, so it can seldom radiate electromagnetic energy.

• T. True
• F. False

Answer: False

A vanishing dipole moment eliminates *dipole* radiation, but not all radiation. Such a system would still radiate through electric quadrupole (and possibly magnetic dipole) contributions, though suppressed by a factor (ka)² relative to dipole. Only a system with zero for *all* multipole moments would be completely non-radiating — which does not occur for oscillating charge distributions. The statement reflects the misconception that dipole radiation exhausts all radiation mechanisms.

Question 5

Why do gravitational waves from merging black holes radiate only through the quadrupole moment, and what does this tell you about the fundamental constraints on gravitational radiation?

Model answer: Monopole gravitational radiation is forbidden by energy conservation — a changing mass monopole would imply changing total mass, which is forbidden. Dipole gravitational radiation is forbidden by momentum conservation — a changing gravitational dipole moment (Σmᵢrᵢ = M·R_cm) would require a net external force on the system. With monopole and dipole both eliminated by conservation laws, the leading gravitational radiation is quadrupole, suppressed by (v/c)² relative to electromagnetic dipole radiation of comparable sources. This is why gravitational waves are extraordinarily weak and required kilometer-scale interferometers to detect.

The conservation law argument is fundamental: each suppressed multipole order corresponds to a conserved quantity (mass, momentum) that forbids lower-order radiation. This hierarchy — monopole and dipole forbidden, quadrupole the first allowed term — explains the extreme weakness of gravitational waves even from spectacular astrophysical events.