Radiation from an Oscillating Electric Quadrupole: Power Distribution and Angular Dependence

Analyzing the Fraction of Radiated Power and Wavelength Dependence in Quadrupole Radiation

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Key Takeaways

The angular distribution of radiated power from an oscillating electric quadrupole is proportional to \(\sin^2(2\theta)\), highlighting a distinct pattern compared to dipole radiation.
At an angle \(\theta = 30^\circ\), the fraction of the maximum power radiated is precisely \(\dfrac{3}{4}\), indicating significant radiation in this direction.
The total radiated power from a quadrupole source scales with the wavelength as \(\lambda^{-6}\), reflecting the higher-order nature of quadrupole radiation.

Introduction

Radiation from oscillating charge distributions is a fundamental phenomenon in electromagnetism with wide-ranging applications in physics and engineering. While dipole radiation is commonly studied, higher-order multipole radiation, such as that from an electric quadrupole, exhibits more complex spatial and spectral characteristics. This comprehensive analysis delves into the angular power distribution and wavelength dependence of radiation from an oscillating spheroidal charge distribution acting as an electric quadrupole, with symmetry along the z-axis.

Electric Quadrupole Radiation

Understanding the Electric Quadrupole Moment

An electric quadrupole moment arises in systems where there is no net charge or dipole moment, but the charge distribution has a specific spatial arrangement that leads to a non-zero quadrupole moment. Mathematically, the quadrupole moment tensor \( Q_{ij} \) is defined as: \[ Q_{ij} = \sum_{k} q_k (3x_{ki} x_{kj} - r_k^2 \delta_{ij}) \] where \( q_k \) is the charge, \( x_{ki} \) is the \( i \)-th component of the position vector of the \( k \)-th charge, \( r_k \) is its magnitude, and \( \delta_{ij} \) is the Kronecker delta.

Radiation Mechanism in Quadrupoles

In oscillating quadrupoles, the time-varying quadrupole moment \( Q(t) \) radiates electromagnetic waves. Unlike dipole radiation, which arises from linear oscillations of charge and scales with the square of the angular frequency \( \omega^2 \), quadrupole radiation involves more intricate oscillations and scales with \( \omega^6 \).

Mathematical Expression for Radiated Power

The time-averaged power radiated by an oscillating quadrupole is given by: \[ P_{\text{quadrupole}} = \dfrac{\mu_0\, \omega^6}{20\pi c^5} \sum_{i,j} |Q_{ij}|^2 \] where \( \mu_0 \) is the permeability of free space, \( c \) is the speed of light, and \( |Q_{ij}|^2 \) represents the magnitude squared of the quadrupole moment tensor components.

Angular Distribution of Radiated Power

Deriving the Angular Dependence

In the far-field or radiation zone, the angular distribution of the radiated power per unit solid angle \( \dfrac{dP}{d\Omega} \) for an electric quadrupole is derived from the quadrupole radiation fields. The power per unit solid angle is given by: \[ \dfrac{dP}{d\Omega} = \dfrac{\mu_0\, \omega^6\, |Q|^2}{640\pi^2 c^5} \sin^2(2\theta) \] where \( |Q|^2 \) is the effective quadrupole moment magnitude, and \( \theta \) is the angle relative to the z-axis.

Comparison with Dipole Radiation

For context, the angular distribution for electric dipole radiation is: \[ \dfrac{dP_{\text{dipole}}}{d\Omega} = \dfrac{\mu_0\, \omega^4\, |p|^2}{32\pi^2 c^3} \sin^2\theta \] with \( |p| \) being the dipole moment magnitude. The quadrupole's \( \sin^2(2\theta) \) dependence indicates a different radiation pattern, exhibiting lobes and nulls distinct from those of dipole radiation.

Calculating the Fraction of Maximum Power at \(\theta = 30^\circ\)

Maximum Radiated Power

The maximum of \( \sin^2(2\theta) \) occurs when \( 2\theta = 90^\circ \), i.e., \( \theta = 45^\circ \): \[ \sin^2(2 \times 45^\circ) = \sin^2(90^\circ) = 1 \] Thus, the maximum power per unit solid angle \( \left( \dfrac{dP}{d\Omega} \right)_{\text{max}} \) occurs at \( \theta = 45^\circ \).

Fraction at \(\theta = 30^\circ\)

At \( \theta = 30^\circ \): \[ \sin^2(2 \times 30^\circ) = \sin^2(60^\circ) \] \[ \sin(60^\circ) = \dfrac{\sqrt{3}}{2} \implies \sin^2(60^\circ) = \left( \dfrac{\sqrt{3}}{2} \right)^2 = \dfrac{3}{4} \]

Therefore, the power per unit solid angle at \( \theta = 30^\circ \) is: \[ \left( \dfrac{dP}{d\Omega} \right)_{\theta=30^\circ} = \left( \dfrac{dP}{d\Omega} \right)_{\text{max}} \times \dfrac{3}{4} \]

The fraction of the maximum power \( A \) radiated at \( \theta = 30^\circ \) is thus: \[ \text{Fraction} = \dfrac{\left( \dfrac{dP}{d\Omega} \right)_{\theta=30^\circ}}{\left( \dfrac{dP}{d\Omega} \right)_{\text{max}}} = \dfrac{3}{4} \]

Interpretation of the Result

This result indicates that at \( \theta = 30^\circ \), the power radiated is \( 75\% \) of the maximum possible radiated power per unit solid angle. The quadrupolar radiation pattern thus exhibits significant radiation at this angle, with the maximum occurring at \( \theta = 45^\circ \).

Wavelength Dependence of Quadrupole Radiation

Total Radiated Power and Frequency Relation

The total radiated power from an oscillating electric quadrupole scales with the sixth power of the angular frequency \( \omega \): \[ P_{\text{quadrupole}} \propto \omega^6 \] This strong frequency dependence reflects the fact that higher-order multipole moments radiate less efficiently at lower frequencies compared to dipoles.

Expressing Frequency in Terms of Wavelength

The angular frequency \( \omega \) is related to the wavelength \( \lambda \) by: \[ \omega = \dfrac{2\pi c}{\lambda} \] Substituting this relation into the expression for radiated power: \[ P_{\text{quadrupole}} \propto \left( \dfrac{2\pi c}{\lambda} \right)^6 \] \[ P_{\text{quadrupole}} \propto \dfrac{(2\pi c)^6}{\lambda^6} \] \] Since \( (2\pi c)^6 \) is a constant, the dependence on \( \lambda \) is: \[ P_{\text{quadrupole}} \propto \lambda^{-6} \]

Functional Form of \( f(\lambda, \theta) \)

Combining the angular and wavelength dependencies, the power per unit solid angle can be expressed as: \[ \dfrac{dP}{d\Omega} = K \lambda^{-6} \sin^2(2\theta) \] where \( K \) is a constant that includes factors like the quadrupole moment magnitude and fundamental constants.

Comparison of Dipole and Quadrupole Radiation

Characteristic	Dipole Radiation	Quadrupole Radiation
Order of Multipole	First-order	Second-order
Angular Dependence	\( \sin^2\theta \)	\( \sin^2(2\theta) \)
Power Scaling with Frequency	\( \omega^4 \) (or \( \lambda^{-4} \))	\( \omega^6 \) (or \( \lambda^{-6} \))
Radiation Pattern	Doughnut-shaped	Four-lobed structure
Efficiency at Low Frequencies	Higher	Lower

Applications and Implications

Understanding quadrupole radiation is essential in fields such as nuclear physics, gravitational wave astronomy, and antenna design:

Nuclear Transitions: Quadrupole radiation plays a role in certain gamma-ray emissions from nuclei where dipole transitions are forbidden.
Gravitational Waves: Gravitational radiation from astronomical sources often involves quadrupole moments, as mass monopoles and dipoles do not radiate gravitational waves.
Antenna Design: Quadrupole antennas are used in specific communication systems where directional radiation patterns are required.

Conclusion

For an oscillating spheroidal charge distribution acting as an electric quadrupole source:

The fraction of the maximum radiated power \( A \) at an angle \( \theta = 30^\circ \) is: \[ \dfrac{3}{4} \] indicating substantial radiation in this direction.
The functional dependence of the radiated power on the wavelength \( \lambda \) is: \[ \lambda^{-6} \] reflecting the strong wavelength dependence characteristic of quadrupole radiation.

Therefore, the correct choice among the provided options is:

Option A: '\(\dfrac{3}{4}, \lambda^{-6}\)'

Further Mathematical Details

Derivation of Radiated Power

Starting from the general expression for the radiated power per unit solid angle for an oscillating quadrupole: \[ \dfrac{dP}{d\Omega} = \dfrac{\mu_0\, \omega^6\, |Q|^2}{640\pi^2 c^5} \sin^2(2\theta) \] Integrating over all solid angles gives the total power: \[ P_{\text{quadrupole}} = \int \dfrac{dP}{d\Omega} d\Omega = \dfrac{\mu_0\, \omega^6\, |Q|^2}{640\pi^2 c^5} \int_0^{2\pi} d\phi \int_0^\pi \sin^2(2\theta) \sin\theta d\theta \] \[ P_{\text{quadrupole}} = \dfrac{\mu_0\, \omega^6\, |Q|^2}{640\pi^2 c^5} (2\pi) \left( \dfrac{\pi}{2} \right) = \dfrac{\mu_0\, \omega^6\, |Q|^2}{160\pi c^5} \]

Significance of the \( \lambda^{-6} \) Dependence

The \( \lambda^{-6} \) dependence implies that as the wavelength increases, the radiated power decreases rapidly. This has practical implications:

Short-Wavelength Radiation: Quadrupole radiation is more significant at shorter wavelengths (higher frequencies), making it relevant in the context of X-rays and gamma rays.
Observational Astronomy: Detection of quadrupole radiation requires sensitive instruments capable of measuring weak signals at high frequencies.

References

For further reading on quadrupole radiation and its characteristics:

Quadrupole Radiation - Wikipedia
Electric Quadrupole Radiation - University of Texas
Radiation from Multipole Moments - Duke University
Multipole Radiation Fields - Springer