Related Experiment Video
Updated: Jun 19, 2026
08:44
Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene
Published on: August 22, 2017
Free-space asymptotic far-field series
Riccardo Borghi1, Miguel A Alonso
1Dipartimento di Elettronica Applicata, Università Roma Tre, I-00146 Rome, Italy. borghi@uniroma3.it
Summary
A new method derives complete asymptotic series for electromagnetic fields in free space. This technique uses a differential operator on the angular spectrum for efficient computation.
Area of Science:
- Electromagnetism
- Mathematical Physics
- Wave Propagation
Background:
- Asymptotic series are crucial for approximating field behavior at large distances.
- Deriving these series for complex fields, especially in 2D and 3D scalar/vectorial cases, presents significant challenges.
- Existing methods may lack completeness or computational efficiency.
Purpose of the Study:
- To develop a systematic procedure for deriving the complete asymptotic series of free-space electromagnetic fields.
- To express each term of the series in a closed form.
- To provide a practical computational tool for analyzing field behavior.
Main Methods:
- A systematic procedure is established for deriving asymptotic series in inverse powers of distance.
- A differential operator acting on the angular spectrum is utilized to express each series term in closed form.
- A simple recursive routine is developed for efficient computation of the necessary derivatives.
Main Results:
- The complete asymptotic series for 2D and 3D, scalar and vectorial, monochromatic electromagnetic fields in free space are derived.
- Each term of the series is expressed in a closed form.
- A recursive routine for computing derivatives is provided, simplifying practical application.
Conclusions:
- The presented systematic procedure offers a powerful and efficient method for analyzing electromagnetic fields at large distances.
- The closed-form expressions and recursive routine facilitate the computation and application of asymptotic series.
- This work provides a valuable tool for researchers in electromagnetism and related fields.
Related Concept Videos
01:21Convergence of Fourier Series
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
01:24Exponential Fourier series
In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
Euler's identity...
01:28Alternating Series and Absolute Convergence
A mass attached to a vertical spring can exhibit oscillatory motion as it moves above and below a central equilibrium point. In an ideal spring, the oscillations would continue indefinitely with constant amplitude. In a damped spring, however, resistive forces such as air resistance or internal friction gradually reduce the size of each swing. This behavior is often modeled by combining a sinusoidal function, which represents the repeated motion, with an exponential decay factor, which reduces...
01:26Gauss's Law: Spherical Symmetry
A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...
01:30Binomial Series
The binomial series extends the familiar binomial theorem from finite polynomial expansions to infinite series expansions. This distinction is important: the binomial theorem applies to positive integer exponents, while the binomial series applies more broadly, including fractional and negative exponents. It is obtained from the Maclaurin series of (1 + x)m, where m is any real exponent, and the expansion converges for |x| < 1.The familiar binomial theorem...
01:10Gauss's Law: Problem-Solving
Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...