[Paper Review] Expressing the power radiated by electric charged systems

C. Vrejoiu, D. Nicmorus|ArXiv.org|Jul 23, 2003

Electromagnetic Compatibility and Measurements1 references3 citations

TL;DR

This paper presents a systematic derivation of the power radiated by localized electric charge and current distributions using the retarded vector potential and multipole expansion. It introduces a simplified formula for radiated power based on the dominant 1/r terms in the vector potential, leading to expressions for radiation fields and power distribution in terms of symmetric, traceless multipole tensors. The key contribution is a unified framework for computing radiation from arbitrary charge-current systems via multipole decomposition with traceless tensor reduction.

ABSTRACT

After a systematic introduction of some formulae for the energy radiated by localized electric charges and currents, one considers the multipole radiation and the reduction of the multipole tensors to the symmetric traceless ones.

Motivation & Objective

• To derive a simplified, exact formula for the power radiated by localized charge and current distributions without computing full electromagnetic fields.
• To establish a systematic method for expressing radiation fields in terms of the retarded vector potential and its time derivatives.
• To reduce general multipole tensors to symmetric, traceless forms to simplify radiation pattern calculations.
• To provide a consistent framework for computing radiation from arbitrary charge-current systems using multipole decomposition.
• To validate the method through application to point charges and higher-order multipole systems, including relativistic corrections.

Proposed method

• Uses the retarded vector potential A(r,t) and retains only dominant 1/r terms to define the radiated vector potential fArad.
• Applies the wave region approximation (r >> λ) to isolate radiation fields from near-field terms.
• Derives the radiation electric and magnetic fields as fErad = ν × (ν × ∂fArad/∂t) and fBrad = (1/c) ∂fArad/∂t × ν.
• Expresses the Poynting vector and radiated power in terms of the time derivative of fArad, leading to dP/dΩ = (1/μ₀c) |ν × ∂fArad/∂t|².
• Performs multipole expansion of the current density and reduces general multipole tensors to symmetric, traceless forms for simplification.
• Applies the method to point charges and higher-order multipoles, deriving radiation patterns with relativistic corrections.

Experimental results

Research questions

• RQ1 How can the radiated power from a localized charge-current system be computed without solving for full electromagnetic fields?
• RQ2 What is the correct form of the radiation field in terms of the retarded vector potential for arbitrary current distributions?
• RQ3 How can general multipole tensors be reduced to symmetric, traceless forms to simplify radiation pattern calculations?
• RQ4 What are the relativistic corrections to the radiation power for higher-order multipoles?
• RQ5 How does the method compare with standard Liénard-Wiechert potential results for point charges?

Key findings

• The radiated power per solid angle is given by dP/dΩ = (1/μ₀c) |ν × ∂fArad/∂t|², where fArad is the dominant 1/r term of the retarded vector potential.
• The radiation fields fErad and fBrad satisfy the relations fErad = c fBrad × ν and fBrad = (1/c) ν × fErad, confirming a transverse, plane-wave-like structure.
• The electric and magnetic radiation fields are derived from the time derivative of the vector potential, with the leading-order terms being O(1/r), ensuring energy flux is carried by radiation.
• Multipole tensors such as Nijkl and Πijkl are systematically reduced to symmetric, traceless forms to eliminate spurious contributions and ensure gauge invariance.
• For point charges, the method reproduces the standard Larmor formula and its relativistic generalization, confirming consistency with known results.
• The framework allows for systematic inclusion of higher-order multipole contributions (up to quadrupole and octupole) with explicit relativistic corrections in the power formula.