Why Does a Moving Charge's Magnetic Field Follow an Inverse Square Law?

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SUMMARY

The magnetic field of a moving charge is defined by the equation B = (μ₀ / 4π) * (q * v × ȓ) / r², illustrating its adherence to an inverse square law. However, localized current distributions, including moving charges, exhibit a dipole moment that results in an inverse cube law at significant distances. The discussion highlights that a moving charge, represented as J = q * v * δ³(x), has a zero dipole moment, necessitating a reevaluation of the applicable formulas for non-stationary current distributions. This distinction is crucial for understanding the behavior of magnetic fields generated by moving charges.

PREREQUISITES
  • Understanding of electromagnetic theory, specifically the Biot-Savart Law
  • Familiarity with vector calculus and cross products
  • Knowledge of dipole moments and their implications in electromagnetism
  • Basic principles of current distributions and their classifications
NEXT STEPS
  • Study the Biot-Savart Law in detail to understand magnetic field generation
  • Explore the concept of dipole moments and their mathematical representation
  • Investigate the implications of non-stationary current distributions in electromagnetic theory
  • Learn about the differences between stationary and non-stationary current distributions
USEFUL FOR

Physicists, electrical engineers, and students of electromagnetism seeking to deepen their understanding of magnetic fields produced by moving charges and the underlying principles of current distributions.

Ahmes
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The magnetic field of a moving charge is:
\boldsymbol{B} = \frac{\mu_0}{4\pi} \frac{q \boldsymbol{v}\times \boldsymbol{\hat{r}}}{r^2}
This is an inverse square law.

But also we know that every localized current distribution (and a moving particle is most obviously a localized current distribution) appears from very far away as a dipole moment - which field is an inverse cube law.

Also using \boldsymbol{m} = \iiint \boldsymbol{x} \times \boldsymbol{J}(\boldsymbol{x}) d^3 x it appears a moving charge, \boldsymbol{J}=q \boldsymbol{v} \delta^3 (x) has a zero dipole moment.

So how could this be explained?
Thank you.
 
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A moving charge is a non stationary current distribution, so the last two formulae are no longer valid to describe its magnetic field.
 
When Jackson develops these formulae he doesn't demand the current distribution to be stationary, although I can see why it is not.
OK, thank you.
 

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