AcademySources of Magnetic Fields
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Fields from Moving Charges

Level 1 - Physics topic page in Sources of Magnetic Fields.

Principle

A moving charge produces a magnetic field. The field circles around the direction of motion and is strongest to the side of the moving charge.

Notation

\(q\)
source charge
\(\mathrm{C}\)
\(\vec v\)
velocity of the source charge
\(\mathrm{m\,s^{-1}}\)
\(\vec r\)
vector from the charge to the field point
\(\mathrm{m}\)
\(\hat r\)
unit vector from the charge to the field point
\(\theta\)
angle between \(\vec v\) and \(\vec r\)
\(\mathrm{rad}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)

Method

Derivation 1: Use the moving-charge field law

For a slowly moving point charge, the magnetic field at a point displaced by \(\vec r\) from the charge is proportional to charge, speed, and the sine of the angle to the line of sight.

Vector field
\[\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times \hat r}{r^2}\]
Magnitude
\[B=\frac{\mu_0}{4\pi}\frac{|q|v\sin\theta}{r^2}\]

Derivation 2: Determine the direction

For a positive charge, point the fingers of your right hand along \(\vec v\) and curl toward \(\hat r\). Your thumb gives \(\vec v\times\hat r\), which is the field direction. A negative source charge reverses the direction.

Derivation 3: Recognize zero-field directions

If the field point lies directly ahead of or behind the moving charge, then \(\theta=0\) or \(\theta=\\pi\). The cross product is zero, so the magnetic field from this motion is zero at that instant.

Rules

Moving charge field
\[\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times\hat r}{r^2}\]
Magnitude
\[B=\frac{\mu_0}{4\pi}\frac{|q|v\sin\theta}{r^2}\]
Field vanishes on the velocity line
\[\theta=0\text{ or }\pi\Rightarrow B=0\]

Examples

Question
A positive charge moves to the right. What is the magnetic field direction at a point directly above the charge?
Answer
Use
\[\vec v\times\hat r\]
Right crossed with up points out of the page, so
\[\vec B\]
is out of the page.

Checks

  • The field is magnetic only because the charge is moving.
  • The field is perpendicular to both \(\vec v\) and \(\hat r\).
  • A negative source charge reverses the right-hand-rule direction.
  • The equation is the low-speed point-charge result, not a full relativistic field model.

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