---
title: Moving-Charge Fields
module: Magnetic Sources
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  Every magnetic field comes from charge in motion, and the simplest source is a
  single point charge drifting past. We work out the field it produces — normal to
  both the velocity and the line of sight, falling off as the inverse square — and
  read off why it vanishes straight ahead of the charge and peaks broadside. Summing
  many such charges is the bridge to steady currents, valid while speeds stay far
  below $c$ and the motion changes little during the time its field takes to
  propagate outward.
topics: [Magnetic Sources]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 27 — Sources of the Magnetic Field; §27-1 The Magnetic Field of Moving Point Charges"
---

A moving point charge produces a magnetic field. In the nonrelativistic
magnetostatic approximation, a charge $q$ moving with velocity $\vec v$ produces
at field point displacement $\vec r$ from the instantaneous source position

$$
\vec B=\frac{\mu_0}{4\pi}q\frac{\vec v\times\hat r}{r^2}
=\frac{\mu_0}{4\pi}q\frac{\vec v\times\vec r}{r^3}.
$$

The field is perpendicular to both source velocity and displacement. Its magnitude,

$$
B=\frac{\mu_0}{4\pi}\frac{|q|v\sin\theta}{r^2},
$$

vanishes on the velocity axis and is largest in the plane perpendicular to
$\vec v$. The sign of $q$ reverses the field direction. The formula describes a
source whose motion changes slowly enough that radiation and finite propagation-time
effects can be neglected.

A positive charge produces $\vec B$ normal to the $\vec v$-$\vec r$ plane.

$$
% caption: Field of a moving positive charge. The velocity $\vec v$ and the displacement $\vec r$ to the field point span a plane; $\vec B$ points out of that plane along $\vec v\times\hat r$, largest broadside to $\vec v$ and vanishing along it. A negative charge reverses $\vec B$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=1.15]
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$$

Field strength follows the same inverse-square distance scaling as Coulomb's law,
but its angular factor distinguishes magnetic sources from static electric charge.
At a fixed transverse distance, doubling source speed doubles $B$. A stationary
charge has no magnetic field in this approximation, although it may still have an
electric field.

A charge moving along $+x$ and observed at $(0,y,0)$ has displacement
$y\hat\jmath$. Therefore

$$
\vec v\times\vec r
=v\hat\imath\times y\hat\jmath
=vy\hat k,
\qquad
B=\frac{\mu_0}{4\pi}\frac{qv}{y^2}.
$$

The result points along $+z$ for $q>0$, $y>0$, and reverses below the path or for a
negative charge. Component calculation is preferable to a memorized right-hand rule
when signs and axes are specified explicitly.

> **Worked example (Proton and electron at the same point).** A proton and an electron
> each move at $v=1.0\times10^7\ \mathrm{m/s}$ along $+x$. Find $\vec B$ each produces
> at $y=2.0\ \mathrm{mm}$ directly above the path. The geometry is transverse
> $(\sin\theta=1)$, so
>
> $$
> B=\frac{\mu_0}{4\pi}\frac{|q|v}{y^2}
> =(10^{-7})\frac{(1.602\times10^{-19})(1.0\times10^7)}{(2.0\times10^{-3})^2}
> =4.0\times10^{-14}\ \mathrm T.
> $$
>
> Both give the same $40\ \mathrm{fT}$ magnitude, but the directions oppose: for the
> proton $\vec B=+B\,\hat k$ (out of the page), and reversing the charge sign flips the
> electron's to $-B\,\hat k$ (into the page). Speed and geometry set the magnitude;
> the sign of $q$ sets the direction.

$$
% caption: Transverse field of a rightward-moving charge. Directly above the trajectory $\vec B$ points out of the page; directly below, into it. On the velocity axis ahead of the charge $\vec v\parallel\vec r$, so $\vec B=0$.
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\filldraw[black] (1.9,0) circle (.05);
\node[below] at (1.9,-.12) {$B=0$};
\end{tikzpicture}
$$

## Observation angle and validity

The angular factor can be separated from distance dependence by resolving source
velocity into a component transverse to the observation displacement:

$$
v_\perp=v\sin\theta,
\qquad B=\frac{\mu_0}{4\pi}\frac{|q|v_\perp}{r^2}.
$$

An axial observation point has $v_\perp=0$ and therefore no magnetic field in the
low-speed expression. At the same distance in the transverse plane, $v_\perp=v$ and
field magnitude is maximal. The comparison distinguishes an angular zero from a
distance-law zero.

The expression assumes $v\ll c$ and a source velocity that remains nearly constant
over the electromagnetic propagation time to the observation point. Accelerated
charges require retarded fields and radiation terms. These corrections change the
complete field calculation but not the cross-product direction of the low-speed
near-field term.

## Transverse-speed measurement geometry

Only the transverse component of source motion contributes to the magnetic field.
At the observation point, decompose velocity into radial and transverse parts,

$$
\vec v=v_\parallel\hat r+\vec v_\perp,
\qquad
\vec v\times\hat r=\vec v_\perp\times\hat r.
$$

The radial term is absent from the cross product. A source moving directly toward a
sensor can have substantial speed while producing zero magnetic field at that sensor
in the nonrelativistic formula. A transverse sensor geometry is therefore required
when magnetic field is used to infer source speed. The known distance, source charge,
and angle must all be part of the measurement model; $B$ alone does not determine
$v$ unless the observation geometry has fixed $\sin\theta$.

Resolve the source velocity into radial and transverse components at the observation
point.

$$
% caption: Source velocity resolved at the field point. The radial part $v_r$ (along $\vec r$, written $v_\parallel$ in the text) drops out of $\vec v\times\hat r$; only the transverse part $v_t=v\sin\theta$ sets the field magnitude.
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$$

> **Worked example (Field of a moving charge at an angle).** A proton
> $(q=+e)$ moves at $v=3.0\times10^6\ \mathrm{m/s}$. Find the magnetic field it
> produces at a point $r=0.040\ \mathrm m$ away, on a line making $\theta=30^\circ$
> with the velocity. Only the transverse component $v\sin\theta$ enters the
> magnitude:
>
> $$
> B=\frac{\mu_0}{4\pi}\frac{|q|v\sin\theta}{r^2}
> =(10^{-7}\ \tfrac{\mathrm{T\cdot m}}{\mathrm A})
> \frac{(1.602\times10^{-19}\ \mathrm C)(3.0\times10^6\ \tfrac{\mathrm m}{\mathrm s})(0.500)}
> {(0.040\ \mathrm m)^2}
> =1.5\times10^{-17}\ \mathrm T.
> $$
>
> Directly ahead of the proton $\theta\to0$ and the field vanishes; broadside
> $(\theta=90^\circ)$ it is largest, $3.0\times10^{-17}\ \mathrm T$. The angle
> alone moves the field between these limits at fixed $r$.

Measurement uncertainty grows when $\theta$ approaches zero because a small angular
error produces a large fractional error in $\sin\theta$. The transverse arrangement
is therefore preferable: at $\theta=90^\circ$, the first-order angular sensitivity
of $\sin\theta$ vanishes and the field is largest for the same source speed and
distance.

## Direction from the vector cross product

The direction of a moving charge's magnetic field is determined algebraically by
$q\vec v\times\vec r$, not by a scalar magnitude rule. With source velocity
along $+x$, an observation point above the path has displacement along $+y$;
$\hat\imath\times\hat\jmath=\hat k$, so a positive source
produces field out of the page. A point below the path has displacement along $-y$
and the field reverses. Replacing the positive source with an electron reverses both
directions again. This component rule remains reliable when a sketch is rotated or
when several axes have nonstandard orientations.

$$
\vec B(0,y,0)=\frac{\mu_0qv}{4\pi y^2}\sgn(y)\hat k
$$

for a positive charge moving along $+x$. The factor $\sgn(y)$ reverses the field
across the path. On the path axis, this idealized expression has zero magnitude and
therefore no magnetic-field direction, except at the singular source position.

An on-axis sensor and a side-axis sensor at the same source distance answer different
questions. The axial sensor measures no magnetic field from this ideal motion, while
the transverse sensor measures the maximum available magnitude. The comparison is
geometric rather than a change in source charge or speed.

## Source point, field point, and signed geometry

The moving-charge expression has three distinct geometric objects: the source
position, the field point, and the displacement from source to field point. The
displacement vector is not the particle's velocity and it is not an arbitrary radius
drawn from a coordinate origin. With source at $\vec r_s(t)$ and field point
$\vec r_P$, use $\vec R=\vec r_P-\vec r_s$. The magnetostatic
nonrelativistic term is then proportional to $q\vec v\times\vec R/R^3$.
Writing the source-to-field displacement explicitly prevents a common reversal in
which the cross product is formed with a vector pointing from the field point back to
the source.

The sign of charge belongs outside the cross product. A positive source moving along
$+x$ gives a field along $+z$ at a point with positive $y$ displacement. An electron
with the same velocity and observation point gives field along $-z$. Reversing source
velocity also reverses the field. Reversing both charge and velocity leaves the field
direction unchanged. These sign transformations can be checked without a diagram by
using the antisymmetry of the cross product,
$\vec a\times\vec b=-\vec b\times\vec a$.

The field point is held fixed while the source velocity and displacement are evaluated
at the source. This local construction is the point-particle precursor to the
Biot--Savart procedure for a current element. A current element has a directed length
$\d\vec\ell$ in the direction of conventional current, and its contribution
has the same geometric structure $\d\vec\ell\times\hat R$. The point
charge formula should not be converted into a complete-wire result by replacing $q$
with a total wire charge: a steady wire is ordinarily neutral, and its magnetic field
arises from the motion of its charge carriers distributed along the path.

Observation geometry controls which source motion is visible. At fixed distance,
rotating the field point around the velocity axis changes only $\sin\theta$. An
angular sweep therefore measures a sinusoidal magnetic-field envelope, with zeros on
the forward and backward axes. At fixed angle, increasing distance changes the field
as $R^{-2}$. These two dependencies should be varied separately in an experiment;
otherwise an angular change can be incorrectly attributed to an inverse-square
distance change.

The nonrelativistic formula is a near-field, slowly varying-source approximation. It
uses the source's instantaneous velocity as a practical approximation only when
$v/c$ is small and the source changes little during the propagation time $R/c$.
Rapidly accelerated or relativistic sources have electromagnetic fields that depend
on retarded source data and contain radiation contributions. The field remains a
vector field satisfying Maxwell's equations, but the elementary expression in this
lesson is no longer sufficient for quantitative prediction.

The equal-radius comparison removes the inverse-square distance factor. The axial
sensor and side sensor differ only in the transverse projection of the same source
velocity. A sensor arrangement that instead changes both angle and distance cannot
separate the $\sin\theta$ dependence from the $R^{-2}$ dependence without an
additional model or calibration.

The source-to-field displacement must point from the moving charge toward the
observation point. The diagram below labels this directed vector explicitly.

$$
% caption: The directed source-to-field displacement. $\vec R=\vec r_P-\vec r_s$ runs from the instantaneous source position to the field point; the field uses $\vec v\times\vec R$, so reversing $\vec R$ reverses the inferred direction.
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\draw[->,black,thick] (.15,.09)--(2.95,1.41);
\node[black,above left] at (1.45,.9) {$\mathbf R$};
\end{tikzpicture}
$$

Translation of the coordinate origin changes $\vec r_s$ and $\vec r_P$ by the
same vector but leaves $\vec R=\vec r_P-\vec r_s$ unchanged. The field
depends on this relative geometry rather than on an arbitrary coordinate origin.

Field direction can be checked by holding source velocity fixed and comparing points
on opposite sides of the trajectory. The displacement changes sign in the transverse
component, so the cross product and the magnetic field reverse. This comparison is
independent of field magnitude and exposes a sign error immediately.

The opposite-side figure holds speed, distance, and $\lvert\theta\rvert$ fixed.
Only the transverse displacement changes sign. The two field magnitudes are therefore
equal; the field directions are opposite. Reversing the source charge reverses both
field directions again without changing either magnitude.

At fixed source charge magnitude and speed, the angular and distance tests are

$$
B(R,\theta)\propto\frac{\sin\theta}{R^2},
\qquad
\frac{B(2R,\theta)}{B(R,\theta)}=\frac14,
\qquad
\frac{B(R,\pi/2)}{B(R,\theta)}=\frac{1}{\sin\theta}.
$$

The ratios assume the same point-source model, the same source speed, and a
nonzero reference angle. A measurement that changes both $R$ and $\theta$ cannot
assign a discrepancy to one factor without an additional geometric model.

| observation geometry at fixed $R$ | transverse velocity factor | magnetic result |
|---|---:|---|
| forward or rear velocity axis | $v\sin\theta=0$ | zero low-speed magnetic term |
| intermediate angle | $0<v\sin\theta<v$ | intermediate magnitude |
| transverse plane | $v\sin\theta=v$ | maximum magnitude |
| opposite transverse point | same magnitude | opposite field direction |

A right-hand rule is shorthand for the ordered product $q(\vec v\times\hat r)$: form
the cross product of the velocity with the source-to-observer unit vector, then let
the sign of $q$ fix the direction. A component calculation is safer when the axes or
the charge sign are unusual, because interchanging the two vectors flips the sign.

For several moving point charges at one observation point, keep the source index
until after the vector sum:

$$
\vec B_{\mathrm{tot}}=\sum_k\vec B_k,
\qquad
\vec B_k\propto
q_k\,\frac{\vec v_k\times\hat R_k}{R_k^2}.
$$

Each $\hat R_k$ points from source $k$ to the common observation point.
Add Cartesian components, not magnitudes. A reflected pair can cancel one component
on a symmetry plane while reinforcing another; changing one charge sign or one source
velocity reverses that contribution before the sum is taken.

Two charges with equal speeds and equal distances from the probe can still cancel,
because their source-to-observer vectors or charge signs differ. A symmetry plane can
therefore carry a small resultant even when each contribution is large. Close to a
compact cloud, the separate carriers resolve; far away, only the combined distribution
survives in the leading field.

Reversing the source charge and reflecting the observer across the velocity axis are
independent sign flips: either one reverses $\vec B$, and applying both restores the
original direction.

$$
% caption: Charge sign reverses $\vec B$. Equal positive and negative sources share velocity and field point $P$; since $\vec B\propto q\,\vec v\times\hat r$, the fields are equal in magnitude and opposite in direction — out of the page for $q>0$, into it for $q<0$.
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\node[right,acc] at (2.23,1.15) {B in};
\end{tikzpicture}
$$

The result is a field in tesla, not a force. What a later test particle feels comes
from the Lorentz force, which needs that particle's own charge and velocity; the
source charge and any test charge stay distinct. Several specified charges combine at
one point by adding their signed field vectors component by component — equal
magnitudes cancel only when the vectors oppose.

$$
% caption: Superposition at one field point. Two moving charges give magnetic vectors $\vec B_1$ and $\vec B_2$ at $P$; the resultant $\vec B_{\text{sum}}$ is their vector sum, added component by component, not a sum of magnitudes.
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\node[above] at (0,1.75) {$P$};
\draw[->,black,thick] (0,1.75)--(-.78,1.05) node[left] {$B_1$};
\draw[->,black,thick] (0,1.75)--(.78,1.05) node[right] {$B_2$};
\draw[->,acc,very thick] (0,1.75)--(0,.72) node[right] {$B_{\rm sum}$};
\end{tikzpicture}
$$

At fixed speed and angle the field falls as $R^{-2}$: doubling the distance quarters
it, tripling it cuts it to a ninth. Far from a compact cloud of moving charge — at
distances large compared with the cloud's size — the leading field is that of a point
source; nearer in, each carrier's geometry must be summed. That sum, in the continuum
limit, is the Biot--Savart integral.

$$
% caption: Inverse-square falloff at fixed speed and transverse angle. Doubling the distance from $r$ to $2r$ quarters $\vec B$: the marked ordinates stand in ratio $4:1$, tracing the point-source $1/r^2$ law.
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\draw[->,black] (0,0)--(0,2.7) node[above] {$B$};
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\node[below] at (2,0) {$2r$};
\node[acc] at (2.7,1.2) {$\tfrac{1}{r^2}$};
\end{tikzpicture}
$$

### Vector form and error propagation

Work with the vector expression before reducing to a magnitude. With
$\vec R=\vec r_P-\vec r_s$,

$$
\vec B=\frac{\mu_0}{4\pi}\frac{q\,\vec v\times\vec R}{R^3},
\qquad
B=\frac{\mu_0}{4\pi}\frac{|q|v_\perp}{R^2},
\qquad
v_\perp=v\sin\theta.
$$

The ordered cross product keeps the direction and makes the axial limit automatic:
$v_\perp=0$ sends the low-speed term to zero with no separate rule. Away from the
axis, independent uncertainties in charge, speed, distance, and angle combine as

$$
\left(\frac{\sigma_B}{B}\right)^2\simeq
\left(\frac{\sigma_q}{q}\right)^2+
\left(\frac{\sigma_v}{v}\right)^2+
\left(2\frac{\sigma_R}{R}\right)^2+
\left(\cot\theta\,\sigma_\theta\right)^2.
$$

The distance term is doubled by the inverse square. As $\theta\to0$ the $\cot\theta$
factor diverges while the field itself vanishes, so near the axis report Cartesian
components rather than a relative error on a near-zero magnitude.

> **Worked example (Field of a single carrier, in components).** Take $q=+e$,
> $\vec v=(2.00\times10^6,0,0)\ \mathrm{m/s}$, and a field point $5.00\ \mathrm{cm}$
> above the path, $\vec R=(0,5.00\times10^{-2},0)\ \mathrm m$. Here
> $\vec v\times\vec R=(0,0,vR)$, so
>
> $$
> \vec B=\frac{\mu_0}{4\pi}\,q\frac{\vec v\times\vec R}{R^3}
> =(10^{-7})\frac{(1.602\times10^{-19})(2.00\times10^6)}{(5.00\times10^{-2})^2}\,\hat k
> =(0,0,\,1.28\times10^{-17})\ \mathrm T.
> $$
>
> Flipping the field point to $\vec R=(0,-5.00\times10^{-2},0)$ reverses the $z$
> sign; moving it onto the axis, $\vec R=(5.00\times10^{-2},0,0)$, sends the cross
> product and the field to zero. One elementary charge yields
> $\sim10^{-17}\ \mathrm T$: measurable magnetism comes from the combined motion of
> enormous numbers of carriers, which is why a steady current, not a lone charge, is
> the practical source.

## Source-model and reference-frame limits

The low-speed point-charge expression assumes that the source speed changes slowly
enough that retardation and radiation can be neglected over the observation region.
It also treats the observation geometry at one stated time. A charge with appreciable
acceleration produces radiation fields with a different distance dependence and
requires a retarded-time description. The inverse-square magnetic term remains a
near-source approximation only within its declared speed, acceleration, and
distance regime.

Reference-frame language must also be stated carefully. Magnetic and electric fields
transform together between frames. A charge at rest in one frame has no magnetic
field from its own translational motion there, while an observer moving relative to
that charge can measure a magnetic component together with a transformed electric
field. The low-speed formula describes one chosen laboratory frame; it does not
assign a frame-independent magnetic field to a moving charge in isolation.

For several point charges, calculate each source-to-observer displacement separately:

$$
\vec B(\vec r_P,t)
=\frac{\mu_0}{4\pi}\sum_k
q_k\frac{\vec v_k(t)\times
\left[\vec r_P-\vec r_k(t)\right]}
{\lvert\vec r_P-\vec r_k(t)\rvert^3}.
$$

The vector sum must precede the magnitude. A continuous current distribution is the
limiting form of that sum; in the quasistatic regime a volume current density gives

$$
\vec B(\vec r_P)
=\frac{\mu_0}{4\pi}\int
\frac{\vec J(\vec r')\times
\left(\vec r_P-\vec r'\right)}
{\lvert\vec r_P-\vec r'\rvert^3}\,\d^3 r'.
$$

The transverse current density of a thin wire carrying steady current reduces to the
familiar current-element integral
$\frac{\mu_0}{4\pi}\int I\,\d\vec\ell'\times\vec R/R^3$. The conversion
does not permit an arbitrary point charge to
be replaced by a current: it requires a stated carrier distribution, a cross section,
and a time scale over which the current is effectively steady. That distinction
matters near contacts, in a pulsed beam, and wherever charge density changes along a
conductor. The point-charge result survives inside every current element: each
inherits the transverse-velocity factor, the inverse-square falloff, and the signed
cross-product direction before the source integral is taken.