Moving-Charge Fields

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

B = \frac{μ _{0}}{4 π} q \frac{v \times r ^}{r ^{2}} = \frac{μ _{0}}{4 π} q \frac{v \times r}{r ^{3}} .

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

B = \frac{μ _{0}}{4 π} \frac{∣ q ∣ v sin θ}{r ^{2}},

vanishes on the velocity axis and is largest in the plane perpendicular to $v$ . 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 $B$ normal to the $v$ - $r$ plane.

Field of a moving positive charge. The velocity

v

and the displacement

r

to the field point span a plane;

B

points out of that plane along

v \times \overset{r}{^}

, largest broadside to

v

and vanishing along it. A negative charge reverses

B

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 $y \overset{}{^}$ . Therefore

v \times r = v \overset{}{^} \times y \overset{}{^} = v y \hat{k}, B = \frac{μ _{0}}{4 π} \frac{q v}{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.

Transverse field of a rightward-moving charge. Directly above the trajectory

B

points out of the page; directly below, into it. On the velocity axis ahead of the charge

v ∥ r

, so

B = 0

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_{⊥} = v sin θ, B = \frac{μ _{0}}{4 π} \frac{∣ q ∣ v _{⊥}}{r ^{2}} .

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

The expression assumes $v ≪ c$ 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,

v = v_{∥} \overset{r}{^} + v_{⊥}, v \times \overset{r}{^} = v_{⊥} \times \overset{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 $sin θ$ .

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

Source velocity resolved at the field point. The radial part

v_{r}

(along

r

, written

v_{∥}

in the text) drops out of

v \times \overset{r}{^}

; only the transverse part

v_{t} = v sin θ

sets the field magnitude.

Measurement uncertainty grows when $θ$ approaches zero because a small angular error produces a large fractional error in $sin θ$ . The transverse arrangement is therefore preferable: at $θ = 9 0^{\circ}$ , the first-order angular sensitivity of $sin θ$ 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 $q v \times r$ , not by a scalar magnitude rule. With source velocity along $+ x$ , an observation point above the path has displacement along $+ y$ ; $\overset{}{^} \times \overset{}{^} = \hat{k}$ , 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.

B (0, y, 0) = \frac{μ _{0} q v}{4 π 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 $r_{s} (t)$ and field point $r_{P}$ , use $R = r_{P} - r_{s}$ . The magnetostatic nonrelativistic term is then proportional to $q v \times R / R^{3}$ . 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, $a \times b = - b \times a$ .

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 $d ℓ$ in the direction of conventional current, and its contribution has the same geometric structure $d ℓ \times \hat{R}$ . 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 $sin θ$ . 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 $sin θ$ 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.

The directed source-to-field displacement.

R = r_{P} - r_{s}

runs from the instantaneous source position to the field point; the field uses

v \times R

, so reversing

R

reverses the inferred direction.

Translation of the coordinate origin changes $r_{s}$ and $r_{P}$ by the same vector but leaves $R = r_{P} - r_{s}$ 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 $∣ θ ∣$ 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, θ) \propto \frac{sin θ}{R ^{2}}, \frac{B ( 2 R , θ )}{B ( R , θ )} = \frac{1}{4}, \frac{B ( R , π /2 )}{B ( R , θ )} = \frac{1}{sin θ} .

The ratios assume the same point-source model, the same source speed, and a nonzero reference angle. A measurement that changes both $R$ and $θ$ 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	$v sin θ = 0$	zero low-speed magnetic term
intermediate angle	$0 < v sin θ < v$	intermediate magnitude
transverse plane	$v sin θ = v$	maximum magnitude
opposite transverse point	same magnitude	opposite field direction

A right-hand rule is shorthand for the ordered product $q (v \times \overset{r}{^})$ : 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:

B_{tot} = k \sum B_{k}, B_{k} \propto q_{k} \frac{v _{k} \times 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 $B$ , and applying both restores the original direction.

Charge sign reverses

B

. Equal positive and negative sources share velocity and field point

P

; since

B \propto q v \times \overset{r}{^}

, the fields are equal in magnitude and opposite in direction — out of the page for

q > 0

, into it for

q < 0

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.

Superposition at one field point. Two moving charges give magnetic vectors

B_{1}

and

B_{2}

P

; the resultant

B_{sum}

is their vector sum, added component by component, not a sum of magnitudes.

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.

Inverse-square falloff at fixed speed and transverse angle. Doubling the distance from

r

2 r

quarters

B

: the marked ordinates stand in ratio

4 : 1

, tracing the point-source

1/ r^{2}

law.

Vector form and error propagation

Work with the vector expression before reducing to a magnitude. With $R = r_{P} - r_{s}$ ,

B = \frac{μ _{0}}{4 π} \frac{q v \times R}{R ^{3}}, B = \frac{μ _{0}}{4 π} \frac{∣ q ∣ v _{⊥}}{R ^{2}}, v_{⊥} = v sin θ .

The ordered cross product keeps the direction and makes the axial limit automatic: $v_{⊥} = 0$ 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

(\frac{σ _{B}}{B})^{2} ≃ (\frac{σ _{q}}{q})^{2} + (\frac{σ _{v}}{v})^{2} + (2 \frac{σ _{R}}{R})^{2} + (cot θ σ_{θ})^{2} .

The distance term is doubled by the inverse square. As $θ \to 0$ the $cot θ$ factor diverges while the field itself vanishes, so near the axis report Cartesian components rather than a relative error on a near-zero magnitude.

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:

B (r_{P}, t) = \frac{μ _{0}}{4 π} k \sum q_{k} \frac{v _{k} ( t ) \times [ r _{P} - r _{k} ( t ) ]}{∣ r _{P} - r _{k} ( t ) ∣ ^{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

B (r_{P}) = \frac{μ _{0}}{4 π} \int \frac{J ( r ^{'} ) \times ( r _{P} - r ^{'} )}{∣ r _{P} - r ^{'} ∣ ^{3}} d^{3} r^{'} .

The transverse current density of a thin wire carrying steady current reduces to the familiar current-element integral $\frac{μ _{0}}{4 π} \int I d ℓ^{'} \times R / R^{3}$ . 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.