---
title: How E and B Transform
module: Covariant Electromagnetism
moduleNumber: 4
lessonNumber: 3
order: 403
summary: >
  Transforming the field tensor under a boost gives explicit rules for the
  electric and magnetic fields: components along the motion are unchanged,
  transverse components mix and pick up a gamma. The field of a uniformly moving
  charge compresses transversely, and the force between a current and a moving
  charge shows that magnetism is the relativistic shadow of electrostatics.
topics: [Covariant Electromagnetism]
draft: false
sources:
  - book: Schutz
    ref: "Ch. 4 — transformation of the electromagnetic field"
  - book: Tipler & Llewellyn
    ref: "Ch. 2 — fields between frames (conceptual); the relativistic origin of magnetism"
---

The [field-strength
tensor](/relativity/covariant-electrodynamics/the-electromagnetic-field-tensor)
transforms as a rank-two tensor, $F'^{\mu\nu} = \Lambda^\mu{}_\alpha
\Lambda^\nu{}_\beta F^{\alpha\beta}$. Because $\vec E$ and $\vec B$ are its
components, this single rule fixes how both fields change between frames.
Carrying it out turns the abstract statement "the fields mix" into explicit
formulas, and those formulas explain two things at once: why a moving charge's
field is squashed, and why magnetism exists at all.[^tl2]

Conventions as before: signature $(-,+,+,+)$, and the primed frame $S'$ moves at
velocity $\vec v = v\hat x$ relative to $S$, with $\beta = v/c$ and $\gamma = (1 -
\beta^2)^{-1/2}$.

## The transformation rules

Apply the boost matrix

$$
\Lambda^\mu{}_\nu = \begin{pmatrix}
\gamma & -\gamma\beta & 0 & 0 \\
-\gamma\beta & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$$

to $F^{\alpha\beta}$. Each primed component is a sum of two unprimed ones. For
the electric field,

$$
E'_x = E_x, \qquad E'_y = \gamma(E_y - v B_z), \qquad E'_z = \gamma(E_z + v B_y),
$$

and for the magnetic field,

$$
B'_x = B_x, \qquad B'_y = \gamma\!\left(B_y + \frac{v}{c^2} E_z\right), \qquad
B'_z = \gamma\!\left(B_z - \frac{v}{c^2} E_y\right).
$$

The component along the boost is unchanged; the transverse components mix and
carry a factor $\gamma$. Written in terms of parts parallel and perpendicular to
$\vec v$, the rules are frame-direction independent:

$$
\vec E'_\parallel = \vec E_\parallel, \qquad \vec E'_\perp = \gamma\big(\vec
E_\perp + \vec v \times \vec B\big),
$$

$$
\vec B'_\parallel = \vec B_\parallel, \qquad \vec B'_\perp = \gamma\!\left(\vec
B_\perp - \frac{1}{c^2}\, \vec v \times \vec E\right).
$$

> **Result.** Field components along the boost direction are invariant.
> Transverse components transform into a mixture of $\vec E$ and $\vec B$, scaled
> by $\gamma$. A pure electric field in one frame therefore carries a magnetic
> field in any other frame moving transverse to it, and vice versa. The mixing
> terms $\vec v \times \vec B$ and $-c^{-2}\vec v \times \vec E$ are what tie the
> two fields into one object.

At low speed, $\gamma \to 1$ and the rules become $\vec E' \approx \vec E + \vec v
\times \vec B$ and $\vec B' \approx \vec B - c^{-2}\vec v \times \vec E$, the
Galilean field transformations that underlie motional emf.

$$
% caption: Under a boost the field splits into a part along the velocity, which
% is unchanged, and a transverse part, which is scaled by gamma and mixed with
% the other field; the parallel and perpendicular projections transform by
% different rules.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % boost direction
  \draw[->, black, very thick] (0,0) -- (2.4,0) node[right, black!70] {v};
  % a field vector
  \draw[->, acc, very thick] (0,0) -- (2.0,1.9) node[above right, acc] {E or B};
  % parallel projection
  \draw[->, black, thick] (0,0) -- (2.0,0);
  \draw[black, dashed] (2.0,1.9) -- (2.0,0);
  \node[black!70, anchor=north] at (1.0,-0.15) {parallel: unchanged};
  % perpendicular projection
  \draw[->, black, thick] (2.0,0) -- (2.0,1.9);
  \node[black!70, anchor=west] at (2.1,1.0) {perpendicular: scaled by gamma, mixed};
\end{tikzpicture}
$$

## The field of a uniformly moving charge

Take a charge $q$ at rest at the origin of $S'$. There it produces a pure
Coulomb field, $\vec B' = 0$ and

$$
\vec E' = \frac{q}{4\pi\epsilon_0}\frac{\hat r\,'}{r'^2},
$$

isotropic and radial. View it from $S$, in which the charge moves at $v\hat x$.
Transforming the fields and re-expressing them in terms of the charge's
_present_ position gives, at the moment the charge passes the origin,

$$
\vec E = \frac{q}{4\pi\epsilon_0}\,\frac{1 - \beta^2}{(1 - \beta^2 \sin^2\theta)^{3/2}}\,
\frac{\hat r}{r^2},
$$

where $\theta$ is the angle between $\vec r$ and the direction of motion. The
field still points radially from where the charge is _now_, but its strength is
no longer isotropic:

- **Along the motion** ($\theta = 0$): the field is weakened, $E_\parallel =
  \dfrac{1}{\gamma^2}\dfrac{q}{4\pi\epsilon_0 r^2}$.
- **Transverse to the motion** ($\theta = \tfrac{\pi}{2}$): the field is
  strengthened, $E_\perp = \gamma\,\dfrac{q}{4\pi\epsilon_0 r^2}$.

The field lines, uniformly spaced for a charge at rest, are swept out of the
direction of motion and compressed into a transverse pancake. As $v \to c$ the
compression becomes extreme, and the field approaches a sheet concentrated in the
plane perpendicular to the motion — the limiting field of a light-like charge.
The moving charge also carries a magnetic field circling the line of motion,

$$
\vec B = \frac{1}{c^2}\,\vec v \times \vec E,
$$

which is the field a current element produces, here derived purely by boosting a
static Coulomb field.

$$
% caption: The radial field of a charge at rest (left) is compressed transverse
% to the motion once the charge is moving (right): field lines thin out ahead and
% behind and bunch up in the perpendicular plane, strengthened by gamma there and
% weakened by one over gamma-squared along the motion.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % rest charge: even radial lines
  \fill (0,0) circle (2pt);
  \foreach \a in {0,30,...,330} \draw[black!45] (0,0) -- (\a:1.3);
  \node[black!70, anchor=north] at (0,-1.7) {at rest: isotropic};
  % moving charge: compressed
  \begin{scope}[xshift=5.3cm]
    \fill (0,0) circle (2pt);
    % dense transverse lines
    \foreach \a in {70,80,90,100,110,250,260,270,280,290}
      \draw[acc] (0,0) -- (\a:1.4);
    % sparse longitudinal lines
    \foreach \a in {0,35,145,180,215,325}
      \draw[black!55] (0,0) -- (\a:0.8);
    \draw[->, black, very thick] (0.2,-0.05) -- (1.6,-0.05) node[right, black!70] {v};
    \node[black!70, anchor=north] at (0,-1.7) {moving: pancake};
  \end{scope}
\end{tikzpicture}
$$

> **Worked example (Anisotropy of a fast charge's field).** A charge moves at
> $\beta = 0.90$, so $\gamma = 2.29$. At a fixed distance $r$, the transverse
> field is $E_\perp = \gamma\, q/(4\pi\epsilon_0 r^2)$ and the forward field is
> $E_\parallel = \gamma^{-2}\, q/(4\pi\epsilon_0 r^2)$. Their ratio is $E_\perp /
> E_\parallel = \gamma^3 = 12$: the field to the side is an order of magnitude
> stronger than the field ahead. As $\beta \to 1$ this ratio grows without bound
> and the field collapses onto the transverse plane. The accompanying magnetic
> field at the side has magnitude $B_\perp = v E_\perp / c^2 = \beta E_\perp / c$,
> so $cB_\perp / E_\perp = \beta = 0.90$; the electric and magnetic parts are
> nearly equal in the ultrarelativistic limit, the signature of a field
> approaching that of radiation.

## A pure field boosted

The transformation converts field types. A capacitor at rest sets up a uniform
$\vec E$ between its plates and no magnetic field. An observer moving parallel to
the plates sees the same $\vec E_\parallel$ but, moving perpendicular to the
field, measures a magnetic field $\vec B'_\perp = -\gamma\, c^{-2}\,\vec v \times
\vec E$. The moving charges on the plates constitute surface currents in that
frame, and those currents are the source of $\vec B'$. Nothing changed about the
physical setup; the magnetic field is the same electric field, sliced by a
different observer.

$$
% caption: A frame at rest with a charged capacitor measures only an electric
% field; a frame moving parallel to the plates measures the same field along the
% motion plus a magnetic field transverse to it, sourced by the surface currents
% the moving charges now form.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % rest frame capacitor
  \draw[black, very thick] (0,2.0) -- (2.4,2.0);
  \draw[black, very thick] (0,0.4) -- (2.4,0.4);
  \foreach \x in {0.3,0.9,1.5,2.1} \draw[->, thick] (\x,2.0) -- (\x,0.4);
  \node at (1.2,2.35) {E only};
  \node[black!70, anchor=north] at (1.2,0.1) {rest frame};
  % arrow
  \draw[->, very thick] (2.9,1.2) -- (4.3,1.2) node[above, midway, black!70] {boost};
  % moving frame
  \begin{scope}[xshift=4.8cm]
    \draw[black, very thick] (0,2.0) -- (2.4,2.0);
    \draw[black, very thick] (0,0.4) -- (2.4,0.4);
    \foreach \x in {0.3,0.9,1.5,2.1} \draw[->, thick] (\x,2.0) -- (\x,0.4);
    % B into page markers
    \foreach \p in {(0.6,1.2),(1.2,1.2),(1.8,1.2)} \node[acc] at \p {$\otimes$};
    \node at (1.2,2.35) {E and B};
    \node[black!70, anchor=north] at (1.2,0.1) {moving frame};
  \end{scope}
\end{tikzpicture}
$$

## Magnetism as relativistic electrostatics

The most direct demonstration that magnetism is a relativistic effect is the
force between a current-carrying wire and a moving charge. Take a long neutral
wire: a stationary lattice of positive ions with linear density $+\lambda_0$, and
conduction electrons with density $-\lambda_0$ drifting at speed $v_d$ to
produce a current $I = \lambda_0 v_d$. The wire is neutral, so in the lab frame a
test charge $q$ outside it feels no electric force. Let the charge move parallel
to the wire at speed $v$. In the lab, the only force is magnetic: the current's
field $B = \mu_0 I / (2\pi d)$ at distance $d$ acts on the moving charge with
force $F = qvB$.

Now view the situation from the test charge's rest frame. The positive ions were
at rest in the lab and now stream backward at speed $v$, so their spacing
contracts and their density rises in magnitude to $\gamma_v \lambda_0$. The
electrons drifted at $v_d$ in the lab, so their speed in the new frame follows
from velocity addition and differs from $v$; their spacing changes by a different
factor. The two densities no longer cancel. The wire carries a net charge density in the test
charge's frame,

$$
\lambda_{\text{net}} = -\frac{\lambda_0\, v\, v_d}{c^2}\,\gamma_v,
$$

producing an electric field that exerts a force on the stationary test charge.
Working out the magnitude and transforming back gives exactly the magnetic force
$F = qvB$ of the lab frame. What one observer calls a magnetic force, another
calls the electric force of a wire that length contraction has left charged.

> **Result.** Magnetism is not an independent interaction. The magnetic force on
> a charge moving near a current is the electric force seen in the charge's own
> frame, where unequal length contraction of the positive and negative carriers
> leaves the wire electrically charged. Special relativity plus Coulomb's law
> imply the existence of magnetic forces; the field tensor is the bookkeeping
> that makes this automatic.

$$
% caption: In the lab the wire is neutral and the moving charge feels a magnetic
% force; in the charge's rest frame the positive ions (plus signs) and the
% conduction electrons (filled dots) contract by different factors, leaving a net
% positive charge on the wire whose electric field supplies the same force.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % lab frame
  \node[black!70] at (-1.5,2.3) {lab frame};
  \draw[black] (-0.3,2.3) -- (5.3,2.3);
  \foreach \x in {0,0.7,1.4,2.1,2.8,3.5,4.2,4.9} \node at (\x,2.55) {$+$};
  \foreach \x in {0,0.7,1.4,2.1,2.8,3.5,4.2,4.9} \fill[black!70] (\x,2.05) circle (1.7pt);
  \node[black, anchor=west] at (5.4,2.3) {neutral};
  \fill[acc] (2.5,1.3) circle (2pt) node[below, black!70] {q};
  \draw[->, black, thick] (2.7,1.3) -- (3.7,1.3) node[right, black!70] {v};
  % charge frame
  \node[black!70] at (-1.5,0.3) {charge frame};
  \draw[black] (-0.3,0.3) -- (5.3,0.3);
  % contracted positives (denser)
  \foreach \x in {0,0.55,1.1,1.65,2.2,2.75,3.3,3.85,4.4,4.95} \node at (\x,0.55) {$+$};
  % expanded negatives (sparser)
  \foreach \x in {0,0.9,1.8,2.7,3.6,4.5} \fill[black!70] (\x,0.05) circle (1.7pt);
  \node[black, anchor=west] at (5.4,0.3) {net charge};
\end{tikzpicture}
$$

## Summary

- Under a boost, field components along the velocity are unchanged; transverse
  components mix $\vec E$ and $\vec B$ and gain a factor $\gamma$: $\vec E'_\perp =
  \gamma(\vec E_\perp + \vec v \times \vec B)$ and $\vec B'_\perp = \gamma(\vec
  B_\perp - c^{-2}\vec v \times \vec E)$.
- Boosting a static Coulomb field gives the field of a uniformly moving charge:
  radial from the present position but compressed transverse to the motion,
  weakened by $1/\gamma^2$ ahead and strengthened by $\gamma$ to the side, with a
  circulating magnetic field $\vec B = c^{-2}\vec v \times \vec E$.
- A pure electric field acquires a magnetic part for a transversely moving
  observer; the moving source charges are the currents that produce it.
- The magnetic force between a current and a moving charge is the electric force
  of a differentially contracted, net-charged wire in the charge's frame.
  Magnetism is the relativistic consequence of electrostatics.

[^tl2]: Tipler & Llewellyn, _Modern Physics_, Ch. 2 (the relativistic relation
between electric and magnetic fields, discussed conceptually); Schutz, _A First
Course in General Relativity_, Ch. 4 (the tensor transformation of the fields).
The moving-charge field and the current-and-charge argument follow the standard
special-relativistic treatment.