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Topic 10 of 10

Relativistic Electrodynamics

Maxwell's equations are already Lorentz-covariant — they were special relativity before Einstein. Electric and magnetic fields are not separately invariant: a pure electric field in one frame has a magnetic component in another. The electromagnetic field tensor $F^{\mu\nu}$ unifies $\vec E$ and $\vec B$ into a single Lorentz covariant object.

Key Concepts

Field Transformations

Under a boost along

x

at speed

v

E'_x=E_x

E'_y=\gamma(E_y-vB_z)

B'_z=\gamma(B_z-vE_y/c^2)

. A pure electric field in one frame appears as both

E

and

B

in a moving frame.

Electromagnetic Field Tensor

F^{\mu\nu}

: a

4\times4

antisymmetric tensor. Its upper rows contain

E_x,E_y,E_z

(in units of

c

) and the remaining entries contain

B_x,B_y,B_z

. Maxwell's equations become

\partial_\nu F^{\mu\nu}=\mu_0 J^\mu

Lorentz Invariants of EM Fields

\vec E^2-c^2\vec B^2

and

\vec E\cdot\vec B

are the two independent Lorentz invariants of the electromagnetic field. They can be used to classify fields: if both vanish, the field is a pure radiation field.

Four-Current

J^\mu=(c\rho,\vec J)

. In a frame moving with velocity

v

relative to a stationary charge distribution, the charge density transforms as

\rho'=\gamma\rho

and a current

\vec J'=\gamma\rho\vec v

appears.

Key Equations

E-field boost (transverse)

E'_y = \gamma(E_y - vB_z)

Transverse electric field in S'; mixes with B-field. Longitudinal E field is unchanged.

B-field boost (transverse)

B'_z = \gamma\left(B_z - \frac{v E_y}{c^2}\right)

Transverse B field in S'; mixes with E-field.

EM invariants

I_1 = E^2 - c^2B^2, \quad I_2 = \vec{E}\cdot\vec{B}

Two Lorentz scalars built from the EM field; same in all inertial frames.

Worked Example

Magnetic Force as a Relativistic Effect

Problem

A wire carries current. In the wire's frame, electrons move at $v_d$ and positive ions are stationary (net charge neutral). An electron outside the wire moves at $v_d$ parallel to it. Show that the magnetic force in the lab frame is a Lorentz-contracted electric force in the electron's frame.

Solution

In the lab: the wire is neutral, current creates $B$ , the moving external electron feels force $F=qv_dB=\mu_0qv_d I/(2\pi r)$ .

In the electron's frame: positive ions are stationary but electrons in wire are at rest (from the external electron's view, they have relative velocity zero). The ion spacing contracts (ions were at rest, now they move at $-v_d$ ): $\lambda^+_{\rm frame}=\gamma\lambda_+$ .

So in the electron's frame, the wire appears positively charged, creating an electric field that attracts the external electron. The two forces are equal by Lorentz covariance.

Answer Magnetic force in the lab = electric force in the electron's frame — magnetism is a relativistic effect of electricity.

Practice

Exercises

7 problems

1 of 7

A uniform electric field $E_y=1000$ N/C, $B_z=0$ in frame $S$ . Frame $S'$ moves at $v=0.60c$ ( $\gamma=1.25$ ) along $x$ . Find $E'_y$ (in N/C).

Your answer

N/C

2 of 7

Same field. Find $B'_z$ (in T) in frame $S'$ . $v=0.60c=1.8\times10^8$ m/s, $E_y=1000$ N/C.

Your answer

3 of 7

An EM field has $E=300$ N/C and $B=1.0\times10^{-6}$ T (both perpendicular). Find the invariant $I_1=E^2-c^2B^2$ (in N²/C²).

Unlock Exercise 3