How E and B Fields Transform

The Electromagnetic Field as a 4D Object

In pre-relativistic physics, electric and magnetic fields were treated as separate entities. But Einstein's special relativity revealed a profound truth: E and B are not independent. They are components of a single electromagnetic field tensor, and they mix when you change reference frames.

What one observer calls a pure electric field, another observer in motion may see as a combination of electric and magnetic fields. This mixing is a direct consequence of the Lorentz transformation, and it explains why magnetism exists at all.

The Transformation Laws

Consider a boost along the x-axis with velocity β. The electromagnetic fields transform as:

Parallel Components (Along Boost Direction)

E_x' = E_x B_x' = B_x

Components parallel to the boost direction are unchanged. This is analogous to how spatial coordinates along the boost axis transform differently from perpendicular coordinates.

Perpendicular Components (Transverse to Boost)

E_y' = γ(E_y - βB_z) E_z' = γ(E_z + βB_y) B_y' = γ(B_y + βE_z) B_z' = γ(B_z - βE_y)

These are the key mixing terms. Notice that E picks up contributions from B, and vice versa. The cross product structure (β × B and β × E) encodes the spatial relationships.

Why Do E and B Mix?

The electromagnetic field is really a rank-2 antisymmetric tensor, F_μν, in 4D spacetime. When you perform a Lorentz transformation, this tensor transforms just like any other tensor. But because it couples space and time coordinates, components that were purely "electric" (coupling time to space) get mixed with "magnetic" components (coupling space to space).

In other words: E and B are frame-dependent projections of a single 4D object. There's no fundamental distinction between them — it depends on your state of motion.

Lorentz Invariants

While E and B individually change with your frame, certain combinations are invariant:

E·B Invariant

E·B = E_xB_x + E_yB_y + E_zB_z

This dot product is the same in all frames. If E and B are perpendicular in one frame, they're perpendicular in all frames. If they're parallel in one frame, they're parallel in all frames.

E² - B² Invariant

E² - B² = |E|² - |B|²

This difference is also invariant. If E² > B² in one frame, it's true in all frames (electric-like field). If B² > E² in one frame, it's true in all frames (magnetic-like field). If E² = B², the field is null (like an electromagnetic wave).

Special Cases

Pure Electric Field

Suppose in the rest frame you have E = (0, E₀, 0) and B = 0. Boost along x with velocity β:

E_y' = γE₀ B_z' = -γβE₀

A pure electric field in the rest frame becomes E + B in the moving frame! The magnetic field appears as a relativistic effect.

Pure Magnetic Field

Now suppose B = (0, B₀, 0) and E = 0. Same boost:

B_y' = γB₀ E_z' = γβB₀

A pure magnetic field generates an electric field in the moving frame. This is the basis of electromagnetic induction.

Electromagnetic Wave

For a plane wave propagating in the x-direction: E and B are perpendicular to each other and to the propagation direction, with |E| = |B|. This means E² - B² = 0 (null field). Boosting along x doesn't change the wave structure, just its frequency and amplitude (Doppler shift and beaming).

Magnetism from Relativity

Consider a current-carrying wire. In the wire's rest frame, positive charges are stationary and negative charges (electrons) drift with velocity v. The wire is electrically neutral, so there's no electric field. But the moving electrons create a magnetic field B.

Now boost to the frame of the electrons. In this frame, the electrons are stationary and the positive charges move backward. Due to length contraction, the density of moving charges changes differently from the stationary charges. The wire is no longer neutral — there's now an electric field!

The force on a nearby moving charge can be computed in either frame: one frame says it's a magnetic force (F = qv × B), the other says it's an electric force (F = qE). Both give the same answer when you account for how forces transform. This is the deep reason why magnetism exists: it's the electric field as seen from a moving frame, plus length contraction effects on charge density.

Things to Try

1. Pure E to Mixed E+B: Use the "Pure E Field" preset. Increase the boost velocity β. Watch how a magnetic field appears perpendicular to both E and β. The stronger the boost, the larger the B field.
2. Pure B to Mixed B+E: Use the "Pure B Field" preset. Boost to the right. An electric field emerges. This is how electric generators work: move a conductor through a magnetic field, and the conduction electrons see an electric field (motional EMF).
3. Check Invariants: Set any field configuration and boost. Watch the invariants E·B and E² - B². No matter how you change β, these quantities stay constant. That's the signature of Lorentz covariance.
4. EM Wave: Use the "EM Wave" preset. This sets E and B perpendicular with equal magnitude (E² - B² = 0). Boost in any direction and confirm the wave structure is preserved (though field magnitudes change due to Doppler effects).
5. Parallel E and B: Set E_y = B_y = 1, all others zero. Now E·B ≠ 0. Boost along x. Notice E·B doesn't change, even though E and B individually do.
6. Boost Reversal: Set up any configuration and boost to β = +0.8. Note the primed fields. Now boost to β = -0.8 (opposite direction). The fields change, but in a predictable way. This is the origin of reciprocity in electromagnetism.

Physical Consequences

Unification of Forces

Before relativity, electricity and magnetism were thought to be separate forces. Maxwell unified them into electromagnetism, but it was Einstein who showed they were different aspects of the same field. A charge at rest sources an electric field; a moving charge sources electric and magnetic fields. But "at rest" and "moving" are frame-dependent, so E and B must mix.

Electromagnetic Induction

Faraday's law (∇ × E = -∂B/∂t) says a changing magnetic field creates an electric field. But in the rest frame of a magnet, the B field is static. Move a conductor through it, and from the conductor's frame, the B field is changing. That's why a voltage appears. Field transformations make induction inevitable.

Charge Conservation and Current

Charge density ρ and current density J form a 4-vector (ρ, J). When you boost, ρ and J mix, just like how E and B mix. This is why current appears when you boost past static charges. The continuity equation ∂ρ/∂t + ∇·J = 0 is automatically Lorentz covariant.

Historical Note

In 1905, Einstein's first relativity paper was titled "On the Electrodynamics of Moving Bodies." His motivation was to resolve puzzles in electromagnetism, particularly the asymmetry between magnetic induction (moving magnet, stationary conductor) and motional EMF (moving conductor, stationary magnet). Classical physics treated these as different phenomena. Einstein showed they were the same: electromagnetic field transformation under Lorentz boosts.

The mixing of E and B isn't an add-on to Maxwell's equations — it's built into their structure. When Maxwell's equations are written in covariant form (using the field tensor F_μν), Lorentz invariance is manifest. The equations have the same form in all inertial frames. This deep symmetry is one of the most beautiful results in physics, connecting spacetime geometry to the fundamental forces of nature.