How E and B Transform Between Frames

The Geometric View

Imagine the electromagnetic field as a single geometric object, much like a vector in space. Just as a vector’s components change when you rotate your coordinate axes, the components of the electromagnetic field change when you change your frame of reference via a Lorentz boost. In one frame, you might see only an electric field. In another frame moving relative to the first, that same physical situation will appear to have both electric and magnetic components. There is no absolute distinction between "electric" and "magnetic"; the split depends entirely on the observer’s velocity. This perspective dissolves the historical separation between the two forces and shows that they are two faces of the same coin.

Transformation of Parallel Components

Consider a Lorentz boost along the \( x \)-axis with velocity \( v \). The components of the fields parallel to the direction of motion remain invariant. In tensor language, \( E_x \) and \( B_x \) sit in the entries \( F^{01} \) and \( F^{23} \). A boost along \( x \) mixes only the indices 0 and 1, and an entry carrying both of them (or neither) is unchanged.

This invariance is physically intuitive. If you move along the line of a uniform electric field generated by a parallel plate capacitor, the charge density and the geometry of the plates do not change in a way that alters the field strength in the direction of motion. Similarly, a solenoid moving along its own axis keeps its axial field: length contraction packs the turns more closely by a factor \( \gamma \), while time dilation lowers the current in each turn by the same factor, so \( nI \) and hence \( B_x \) are unchanged. The parallel components are "decoupled" from the boost.

Mixing of Perpendicular Components

The components perpendicular to the boost direction mix. This is where the relativistic nature of the fields becomes most apparent. The electric field in the new frame depends on both the electric and magnetic fields in the original frame, and vice versa. The mixing is governed by the Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \).

Notice the symmetry: the electric field transforms into a magnetic field and vice versa, scaled by \( v/c^2 \) for the magnetic part. This reciprocal relationship ensures that the speed of light remains invariant and that the Lorentz force law \( \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \) is consistent across all inertial frames. The term \( \vec{v} \times \vec{B} \) in the electric transformation is the familiar motional EMF, while the term \( \vec{v} \times \vec{E}/c^2 \) in the magnetic transformation is less commonly encountered but equally fundamental.

Physical Interpretation: Magnetism as Relativity

A powerful way to see the unity of electricity and magnetism is to consider a pure electric field in one frame. Suppose a point charge is at rest in frame \( S \). It produces a radial electric field \( \vec{E} \) and no magnetic field \( \vec{B} = 0 \). Now consider an observer in frame \( S' \) moving with velocity \( \vec{v} \) relative to the charge. According to the transformation laws, this observer will measure a magnetic field:

Since \( \vec{E} \) is radial and \( \vec{v} \) points along a fixed direction, the cross product \( \vec{v} \times \vec{E} \) is perpendicular to both, resulting in a magnetic field that circulates around the direction of motion. This is precisely the magnetic field of a moving charge. In this sense magnetism is the relativistic companion of electrostatics, an effect arising from the transformation of the electric field when the source charge is in motion relative to the observer. This insight resolves the historical puzzle of why magnetic forces exist: they are the "shadow" of electric forces cast by the geometry of spacetime.