Questions: Lorentz Transformations of Electromagnetic Fields

Question 1

In frame S, a point charge is at rest, producing E⃗ ≠ 0 and B⃗ = 0 everywhere. An observer in frame S' moves at velocity v relative to S. What does the S' observer measure?

• A. The same pure electric field E⃗ and B⃗ = 0, because the charge itself is unchanged
• B. Both a modified electric field and a nonzero magnetic field, as a consequence of the Lorentz transformation
• C. Only a magnetic field B⃗ ≠ 0, because the charge appears to be moving in S'
• D. Zero field, because the Lorentz transformation preserves the vacuum

Answer: (B)

The Lorentz transformation mixes E and B: transverse field components transform as E_y′ = γ(E_y − vB_z) and B_y′ = γ(B_y + vE_z/c²). Starting from a pure E field (B = 0 in S), the S' observer finds a modified E field and a new nonzero B field — exactly the fields of a moving charge calculated by Biot-Savart. Option A is the Newtonian intuition that fields are absolute. Option C overcorrects by eliminating E. This demonstrates that the magnetic field of a moving charge is simply the electric field of that charge as seen from another frame.

Question 2

A physicist argues: 'The magnetic force on a particle moving near a current-carrying wire is a distinct physical effect from electric attraction — they have different causes.' What does the relativistic treatment of field transformations reveal about this claim?

• A. The claim is correct — electric and magnetic forces are fundamentally distinct phenomena with independent origins
• B. The claim is misleading — the magnetic force in one frame is the electric (Coulomb) force in another frame; both are expressions of the same electromagnetic interaction
• C. The claim is partially correct — the forces are equivalent only at relativistic speeds
• D. The claim is correct for static configurations but wrong for time-varying fields

Answer: (B)

The wire-force example illustrates this directly: in the lab frame, a nearby moving charge experiences a magnetic force. In the charge's rest frame, Lorentz contraction increases the positive ion density of the wire, producing a net electric force. Same physical force, different descriptions in different frames. The 'distinction' between E and B is frame-dependent, not fundamental — both are components of the single electromagnetic field tensor Fᵘᵛ.

Question 3

If E⃗ · B⃗ = 0 in one inertial frame, there should exist another inertial frame where E⃗ · B⃗ ≠ 0.

• T. True
• F. False

Answer: False

E⃗ · B⃗ is a Lorentz invariant — all inertial observers agree on its value. If it is zero in one frame, it is zero in all frames. This is one of two key invariants (the other being E² − c²B²). If E and B are perpendicular in one frame, they remain perpendicular in every frame. Invariants carry observer-independent physical information, in contrast to the individual values of E and B which are frame-dependent.

Question 4

The distinction between 'electric field' and 'magnetic field' is physically meaningful only in the context of a specific reference frame.

• T. True
• F. False

Answer: True

E and B are not independently invariant physical objects — they are components of the electromagnetic field tensor Fᵘᵛ that mix under Lorentz boosts. The same field configuration can appear as a pure electric field in one frame and as a combination of electric and magnetic fields in another. Two observers disagree on how much of the field is 'electric' and how much is 'magnetic,' yet they agree on the physical effects (forces, energy). The observer-independent object is Fᵘᵛ itself.

Question 5

Why does the existence of the electromagnetic field tensor Fᵘᵛ imply that the separation of fields into 'electric' and 'magnetic' parts is frame-dependent?

Model answer: E and B are not separate 4-vectors but components of the same antisymmetric rank-2 tensor Fᵘᵛ. Under a Lorentz boost, the tensor transforms by the standard tensor transformation law, which mixes the components corresponding to E with those corresponding to B. Just as a Lorentz boost mixes time and space components of a 4-vector (so 'time' and 'space' are frame-dependent), a boost mixes the E and B components of Fᵘᵛ. What one observer labels 'electric' another calls a combination of electric and magnetic. The physical object — Fᵘᵛ — is the same in all frames; only the decomposition differs.

The analogy to spacetime mixing is precise: just as there is no absolute 'spatial' or 'temporal' component of a displacement (only the spacetime interval is invariant), there is no absolute 'electric' or 'magnetic' part of an electromagnetic field — only E·B and E²−c²B² are observer-independent.