Questions: Electromagnetic Field Tensor and Special Relativity

Question 1

A positive charge sits at rest in its own reference frame, producing only an electric field. A second observer moves past that charge at high speed. What does the second observer detect?

• A. Only an electric field — the field is the same in all inertial frames
• B. Both an electric field and a magnetic field
• C. Only a magnetic field — the moving observer sees the charge as a current
• D. No field at all — moving observers lose access to static fields

Answer: (B)

Under a Lorentz boost, the components of F^μν mix. The second observer sees the charge as moving — a moving charge is a current — and currents produce magnetic fields. The electric field components also transform. Neither E alone nor B alone is Lorentz-invariant; only the full electromagnetic field tensor F^μν is the genuine relativistic object. Option C is wrong because the electric field does not vanish; option A expresses the pre-relativistic misconception that E and B are separately observer-independent.

Question 2

The covariant form of Maxwell's sourced equations is ∂_μF^μν = μ₀J^ν. How many independent scalar equations does this single tensor equation represent?

• A. 1 — it is a single equation
• B. 4 — one for each value of the free index ν
• C. 6 — one for each independent component of F^μν
• D. 16 — one for each entry of the 4×4 matrix

Answer: (B)

The free index ν runs over four values (0, 1, 2, 3), so the equation ∂_μF^μν = μ₀J^ν is really four equations. These four equations encode exactly Gauss's law for electricity and Ampère's law with Maxwell's correction — the two sourced Maxwell equations in their vector form. The other two Maxwell equations (Faraday's law and Gauss's law for magnetism) are encoded in the separate Bianchi identity ∂_[μF_νλ] = 0.

Question 3

Magnetic forces and electric forces are fundamentally distinct phenomena that can seldom be converted into each other by changing reference frame.

• T. True
• F. False

Answer: False

This is exactly the misconception that covariant electrodynamics refutes. Electric and magnetic fields are frame-dependent components of the same electromagnetic field tensor. A Lorentz boost mixes them: what is purely electric in one frame has both electric and magnetic components in another. The famous example is the magnetic force on a charge moving near a wire — in the wire's rest frame this is a Coulomb force from charge density imbalances due to length contraction. There is one electromagnetic field, viewed from different frames.

Question 4

Maxwell published his equations in 1865, and Einstein published special relativity in 1905. This means Maxwell's original equations were inconsistent with special relativity and had to be reformulated.

• T. True
• F. False

Answer: False

Maxwell's equations were already Lorentz-covariant when Maxwell wrote them — they just didn't look obviously covariant in their original vector form. Special relativity was largely motivated by the fact that Newtonian mechanics was inconsistent with Maxwell's equations, not the other way around. The tensor formulation makes the covariance manifest: each side of ∂_μF^μν = μ₀J^ν transforms as the same type of geometric object under Lorentz transformations. Maxwell did not need updating; Newtonian mechanics did.

Question 5

The electromagnetic field tensor F^μν is a 4×4 matrix but has only 6 independent components. Why?

Model answer: F^μν is antisymmetric: F^μν = −F^νμ. Antisymmetry forces all diagonal elements to zero (F^μμ = −F^μμ implies F^μμ = 0) and means the lower triangle is just the negative of the upper triangle. A 4×4 antisymmetric matrix therefore has 4×3/2 = 6 independent off-diagonal entries. These 6 components encode exactly the three components of E and the three components of B.

The antisymmetry is not just a mathematical convenience — it encodes the physics that the electromagnetic field has no scalar 'self-interaction' term and that the field strength is purely about differences. Understanding why antisymmetry reduces 16 entries to 6 is key to working fluently with the tensor, and it directly explains why the unified tensor packages the six field components so efficiently.