Questions: Plane Electromagnetic Waves

Question 1

A student claims that an electromagnetic wave in free space could have its electric field pointing partly in the direction of propagation. Which of Maxwell's equations directly contradicts this claim?

• A. Faraday's law (∇ × E = −∂B/∂t)
• B. Gauss's law for electric fields (∇ · E = 0 in free space)
• C. Ampère's law with Maxwell's correction (∇ × B = μ₀ε₀ ∂E/∂t)
• D. Gauss's law for magnetic fields (∇ · B = 0)

Answer: (B)

In free space, Gauss's law states ∇ · E = 0. For a plane wave E = E₀ exp(i(k·r − ωt)), the divergence condition becomes ik · E₀ = 0, which forces E₀ to be perpendicular to k. This is transversality — the electric field cannot have any component along the propagation direction k̂. Faraday's law constrains the relationship between E and B; Gauss's law for E is what directly enforces transversality of the electric field.

Question 2

A plane electromagnetic wave travels in the +z direction. At a given point and time, the electric field is E = E₀ x̂. What is the direction of the magnetic field at the same point and time?

• A. +x̂ (parallel to E)
• B. +ẑ (along the direction of propagation)
• C. +ŷ (perpendicular to both E and k̂)
• D. −ẑ (antiparallel to the propagation direction)

Answer: (C)

The magnetic field of a plane wave is given by B = (k̂ × E)/c. With k̂ = ẑ and E = E₀ x̂, we get B = (ẑ × x̂)E₀/c = ŷ E₀/c. The result is +ŷ, perpendicular to both E (which points in x̂) and the propagation direction (ẑ). This three-way mutual perpendicularity — E ⊥ B ⊥ k̂ — is the defining geometric structure of a plane wave in free space. Option A (E ∥ B) and B/D (either field along k̂) both violate transversality.

Question 3

Treating sunlight as a plane wave is an excellent approximation for objects on Earth, even though the Sun emits spherical wavefronts.

• T. True
• F. False

Answer: True

A point source emits spherical wavefronts, but at distances much larger than the source, any small patch of a sphere is locally indistinguishable from a flat plane. Since the Earth is ~150 million km from the Sun, the curvature of the wavefront across any human-scale apparatus is negligible. The plane wave approximation holds whenever the observation region is much smaller than the distance to the source — which is almost always true for astrophysical sources. This is why plane waves are the workhorse of optics and antenna theory.

Question 4

In a plane electromagnetic wave, the electric and magnetic fields are 90° out of phase with each other — E peaks when B is zero, and vice versa.

• T. True
• F. False

Answer: False

This is a common confusion imported from LC circuits, where voltage and current are 90° out of phase. In an electromagnetic plane wave, E and B oscillate in phase — they peak and cross zero simultaneously. This follows from B = (k̂ × E)/c: B is directly proportional to E at every point and time, with no phase lag. The 90° out-of-phase relationship is a feature of standing waves in cavities, not of traveling plane waves in free space.

Question 5

Why must the electric field of an electromagnetic plane wave in free space be perpendicular to the direction of propagation? What does this constraint imply about polarization?

Model answer: In free space, Gauss's law requires ∇ · E = 0. For a plane wave with wave vector k, this condition translates to k · E₀ = 0, meaning E₀ must be perpendicular to k — the wave is transverse. Because E is confined to the plane perpendicular to k (a two-dimensional plane), the electric field can point in any direction within that plane. This two-dimensional freedom is exactly polarization: linear polarization occurs when E oscillates along a fixed direction in that plane, circular polarization when E rotates with constant magnitude, and elliptical polarization for the general case.

Transversality does not come from an ad-hoc assumption — it is a direct consequence of Maxwell's equations in free space. The constraint that E lies in a plane perpendicular to k is what opens up the rich phenomenology of polarization states, with applications in optics, communications, and astronomy (polarimetry of starlight, for instance).