Questions: Maxwell's Equations in Differential Form

Question 1

The equation ∇·B = 0 holds everywhere in space. What does this imply about magnetic fields?

• A. Magnetic fields are zero in a vacuum with no current sources
• B. Magnetic monopoles do not exist — magnetic field lines never begin or end at any point, so they always form closed loops
• C. Magnetic fields only exist near electric currents or permanent magnets
• D. The divergence of B is zero only far from any current source, approaching zero asymptotically

Answer: (B)

∇·B = 0 holds everywhere — this is what makes it a fundamental local equation of electromagnetism. Divergence measures whether field lines begin or end at a point. ∇·B = 0 means B field lines never start or terminate anywhere, which forces them to always form closed loops. This is the mathematical statement of the absence of magnetic monopoles — there are no isolated magnetic charges analogous to electric charges. No magnetic monopole has ever been observed, consistent with this equation.

Question 2

Before Maxwell added the displacement current term μ₀ε₀∂E/∂t to Ampère's law, Ampère's law was inconsistent for time-varying fields. The displacement current was needed to:

• A. Ensure Ampère's law gives the same result as Gauss's law in electrostatic situations
• B. Mathematically resolve the inconsistency and enable prediction of electromagnetic waves propagating at the speed of light
• C. Account for the polarization of dielectric materials in the presence of electric fields
• D. Describe how changing magnetic fields drive currents in conductors, as in Faraday's law

Answer: (B)

Maxwell's displacement current was a theoretical insight, not an empirical discovery. Without it, taking the divergence of ∇×B = μ₀J yields a contradiction in time-varying situations. With it, the equations become self-consistent, and combining Faraday's law and the modified Ampère's law in vacuum yields a wave equation with speed c = 1/√(μ₀ε₀). When Maxwell found this equaled the measured speed of light, it demonstrated that light is an electromagnetic wave — unifying electricity, magnetism, and optics in a single framework.

Question 3

The differential form of Maxwell's equations and their integral form describe the same physical content — they are mathematically equivalent via the divergence theorem and Stokes' theorem.

• T. True
• F. False

Answer: True

The integral and differential forms encode identical physics. The divergence theorem converts ∇·E = ρ/ε₀ to Gauss's law in integral form (surface integrals of E equal enclosed charge); Stokes' theorem converts the curl equations to their integral loop forms. The differential form is often more useful because it holds pointwise at every location in space, avoiding the need to choose specific surfaces or loops. The integral form is convenient when symmetry simplifies the integrals. Neither contains information absent from the other.

Question 4

In Faraday's law ∇×E = −∂B/∂t, the negative sign is a convention with no physical consequence — the sign could be positive without changing observable predictions.

• T. True
• F. False

Answer: False

The negative sign encodes Lenz's law, which has direct and observable physical significance. It means the induced electric field circulates in a direction that opposes the change in magnetic flux. If B is increasing in a given direction, the induced E curls so that if it drove current in a conducting loop, that current would produce a magnetic field opposing the increase. Remove the negative sign and electromagnetic feedback becomes destabilizing rather than self-limiting — generators, transformers, and virtually all electromagnetic induction devices depend on this sign for their correct behavior.

Question 5

What is the physical significance of the displacement current term μ₀ε₀∂E/∂t in Ampère's law, and why was its addition a landmark in physics?

Model answer: The displacement current says that a time-varying electric field generates a circulating magnetic field, even in the absence of any actual moving charges. Physically, this completes the symmetry between E and B: Faraday's law says changing B creates circulating E; Maxwell's addition says changing E creates circulating B. This mutual generation allows each field to sustain the other, enabling self-propagating electromagnetic waves. When Maxwell combined Faraday and the modified Ampère in vacuum to derive a wave equation, its speed c = 1/√(μ₀ε₀) matched the measured speed of light — proving light is an electromagnetic wave and unifying three previously separate fields of physics.

The displacement current is invisible (no real charges move through a capacitor gap, yet a magnetic field circulates around it), which is why it required theoretical rather than experimental discovery. It exemplifies how mathematical self-consistency requirements can lead to genuine physical insight. Without it, Maxwell's equations are internally inconsistent for time-varying fields; with it, they predict the entire spectrum of electromagnetic radiation.