Questions: Maxwell's Equations in Integral Form

Question 1

Maxwell noticed that the original Ampère's law was inconsistent. What was the problem and how did he fix it?

• A. Ampère's law failed to account for magnetic monopoles, so Maxwell added a ∂B/∂t term
• B. Different choices of bounding surface for the same Amperian loop gave different results near a charging capacitor, so Maxwell added the displacement current term ε₀ ∂Φ_E/∂t
• C. Ampère's law predicted infinite B fields near wires, so Maxwell added a correction factor
• D. Ampère's law only worked in vacuum, so Maxwell generalized it for dielectric media

Answer: (B)

For a circuit charging a capacitor, an Amperian surface passing through the capacitor gap encloses no conduction current — it gives zero. But the same loop with a surface that avoids the gap encloses the wire current — it gives non-zero. A single Amperian loop cannot give two different answers. Maxwell resolved this by adding ε₀ ∂Φ_E/∂t: the changing electric field in the capacitor gap contributes the same amount as the conduction current in the wire, making both surface choices consistent.

Question 2

Which of Maxwell's four equations encodes the physical claim that magnetic monopoles do not exist?

• A. Gauss's law for E (∮E·dA = Q_enc/ε₀)
• B. Faraday's law (∮E·dl = −dΦ_B/dt)
• C. Gauss's law for B (∮B·dA = 0)
• D. Ampère-Maxwell law (∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt)

Answer: (C)

Gauss's law for B states that the magnetic flux through any closed surface is zero — no net magnetic field lines ever leave or enter a closed surface. This means magnetic field lines always form closed loops and never start or stop on a 'magnetic charge.' Compare with Gauss's law for E: a nonzero right-hand side (Q_enc/ε₀) means electric field lines can start or stop on electric charges. The absence of an analogous magnetic source term is the mathematical statement that magnetic monopoles don't exist.

Question 3

The displacement current term in the Ampère-Maxwell law implies that a changing electric field can produce a magnetic field, even with no conduction current present.

• T. True
• F. False

Answer: True

This is exactly Maxwell's key insight. The term μ₀ε₀ ∂Φ_E/∂t appears alongside the conduction current on the right-hand side of the Ampère-Maxwell law. It says that a time-varying electric flux through a surface drives B circulation around the surface's boundary, just as a conduction current does. Combined with Faraday's law (changing B produces E), this mutual coupling is what enables self-sustaining electromagnetic waves to propagate through empty space.

Question 4

Faraday's law and the Ampère-Maxwell law play symmetric roles: changing B produces E circulation, and changing E produces B circulation. Together these two couplings enable electromagnetic waves.

• T. True
• F. False

Answer: True

This symmetry is the physical heart of classical electromagnetism. Faraday's law: ∮E·dl = −dΦ_B/dt. Ampère-Maxwell: ∮B·dl = μ₀ε₀ dΦ_E/dt (in free space). In a region with no charges or currents, these equations couple E and B into a self-reinforcing oscillation: changing E generates changing B, which generates changing E... This propagates at 1/√(μ₀ε₀) = c, the speed of light, predicting that light is an electromagnetic wave.

Question 5

Explain in physical terms why the integral form of Maxwell's equations is especially useful for problems with geometric symmetry.

Model answer: The integral form speaks about total flux through surfaces and total circulation around loops — macroscopic quantities over finite regions. When a problem has spherical, cylindrical, or planar symmetry, the field magnitude is constant over a carefully chosen Gaussian surface or Amperian loop, so the field can be pulled outside the integral. This reduces the integral equation to a simple algebraic relation: (field magnitude) × (area or path length) = (enclosed charge or current). Without symmetry, this simplification fails and the differential form is more useful for deriving field equations directly.

The integral form builds physical intuition — you can see that charges create flux, currents circulate B, changing B drives E — while the differential form (∇·E = ρ/ε₀, etc.) is better for wave equations and field problems in continuous media.