Maxwell's equations — four equations and what each describes physically

hard JEE-MAIN JEE-ADVANCED • 4 min read
Tags Maxwell

Question

State all four of Maxwell’s equations in integral form. Explain the physical meaning of each equation. What was Maxwell’s key contribution (displacement current) and how did it lead to the prediction of electromagnetic waves?

(JEE Main + JEE Advanced pattern)

Solution — Step by Step

1. Gauss’s Law (Electric):

∮E⃗⋅dA⃗=Qencϵ0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}

2. Gauss’s Law (Magnetic):

∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0

3. Faraday’s Law:

∮E⃗⋅dl⃗=−dΦBdt\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}

4. Ampere-Maxwell Law:

∮B⃗⋅dl⃗=μ0Ienc+μ0ϵ0dΦEdt\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}

  1. Electric charges produce electric fields. The total electric flux through a closed surface equals the enclosed charge divided by ϵ0\epsilon_0. Positive charges are sources, negative charges are sinks.

  2. Magnetic monopoles do not exist. The total magnetic flux through any closed surface is zero — every magnetic field line that enters must exit. You cannot isolate a north or south pole.

  3. A changing magnetic field produces an electric field. This is electromagnetic induction — the basis of generators, transformers, and inductors.

  4. Electric currents AND changing electric fields produce magnetic fields. The second term (μ0ϵ0dΦEdt\mu_0 \epsilon_0 \frac{d\Phi_E}{dt}) is Maxwell’s addition — the displacement current.

Before Maxwell, Ampere’s law was ∮B⃗⋅dl⃗=μ0Ienc\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}. But this failed for a charging capacitor — no current flows between the plates, yet a magnetic field exists there.

Maxwell proposed that a changing electric field between the plates acts as a “current” (displacement current):

Id=ϵ0dΦEdtI_d = \epsilon_0 \frac{d\Phi_E}{dt}

This completed the symmetry: changing B⃗\vec{B} produces E⃗\vec{E} (Faraday), and changing E⃗\vec{E} produces B⃗\vec{B} (Maxwell). This mutual generation allows self-sustaining electromagnetic waves travelling at c=1μ0ϵ0c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}.

 
flowchart TD
    A["Maxwell's 4 Equations"] --> B["1. Gauss Electric: charges → E field"]
    A --> C["2. Gauss Magnetic: no monopoles"]
    A --> D["3. Faraday: changing B → E field"]
    A --> E["4. Ampere-Maxwell: current + changing E → B field"]
    D --> F["Changing B creates E"]
    E --> G["Changing E creates B"]
    F --> H["Mutual generation → EM waves"]
    G --> H
    H --> I["Speed: c = 1/√(μ₀ε₀)"]

Why This Works

Maxwell’s equations are the complete description of classical electromagnetism. The beauty is in the symmetry between equations 3 and 4: electric and magnetic fields can generate each other. Once you start with a changing electric field, it creates a changing magnetic field, which creates a changing electric field, and so on — the disturbance propagates through space as an electromagnetic wave.

When Maxwell calculated the speed c=1/μ0ϵ0c = 1/\sqrt{\mu_0 \epsilon_0} using the known values of μ0\mu_0 and ϵ0\epsilon_0 (measured from static electric and magnetic experiments), he got 3×1083 \times 10^8 m/s — exactly the speed of light measured by Fizeau. This was the stunning unification: light is an electromagnetic wave.

Alternative Method — Differential Form (For Advanced Students)

The differential forms are more compact:

∇⋅E⃗=ρϵ0,∇⋅B⃗=0\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, \quad \nabla \cdot \vec{B} = 0 ∇×E⃗=−∂B⃗∂t,∇×B⃗=μ0J⃗+μ0ϵ0∂E⃗∂t\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}

For JEE Main, you mainly need the physical meaning and the displacement current concept. For JEE Advanced, you should be able to apply these equations to specific situations (capacitor gap, long wire, etc.). The most commonly tested point: displacement current between capacitor plates equals conduction current in the connecting wires, maintaining continuity of current.

Common Mistake

Students think displacement current involves actual charge flow. It does NOT. Displacement current is not a real current — there are no moving charges. It is a mathematical term ϵ0dΦEdt\epsilon_0 \frac{d\Phi_E}{dt} that has the same magnetic effect as a real current. Between capacitor plates, no charges flow, but the changing electric field produces a magnetic field as if a current were flowing. Maxwell called it “displacement current” by analogy, and the name stuck despite being misleading.

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