Why are Maxwell's equations significant and what is the role of displacement current?

In one line

Maxwell's equations unify electricity, magnetism and light into one theory, and Maxwell's displacement current term made them consistent and predicted electromagnetic waves travelling at \(c = 1/\sqrt{\mu_0\varepsilon_0}\).

Explained simply

Before Maxwell, electricity and magnetism were separate subjects with separate rules. Maxwell noticed a gap in Ampere's law: in a charging capacitor, current flows in the wires but not across the gap, yet a magnetic field exists around the gap. He proposed that a changing electric field acts like a current, the displacement current. With this addition, the equations predicted self-sustaining waves of \(\vec{E}\) and \(\vec{B}\), whose speed matched the measured speed of light.

How it works

1. Ampere's law \(\nabla\times\vec{H} = \vec{J}\). Taking divergence: \(\nabla\cdot(\nabla\times\vec{H}) = 0\), so \(\nabla\cdot\vec{J} = 0\).
2. But the continuity equation says \(\nabla\cdot\vec{J} = -\partial\rho/\partial t\), which is not zero when charge accumulates. Contradiction.
3. Maxwell added \(\vec{J}_d\): \(\nabla\times\vec{H} = \vec{J} + \vec{J}_d\). Taking divergence: \(\nabla\cdot\vec{J}_d = -\nabla\cdot\vec{J} = \partial\rho/\partial t = \partial(\nabla\cdot\vec{D})/\partial t\), so \(\vec{J}_d = \partial\vec{D}/\partial t\).
4. In free space, combining the curl equations gives the wave equation \(\nabla^2\vec{E} = \mu_0\varepsilon_0\,\partial^2\vec{E}/\partial t^2\), with speed \(1/\sqrt{\mu_0\varepsilon_0}\).

Significance:

• Unify electric and magnetic phenomena; changing \(\vec{E}\) and \(\vec{B}\) generate each other.
• Predict EM waves and show light is an EM wave.
• Explain no magnetic monopoles and conservation of charge.
• Basis of radio, TV, mobile phones, radar, optical fibres, antennas, motors and generators.

Key formula

\(\vec{J}_d = \dfrac{\partial\vec{D}}{\partial t} = \varepsilon_0\dfrac{\partial\vec{E}}{\partial t}\) (A/m²); \(c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}\), with \(\mu_0 = 4\pi\times10^{-7}\) H/m and \(\varepsilon_0 = 8.85\times10^{-12}\) F/m.

Quick example

\(c = 1/\sqrt{4\pi\times10^{-7}\times8.85\times10^{-12}} = 1/\sqrt{1.112\times10^{-17}} \approx 3.0\times10^8\) m/s, exactly the speed of light measured experimentally. This calculation is why we know light is electromagnetic.

Common mistakes

• Saying displacement current is a flow of charges; it is a changing electric field.
• Stating Maxwell's equations without the significance part, which is often half the marks.
• Writing \(c = \sqrt{\mu_0\varepsilon_0}\) instead of its reciprocal.

How MU asks it

Asked as "Explain the significance of Maxwell's equations" (3 to 5 marks) or "What is displacement current? Why did Maxwell introduce it?" (5 marks). Show the inconsistency using the continuity equation, derive \(\vec{J}_d = \partial\vec{D}/\partial t\), and list at least four points of significance including EM waves and \(c = 1/\sqrt{\mu_0\varepsilon_0}\).