From Statics to Dynamics: The Road to Maxwell
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You should be comfortable with vector calculus operators (\(\nabla \cdot\), \(\nabla \times\)) and the basic integral forms of Gauss’s and Ampère’s laws for static fields.
Why this matters
Static electromagnetism is incomplete because it cannot describe changing fields.
In Electromagnetism I, we treated electric and magnetic fields as separate entities governed by static sources. However, nature does not distinguish between a steady current and a changing electric field. This lecture outlines the logical necessity of Faraday’s law of induction and the displacement current term. By unifying these concepts, we arrive at the full set of Maxwell’s equations, which predict that electromagnetic fields can propagate through empty space as waves. This transition from statics to dynamics is the foundational step for understanding radiation, optics, and the relativistic nature of the electromagnetic field.
The Incompleteness of Static Laws
Imagine a circuit with a capacitor. When the capacitor is charging, current flows through the wires, creating a magnetic field. However, between the plates, there is no conduction current. If we strictly applied the original Ampère’s law, the magnetic field should vanish in the gap between the plates. This contradicts experimental observation, which shows that a magnetic field does exist in that region. The static laws fail because they assume sources are constant in time. To describe reality, we need laws that account for time-varying quantities. The electric field is no longer just a function of charge density; it is also influenced by the time-varying magnetic field. Similarly, the magnetic field is influenced not only by conduction current but also by the time-varying electric field. This interdependence suggests that electric and magnetic fields are two aspects of a single electromagnetic field.
Faraday’s Law of Induction
Faraday’s law states that a changing magnetic flux through a loop induces an electromotive force (EMF) in that loop. Mathematically, the line integral of the electric field around a closed path is equal to the negative rate of change of the magnetic flux through any surface bounded by that path. This is the first dynamic equation, linking the curl of the electric field to the time derivative of the magnetic field.
In differential form, using Stokes’ theorem, this becomes:
The negative sign is crucial; it represents Lenz’s law, ensuring that the induced field opposes the change in flux that created it, thereby conserving energy. This equation implies that electric fields are no longer conservative in the presence of time-varying magnetic fields. The concept of a scalar electric potential \(\phi\) alone is insufficient; we must introduce a vector potential \(\mathbf{A}\) such that \(\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}\).
The Displacement Current
Ampère’s original law, \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J}\), leads to a contradiction with the continuity equation for charge conservation. Taking the divergence of both sides yields \(\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 \nabla \cdot \mathbf{J}\). Since the divergence of a curl is always zero, this implies \(\nabla \cdot \mathbf{J} = 0\). However, the continuity equation states \(\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}\). These two are only consistent if the charge density \(\rho\) is constant in time. For dynamic situations, such as charging a capacitor, this fails. Maxwell resolved this by adding a "displacement current" term \(\mathbf{J}_d = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) to the current density.
This term ensures that the divergence of the right-hand side is zero, satisfying charge conservation. Physically, a changing electric field acts as a source for the magnetic field, just as a conduction current does. This symmetry between the two dynamic equations is profound: a changing \(\mathbf{B}\) creates a circulating \(\mathbf{E}\), and a changing \(\mathbf{E}\) creates a circulating \(\mathbf{B}\).
The Unified Maxwell Equations
Combining the static laws (Gauss’s law for electricity and magnetism) with the two new dynamic laws gives the complete set of Maxwell’s equations in vacuum. These four equations are the fundamental field equations of classical electromagnetism.
These equations are local, meaning they hold at every point in space. They determine the fields \(\mathbf{E}\) and \(\mathbf{B}\) given the source distributions \(\rho\) and \(\mathbf{J}\). The structure is symmetric and elegant, revealing that electric and magnetic phenomena are inextricably linked. The constants \(\epsilon_0\) and \(\mu_0\) are not independent; their product determines the speed of light, \(c = 1/\sqrt{\mu_0 \epsilon_0}\), linking electromagnetism to optics.
A common misconception is that the displacement current is a real flow of charge. It is not. It is a term in the Ampère-Maxwell law that accounts for the magnetic field generated by a changing electric field. In the gap of a capacitor, there is no conduction current, but the displacement current density \(\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) is non-zero and produces the same magnetic field as the conduction current in the wire. Confusing this with actual charge transport leads to errors in calculating fields in transient circuits. C
Worked example
Problem. A parallel plate capacitor with plate area \(A = 0.01\ \mathrm{m^2}\) is being charged by a current \(I = 10\ \mathrm{A}\). Calculate the magnitude of the displacement current density \(J_d\) between the plates and the magnetic field \(B\) at a radial distance \(r = 0.01\ \mathrm{m}\) from the centre axis, assuming \(r\) is less than the plate radius.
- Determine the rate of change of the electric field. The electric field between the plates is \(E = \frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}\). The displacement current density is \(J_d = \epsilon_0 \frac{\partial E}{\partial t} = \frac{1}{\epsilon_0 A} \frac{\partial Q}{\partial t} = \frac{I}{A}\).
- Calculate \(J_d\). \(J_d = \frac{10\ \mathrm{A}}{0.01\ \mathrm{m^2}} = 1000\ \mathrm{A/m^2}\).
- Use the Ampère-Maxwell law for a circular Amperian loop of radius \(r\). The total current enclosed is the displacement current through the area \(\pi r^2\): \(I_{enc} = J_d (\pi r^2)\).
- Apply the law: \(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\). Since \(B\) is constant along the loop, \(B (2 \pi r) = \mu_0 J_d \pi r^2\).
- Solve for \(B\): \(B = \frac{\mu_0 J_d r}{2}\).
- Substitute values: \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T \cdot m/A}\). \(B = \frac{(4\pi \times 10^{-7}) (1000) (0.01)}{2}\).
- Calculate: \(B = 2\pi \times 10^{-6}\ \mathrm{T} \approx 6.28 \times 10^{-6}\ \mathrm{T}\).
Units check. \(\mu_0\) is \(\mathrm{T \cdot m/A}\), \(J_d\) is \(\mathrm{A/m^2}\), \(r\) is \(\mathrm{m}\). Product: \((\mathrm{T \cdot m/A}) \cdot (\mathrm{A/m^2}) \cdot \mathrm{m} = \mathrm{T}\). The units are consistent.
Three quick tests
- Why does the displacement current term ensure charge conservation?
Answer
It makes the divergence of the total current density (conduction plus displacement) zero, satisfying the continuity equation \(\nabla \cdot \mathbf{J}_{total} = 0\). - What is the physical significance of the negative sign in Faraday’s law?
Answer
It represents Lenz’s law, indicating that the induced electric field opposes the change in magnetic flux, ensuring energy conservation. - How are the speed of light and the constants \(\epsilon_0\) and \(\mu_0\) related?
Answer
The speed of light \(c\) is given by \(c = 1/\sqrt{\mu_0 \epsilon_0}\), showing that electromagnetic waves propagate at the speed of light in vacuum.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.