7.1: Maxwell's Equations

Displacement Current Correction to Ampere's Law

In the historical development of electromagnetic field theory through the nineteenth century, charge and its electric field were studied separately from currents and their magnetic fields. Until Faraday showed that a time varying magnetic field generates an electric field, it was thought that the electric and magnetic fields were distinct and uncoupled. Faraday believed in the duality that a time varying electric field should also generate a magnetic field, but he was not able to prove this supposition.

It remained for James Clerk Maxwell to show that Fara­ day's hypothesis was correct and that without this correction Ampere's law and conservation of charge were inconsistent:

\[ \nabla\times \textbf{H}= \textbf{J}_{f}\Rightarrow \nabla\cdot \textbf{J}_{f}=0 \]

\[ \nabla\cdot \textbf{J}_{f}+\frac{\partial \rho_{f}}{\partial t}=0 \]

for if we take the divergence of Ampere's law in (1), the current density must have zero divergence because the divergence of the curl of a vector is always zero. This result contradicts (2) if a time varying charge is present. Maxwell realized that adding the displacement current on the right-hand side of Ampere's law would satisfy charge conservation, because of Gauss's law relating \(\textbf{D}\) to \(\rho_{f}\left ( \nabla\times \textbf{D} =\rho_{f}\right )\).

This simple correction has far-reaching consequences, because we will be able to show the existence of electro­ magnetic waves that travel at the speed of light \(c\), thus proving that light is an electromagnetic wave. Because of the significance of Maxwell's correction, the complete set of coupled electromagnetic field laws are called Maxwell's equations:

Faraday's Law

\[ \nabla\times\textbf{E}=-\frac{\partial \textbf{B}}{\partial t}\Rightarrow \oint_{\textbf{L}}\textbf{E}\cdot \textbf{dl}=-\frac{d}{dt}\int_{S}\textbf{B}\cdot \textbf{dS} \]

Ampere's law with Maxwell's displacement current correction

\[ \nabla\times \textbf{H}= \textbf{J}_{f}+\frac{\partial \textbf{D}}{\partial t}\Rightarrow \oint_{\textbf{L}}\textbf{H}\cdot \textbf{dl}=\int_{\textbf{S}}\textbf{J}_{f}\cdot \textbf{dS}+\frac{d}{dt}\int_{\textbf{S}}\textbf{D}\cdot \textbf{dS}  \]

Gauss's laws 

\[ \nabla\times \textbf{D}=\rho_{f}\Rightarrow \oint_{\textbf{S}}\textbf{D}\cdot \textbf{dS}=\int_{\textbf{V}}\rho_{f}dV \]

\[ \nabla\times \textbf{B}=0\Rightarrow \oint_{\textbf{S}}\textbf{B}\cdot \textbf{dS}=0 \]

Conservation of charge

\[ \oint_{\textbf{S}}\textbf{J}_{f}\cdot \textbf{dS}+\frac{d}{dt}\int_{\textbf{V}}\rho_{f}dV=0 \]

As we have justified, (7) is derived from the divergence of (4) using (5).

Note that (6) is not independent of (3) for if we take the divergence of Faraday's law, \(\nabla\cdot \textbf{B}\) could at most be a time-independent function. Since we assume that at some point in time \(\textbf{B}=0\), this function must be zero.

The symmetry in Maxwell's equations would be complete if a magnetic charge density appeared on the right-hand side of Gauss's law in (6) with an associated magnetic current due to the flow of magnetic charge appearing on the right-hand side of (3). Thus far, no one has found a magnetic charge or current, although many people are actively looking. Throughout this text we accept (3)-(7) keeping in mind that if magnetic charge is discovered, we must modify (3) and (6) and add an equation like (7) for conservation of magnetic charge.

Circuit Theory as a Quasi-static Approximation

Circuit theory assumes that the electric and magnetic fields are highly localized within the circuit elements. Although the displacement current is dominant within a capacitor, it is negligible outside so that Ampere's law can neglect time vari­ations of \(\textbf{D}\) making the current divergence-free. Then we obtain Kirchoff's current law that the algebraic sum of all currents flowing into (or out of) a node is zero:

\[ \nabla\cdot\textbf{J}=0\Rightarrow \oint_{\textbf{S}}\textbf{J}\cdot \textbf{dS}=0\Rightarrow \sum i_{k}=0 \]

Similarly, time varying magnetic flux that is dominant within inductors and transformers is assumed negligible outside so that the electric field is curl free. We then have Kirchoff's voltage law that the algebraic sum of voltage drops (or rises) around any closed loop in a circuit is zero:

\[ \nabla\times\textbf{E}=0\Rightarrow \textbf{E}=-\nabla V\Rightarrow \oint_{\textbf{L}}\textbf{E}\cdot \textbf{dl}=0\Rightarrow \sum v_{k}=0 \]