Electrodynamics/Maxwell's Four Equations

Introduction

Maxwell's equations along with the Lorentz force completely describe electromagnetism. In previous sections, the electromagnetic theory has been formulated correctly and is ready to be put together all in one place.

Electrodynamics Before Maxwell

1. Gauss's Law

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$

2. No Magnetic Monopole Law

$\nabla \cdot \mathbf {B} =0$

3. Faraday's Law

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

4. Ampère's Law

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J}$

This set of equations represent the state of electromagnetism when James Clark Maxwell started his work. The first equation of the set is Gauss's law. Gauss's law states electric flux begins and ends on charge or at infinity. The second law, which has no name, says magnetic field lines do not begin or end. Faraday' laws states that a changing magnetic field produces an electric field. Last, but not least, Ampère's law states magnetic fields are produced by moving charges.

Electromagnetism as it stood then had a major inconsistency. Upon examination of Maxwell's equations, it is seen that the divergence of the fields are symmetrical. Looking at the curl of the fields, it is seen that Ampère's law may be missing a term. If the divergence of Ampère's law is taken an interesting result arises:

$\nabla \cdot (\nabla \times \mathbf {B} )=\mu _{0}(\nabla \cdot \mathbf {J} )$

The left side of the equation encompasses a well known mathematical fact on divergences and curls: the divergence of a curl is zero. The problem arises with the right side of the equation. In general, the right side is not zero. For steady state currents, the divergence of $\mathbf {J}$ is zero, however outside of magnetostatics; Ampère's law cannot be right. One way to see that Ampère's law fails with non-steady currents is to consider a charging capacitor.

Maxwell's Correction

There is no reason to expect Ampère's law would hold outside of magnetostatics. Ampère's law came from Biot-Savart which applies in the case of steady current. One motivation comes from the continuity equation and Gauss's law. The continuity equation states:

$\nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}$

Using Gauss's law it is trivial to show:

$\nabla \cdot \mathbf {J} =-\nabla \cdot \epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

Adding on this extra term gives Maxwell's correction to Ampère's law. Ampère's law now states:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

Maxwell did not use the continuity equation as a primary motive to arrive at the correction needed for Ampère's law. This modification to Ampère's law does not change anything in regards to magnetostatics. Whenever the electric field is not changing with time, Ampère's law without the additional term is recovered. Maxwell's term was never discovered experimentally by Michael Faraday or others because Maxwell's term is in contest with the current density. Maxwell's term, also coined as the displacement current, brought an aesthetic appeal to the equations. As Faraday's law stated changing magnetic fields gives rise to electric fields, Ampère's law now states changing electric fields induces magnetic fields.

Maxwell's Equations

With the new and improved Ampère's law, it is now time to present all four of Maxwell's equations.

1. Gauss's Law

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$

2. No Magnetic Monopole Law

$\nabla \cdot \mathbf {B} =0$

3. Faraday's Law

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

4. Ampère's Law with displacement current

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

These equations are what are called Maxwell's equations. Maxwell brought these four equations together along with the Lorentz force to completely summarize the theoretical content of electrodynamics. Maxwell's equations tells how charges produce fields and the Lorentz force how fields affect charges.