[NOTES/EM-10006]-Maxwell's Fourth Equation, Displacement Current

We discuss how Maxwell's addition of a displacement current in the fourth equation \begin{equation} \nabla \times \vec{B} = \mu_0 \vec{j} + \epsilon_0\mu_0 \pp[\vec{E}]{t}. \end{equation} resolves problem with the charge conservation.

Charge conservation and displacement current

1.Problem without displacement current

Taking divergence of the fourth equation, we get \begin{equation} \nabla \cdot\vec{j} =0. \end{equation} This together with the equation of continuity implies \[\pp[\rho]{t}=0\], which means that the charge density, must always be a constant. This need not be true in time varying situations.

2 Charge conservation --- restored

Current conservation $\frac{\partial\rho}{\partial t} + \nabla\cdot\vec{j} = 0$ is not satisfied, because $\nabla\cdot\vec{j}=0$ and $\frac{\partial\rho}{\partial t} \ne 0$ \noindent Maxwell's displacement current term restores current conservation. To see this we take the divergence of the fourth equation \begin{align*} \nabla\times\vec{B} = &\mu_0\Big(\vec{j}+\epsilon_0 \frac{\partial\vec{E}}{\partial t}\Big) \end{align*} and use \(\nabla\cdot(\nabla\times\vec{B})=0\)to get \begin{align*} \nabla\cdot\vec{j}+ \epsilon_0 \nabla\cdot\frac{\partial\vec{E}} {\partial t}=0&\\ \nabla\cdot\vec{j}+ \epsilon_0 \frac{\partial}{\partial t} \nabla\cdot\vec{E}=&0\\ \nabla\cdot\vec{j}+ \epsilon_0 \frac{\partial}{\partial t}\left(\frac{\rho}{\epsilon_0}\right)=&0 \end{align*} Therefore, we get the equation of continuity: \begin{equation} \nabla\cdot\vec{j}+\frac{\partial\rho}{\partial t} = 0 \end{equation} and the charge conservation holds.

References

1. Sec 18-1 Maxwell's Equations Sec 18-2 How the New Term WorksR. P. Feynman, Robert B. Leighton and Mathew Sands  Lectures on Physics, vol-II, B.I. Publications (1964)
2. Sec 7.3.1 Electrodynamics Before Maxwell  Sec 7.3.2 How Maxwell Fixed Ampere's LawSec 7.3.3  Maxwell's Equations David Griffiths, Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).