[NOTES/EM-10009]-A Summary Maxwell's Equations

A summary of how Maxwell's equations for static and time varying field is given. The changes needed from static to time varying case are emphasized.

1. Static fields

In the static situation when the fields, do not vary with time, the electric and magnetic field for charges and currents are described by the following equations. \begin{align} &\text{\underline{Electrostatics}} & \qquad &\nabla\cdot\vec{E}=\rho/\epsilon_0,& \qquad &\nabla\times\vec{E}=0\\ &\text{\underline{Magnetostatics}} & \qquad &\nabla\cdot B=0,& \qquad &\nabla\times\vec{B}=\mu_0\vec{j} \end{align}

2. Time varying fields

Including e.m. induction \noindent When the fields are changing with time, we have the phenomena of induction. A correct description of induction phenomena is obtained by modifying the second equation as indicated below. \begin{equation} \nabla\times\vec{E}=0 \longrightarrow \nabla\times \vec{E} = -\frac{\partial\vec{B}}{\partial t}. \end{equation}

Maxwell's fourth equation

What about the fourth equation? 

Is there a need to change the fourth Maxwell's equation \(\nabla \times \vec{B}=\mu_0 \vec{j}\) for time varying fields? Yes, there is a need to modify it. The equation for static becomes inconsistent with several observations in time varying situations. Maxwell modified the static equation by adding a displacement current in the fourth equation. 

Displacement current

To remove the above mentioned anomalies, Maxwell modified the fourth equation by adding an extra term, which contributes only in time dependent situations. The modified the fourth equation reads as follows.

\begin{align} \nabla\times\vec{B} = \mu_0\vec{j} \longrightarrow \nabla\times\vec{B} = \mu_0\vec{j} \ + \ &\mu_0\epsilon_0 \frac{\partial\vec{E}}{\partial t} \label{EQ04}\\\nonumber & \searrow \text{displacement current} \end{align} This modification is important for time varying fields. It does not contribute when fields are not changing with time. Maxwell called the extra term in EQ04 as \underline displacement current.

References: 

1. R. P. Feynman, Robert B. Leighton and Mathew Sands Lectures on Physics, vol-II, B.I. Publications (1964)
2. David Griffiths, Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).