[NOTES/EM-09006]-Faraday's Law and Maxwell's Second Equation

1.Time varying case

Faraday's induction law \begin{eqnarray} \nabla\times\bar{E}=0 \label{eq15} \end{eqnarray} gets modified to \begin{eqnarray} \nabla\times\bar{E}=-\frac{\partial\bar{B}}{\partial{t}} \label{eq16} \end{eqnarray} It implies the flux rule of induction (when applicable) induces emf which is equal to the rate of change of flux. Mathematically ;\\ \begin{eqnarray} emf=-\frac{\partial\phi}{\partial{t}} \label{eq17} \end{eqnarray} This gives the Lenz law.\\ \\ The fourth Maxwell's equation in static case \begin{eqnarray} \nabla\times\bar{B}=\mu_0\bar{j} \label{eq18} \end{eqnarray} gets modified to \begin{eqnarray} \nabla\times\bar{B}=\mu_0\bar{j}+\mu_0\epsilon_0\frac{\partial{\bar{E}}}{\partial{t}} \label{eq19} \end{eqnarray} The justification for this are purely theoretical and comes from charge conservation. The equation of continuity represents the law of charge conservation in a differential form. The second term on the right hand side is called \underline{displacement current}. Addition of this term is Maxwell's contribution and led to many other phenomenons. This opens the door from electricity and magnetism to light.