Module 3 : MAGNETIC FIELD
Lecture 20 : Magnetism in Matter
 
\begin{displaymath}\frac{\partial}{\partial x}f(\vec r, t) = \frac{\partial}{\partial x}\exp(ik_xx+ik_yy+ik_zz-i\omega t)= ik_xf(\vec r, t)\end{displaymath}
  Since
 
\begin{displaymath}\nabla = \hat\imath\frac{\partial}{\partial x}+\hat\jmath \frac{\partial}{\partial y}+\hat k\frac{\partial}{\partial z}\end{displaymath}
  we have,
 
\begin{displaymath}\nabla f(\vec r, t) = i\vec kf(\vec r, t)\end{displaymath}
  In a similar way,
 
\begin{displaymath}\frac{\partial}{\partial t}f(\vec r, t) = -i\omega f(\vec r,t)\end{displaymath}
  Thus for our purpose, the differential operators $\nabla$ and $\partial/\partial t$ may be equivalently replaced by
 
\begin{eqnarray*} \frac{\partial}{\partial t} &\rightarrow& -i\omega\\ \vec\nabla &\rightarrow& i\vec k \end{eqnarray*}
  Using these, the Maxwell's equations in free space become

 
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