Module 3 : MAGNETIC FIELD
Lecture 20 : Magnetism in Matter
  Gauss's law gives
 
\begin{displaymath}\vec\nabla\cdot\vec E = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}=0\end{displaymath}
  Since only $E_x\ne 0$, this implies
 
\begin{displaymath}\frac{\partial E_x}{\partial x} = 0\end{displaymath}
  Thus $E_x$ is independent of $x$ coordinate and can be written as $E_x(y,z,t)$. A similar analysis shows that $B_y$ is independent of $y$ coordinate and can be written explicitly as $B_y(x,z,t)$.
  Consider now the time dependent equations eqns. (3) and (4). The curl equation for $\vec B$ gives, taking z-component
 
\begin{displaymath}\frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}= \frac{\partial E_z}{\partial t}=0 \end{displaymath}
  Since $B_x=0$, this gives
 
\begin{displaymath}\frac{\partial B_y}{\partial x} = 0\end{displaymath}
  showing that $B_y$ is independent of $x$ and hence depends only on $z$ and $t$. In a similar manner we can show that $E_x$ also depends only on $z$ and $t$. Thus the fields $\vec E$ and $\vec B$ do not vary in the plane containing them. Their only variation takes place along the z-axis which is perpendicular to both $\vec E$ and $\vec B$. The direction of propagation is thus $z-$direction.

 
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