Skedbooks
←
Physics for Engineers - 2

Electromagnetic Waves

Electromagnetic Waves: In the absence of any source of charge or current, Maxwell's equations in free space are as follows :

The last two equations couple the electric and the magnetic fields. If $\vec B$ is time dependent, $\vec\nabla\times\vec E$ is non-zero. This implies that $\vec E$ is a function of position. Further, if $\partial\vec B/\partial t$ itself changes with time, so does $\vec\nabla\times\vec E$. In such a case $\vec E$ also varies with time since the $\vec\nabla$ operator cannot cause time variation. Thus, in general, a time varying magnetic field gives rise to an electric field which varies both in space and time. It will be seen that these coupled fields propagate in space.
We will first examine whether the equations lead to transverse waves. For simplicity, assume that the electric field has only x-component and the magnetic field only y-component. Note that we are only making an assumption regarding their directions - the fields could still depend on all the space coordinates $x,y,z$, in addition to time $t$.

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.