5.5 Electromagnetic wave review

Review

Lets take some time to review what we have learned so far. We have derived Maxwell's equations in differential form. $${\displaystyle \begin{array}{c}\vec{\nabla }\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\\ \vec{\nabla }\times \vec{B}={\mu }_0\left(\vec{J}+{\epsilon }_0\frac{\partial \vec{E}}{\partial t}\right)\\ \vec{\nabla }\cdot \vec{E}=\frac{\rho }{{\epsilon }_0}\\ \vec{\nabla }\cdot \vec{B}=0\end{array}}$$ These, in general are much more useful than the integral form you learned in Freshman Physics. These allow one to understand the relationship betweenfields, charges and currents as a function of position. This point by point understanding of what is happening is not obvious in the integral form of theequations.

Another interesting point is that if everything is static, that is nothing is changing with time, then theybecome $${\displaystyle \begin{array}{c}\vec{\nabla }\times \vec{E}=0\\ \vec{\nabla }\times \vec{B}={\mu }_0\vec{J}\\ \vec{\nabla }\cdot \vec{E}=\frac{\rho }{{\epsilon }_0}\\ \vec{\nabla }\cdot \vec{B}=0\end{array}}$$ Notice that for static fields, there is no interplay between electricity and magnetism. If there was just electrostatics, then we would have separateelectric and magnetic fields. Maxwell was able to show that the electricity and magnetism are intimately related, and the theory is unified in that youneed both. (To this day the unification of forces is one of the driving principles of a lot of physics research - I would say the only interestingphysics research but that is perhaps because I do it for a living.)

In free space Maxwell's equations become: $${\displaystyle \begin{array}{c}\vec{\nabla }\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\\ \vec{\nabla }\times \vec{B}={\mu }_0{\epsilon }_0\frac{\partial \vec{E}}{\partial t}\\ \vec{\nabla }\cdot \vec{E}=0\\ \vec{\nabla }\cdot \vec{B}=0\end{array}}$$ We then showed that one can take time derivatives and end up with $${\nabla }^2\vec{B}={\mu }_0{\epsilon }_0\frac{{\partial }^2\vec{B}}{\partial t^2}$$ which is the 3d wave equation! Note that is a second time derivative on one side and a second space derivative on the other side, the hallmarks of awave equation.

It was left as an exercise to show that $${\nabla }^2\vec{E}={\mu }_0{\epsilon }_0\frac{{\partial }^2\vec{E}}{\partial t^2}$$ We also see from this equation that the speed of light in vacuum is $$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$$

A plane wave solution to the electromagnetic wave equation for the $\vec{E}$ field is $$\vec{E}(\vec{r},t)=\overrightarrow{E_0}{e}^{i\left(\vec{k}\cdot \vec{r}-\omega t\right)}$$ In vacuum with no currents present we know that: $\vec{\nabla }\cdot \vec{E}=0$ . Recall that earlier we showed $$\vec{\nabla }\cdot \vec{E}=i\vec{k}\cdot \overrightarrow{E_0}{e}^{i\left(\vec{k}\cdot \vec{r}-\omega t\right)}$$ So $$\vec{\nabla }\cdot \vec{E}=i\vec{k}\cdot \overrightarrow{E_0}{e}^{i\left(\vec{k}\cdot \vec{r}-\omega t\right)}=0$$ implies that the $\vec{E}$ associated with our plane wave is perpendicular to its direction of motion.

Likewise $\vec{\nabla }\cdot \vec{B}=0$ implies that the $\vec{B}$ field is also perpendicular to the direction of motion

The electric and magnetic fields have energy and hence have an energy density. In an EM wave $u=u_E+u_B$ which is $u={\epsilon }_0{E}^2$ or equivalently $u=B^2/{\mu }_0\text{.}$ This is all very amazing when you think about it. Maxwell's equations tell us thatwe can have waves in the electric and magnetic fields. These waves carry energy. That is they are a mechanism to transport energy through free space(or a medium). This is why the sun warms us, which is pretty important.