Page:PoyntingTransfer.djvu/18

This at once gives us the magnetic energy equal to the electric energy, for

${\frac {\mu {\mathfrak {H}}^{2}}{8\pi }}={\frac {\mu K^{2}v^{2}{\mathfrak {E}}^{2}}{8\pi }}={\frac {K{\mathfrak {E}}^{2}}{8\pi }}$

It may be noted that the velocity ${\tfrac {1}{\sqrt {\mu K}}}$ is the greatest velocity with which the two energies can be propagated together, and that they must be equal when travelling with this velocity. For if $v$ be the velocity of propagation and $\theta$ the angle between the two intensities, we have

${\frac {{\mathfrak {EH}}\sin \theta }{4\pi v}}={\frac {K{\mathfrak {E}}^{2}}{8\pi }}+{\frac {\mu {\mathfrak {H^{2}}}}{8\pi }}$

or

$v={\frac {2\sin \theta }{{\frac {K{\mathfrak {E}}}{\mathfrak {H}}}+{\frac {\mu {\mathfrak {H}}}{\mathfrak {E}}}}}$

The greatest value of the numerator is 2 when $\theta$ is a right angle, and the least value of the denominator is $2{\sqrt {\mu K}}$, when the two terms are equal to each other and to ${\sqrt {\mu K}}$.

The maximum value of $v$ therefore is ${\tfrac {1}{\sqrt {\mu K}}}$, and occurs when $\theta ={\frac {\pi }{2}}$ and $K{\mathfrak {E}}^{2}=\mu {\mathfrak {H^{2}}}$.

---

The preceding examples will suffice to show that it is easy to arrange some of the known experimental facts in accordance with the general law of the flow of energy. I am not sure that there has hitherto been any distinct theory of the way in which the energy developed in various parts of the circuit has found its way thither, but there is, I believe, a prevailing and somewhat vague opinion that in some way it has been carried along the conductor by the current. Probably Maxwell's use of the term "displacement" to describe one of the factors of the electric energy of the medium has tended to support this notion. It is very difficult to keep clearly in mind that this "displacement" is, as far as we are yet warranted in describing it, merely a something with direction which has some of the properties of an actual displacement in incompressible fluids or solids. When we learn that the "displacement" in a conductor having a current in it increases continually with the time, it is almost impossible to avoid picturing something moving along the conductor, and it then seems only natural to endow this something with energy-carrying power. Of course it may turn out that there is an actual displacement along the lines of electromotive intensity. But it is quite as likely that the electric "displacement" is only a function of the true displacement, and it is conceivable that many theories may be formed in which this is the case, while they may all account for the observed facts. Mr. Glazebrook has already worked out one such theory in which the component of the electric displacement