783.]
PROPAGATION OF ELECTROMAGNETIC DISTURBANCES.
385
Let
be the specific conductivity of the medium,
its specific capacity for electrostatic induction, and
its magnetic permeability.
To obtain the general equations of electromagnetic disturbance, we shall express the true current
in terms of the vector potential
and the electric potential
.
The true current
is made up of the conduction current
and the variation of the electric displacement
, and since both of these depend on the electromotive force
, we find, as in Art. 611,
|
![{\displaystyle {\mathfrak {C}}=\left(C+{\frac {1}{4\pi }}K{\frac {d}{dt}}\right){\mathfrak {E}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc2e6f5bb5df1e4cbb68fdd45896e85d24d85dd) | (1) |
But since there is no motion of the medium, we may express the electromotive force, as in Art. 599,
|
![{\displaystyle {\mathfrak {E}}=-{\dot {\mathfrak {A}}}-\nabla \Psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcd9d1f8076db61c07752ba89ce68aee61cefb1) | (2) |
Hence |
![{\displaystyle {\mathfrak {C}}=-\left(C+{\frac {1}{4\pi }}K{\frac {d}{dt}}\right)\left({\frac {d{\mathfrak {A}}}{dt}}+\nabla \Psi \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca667a3242a10a72eec021b4cef1d03888af8f8) | (3) |
But we may determine a relation between
and
in a different way, as is shewn in Art. 616, the equations (4) of which may be written
|
![{\displaystyle 4\pi \mu {\mathfrak {C}}=\nabla ^{2}{\mathfrak {A}}+\nabla J,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fb2594bac5d38aafce74b42937e0d1fde4e187) | (4) |
where |
![{\displaystyle J={\frac {dF}{dx}}+{\frac {dG}{dy}}+{\frac {dH}{dz}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d026763a77534eb3a7bc85c559b31af89f168e80) | (5) |
Combining equations (3) and (4), we obtain
|
![{\displaystyle \mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {d{\mathfrak {A}}}{dt}}+\nabla \Psi \right)+\nabla ^{2}{\mathfrak {A}}+\nabla J=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c3b869e9787d7187a314f27c2f6e0030686237) | (6) |
which we may express in the form of three equations as follows―
|
![{\displaystyle \left.{\begin{aligned}\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dF}{dt}}+{\frac {d\Psi }{dx}}\right)+\nabla ^{2}F+{\frac {dJ}{dx}}&=0,\\\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dG}{dt}}+{\frac {d\Psi }{dy}}\right)+\nabla ^{2}G+{\frac {dJ}{dy}}&=0,\\\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dH}{dt}}+{\frac {d\Psi }{dz}}\right)+\nabla ^{2}H+{\frac {dJ}{dz}}&=0.\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908bb1669e43861b74f411b93e7705b23c3cf2e8) | (7) |
These are the general equations of electromagnetic disturbances.
If we differentiate these equations with respect to
,
and
respectively, and add, we obtain
|
![{\displaystyle \mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dJ}{dt}}-\nabla ^{2}\Psi \right)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba57aa53b525dacdddbfa19df79bab90f5dc5173) | (8) |
If the medium is a non-conductor,
, and
, which is proportional to the volume-density of free electricity, is independent of
. Hence
must be a linear function of
, or a constant, or zero, and we may therefore leave
and
out of account in considering periodic disturbances.