Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/368

This page has been proofread, but needs to be validated.
348
The Followers of Maxwell.

or the same way as dynamical energy is carried by water flowing in a pipe; whereas in Maxwell's theory, the storehouse and vehicle of energy is the dielectric medium surrounding the wire. What Poynting achieved was to show that the flux of energy at any place might be expressed by a simple formula in terms of the electric and magnetic forces at the place.

Denoting as usual by E the electric force, by D the electric displacement, by H the magnetic force, and by B the magnetic induction, the energy stored in unit volume of the medium is[1]

;

so the increase of this in unit time is (since in isotropic media D is proportional to E, and B is proportional to H)

or , where S denotes the total current, and ι the current of conduction; or (in virtue of the fundamental electromagnetic equations)

,

or . Now (E. ι) is the amount of electric energy transformed into heat per unit volume per second; and therefore the quantity -(1/4π) div [E.H] must represent the deposit of energy in unit volume per second due to the streaming of energy; which shows that the flux of energy is represented by the vector -(1/4π) div [E.H].[2] This is Poynting's theorem: that the flux of energy at any place is represented by the vector-product of the electric and magnetic forces, divided by 4π.[3]

  1. Cf. pp. 248, 250, 282.
  2. Of course any circuital vector may be added. II. M. Macdonald, Electric Waves, p. 72, propounded a form which differs from Poynting's by a non-circuital vector.
  3. The analogue of Poynting's theorem in the theory of the vibrations of an isotropic elastic solid may be easily obtained; for from the equation of motion of an elastic solid,

    ,

    it follows that

    ,