Note 1.
The fundamental electro-magnetic equations of Maxwell for stationary media are:—
curl H = 1/c (∂D/∂t + ρν) (1)
curl E = -1/c ∂B/∂t (2)
div D = ρ
B = μH
div B = 0
D = kE
According to Hertz and Heaviside, these require modification in the case of moving bodies.
Now it is known that due to motion alone there is a change in a vector R given by
(∂R/∂t) due to motion = u. div R + curl [Ru]
where u is the vector velocity of the moving body and [Ru] the vector product of R and u.
Hence equations (1) and (2) become
c curl H = ∂D/∂t + u div D + curl Vect. [Du] + ρν (1·1)
and
-c curl E = ∂B/∂t + u div B + curl Vect. [Bu] (2·1)
which gives finally, for ρ = 0 and div B = 0,
∂D/∂t + u div D = c curl (H - 1/c Vect. [Du]) (1·2)
∂B/∂t = -c curl (E - 1/c Vect. [uB]) (2·2)
Let us consider a beam travelling along the x-axis, with apparent velocity v (i.e., velocity with respect to the fixed ether) in medium moving with velocity ux = u in the same direction.
Then if the electric and magnetic vectors are proportional to eiA(x - vt), we have
∂/∂x = iA, ∂/∂t = -iAv, ∂/∂y = ∂/∂z = 0, uy = uz = 0
Then ∂D_y/∂t = -c∂Hz/∂x - u∂Dy/∂z ... (1·21)
and ∂Bz/∂t = -c∂Ey/∂x - u∂Bz/∂x (2·21)
Since D = KE and B = μH, we have
iAv(κEy) = -ciA(Hz + uKEy) (1·22)
iAv(μHz) = -ciA(Ey + uμHz) (2·22)
or v(K - u)Ey = cHz (1·23)
μ(v - u)Hz = cEy (2·23)
Multiplying (1·23) by (2·23)
μK(v - u)² = c²
Hence (v - u)² = c²/μk = v₀²
∴ v = v₀ + u,
making Fresnelian convection co-efficient simply unity.
Equations (1·21) and (2·21) may be obtained more simply from physical considerations.
According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is δD/δt.
Consider a slab of matter moving with velocity ux along the x-axis, then even in a stationary field of electrostatic polarisation, that is, for a field in which δD/δt = 0, there will be some change in the polarisation of the body due to its motion, given by ux(δD/δx). Hence we must add this term to a purely temporal rate of change δD/δt. Doing this we immediately arrive at equations (1·21) and (2·21) for the special case considered there.
Thus the Hertz-Heaviside form of field equations gives unity as the value for the Fresnelian convection co-efficient. It has been shown in the historical introduction how this is entirely at variance with the observed optical facts. As a matter of fact, Larmor has shown (Aether and Matter) that 1 - 1/μ² is not only sufficient but is also necessary, in order to explain experiments of the Arago prism type.
A short summary of the electromagnetic experiments bearing on this question, has already been given in the introduction.
According to Hertz and Heaviside the total polarisation is situated in the medium itself and is completely carried away by it. Thus the electromagnetic effect outside a moving medium should be proportional to K, the specific inductive capacity.
Rowland showed in 1876 that when a charged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.
Röntgen (Annalen der Physik 1888, 1890) found that if the dielectric is rotated while the condenser remains stationary, the effect is proportional to K - 1.
Eichenwald (Annalen der Physik 1903, 1904) rotated together both condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K. This is of course quite consistent with Rowland and Röntgen.
Blondlot (Comptes Rendus, 1901) passed a current of air in a steady magnetic field Hy, (H = Hz = 0). If this current of air moves with velocity ux along the x-axis, an electromotive force would be set up along the z-axis, due to the relative motion of matter and magnetic tubes of induction. A pair of plates at z = ±a, will be charged up with density ρ = Dz = KE = K. us Hy/c. But Blondlot failed to detect any such effect.
H. A. Wilson (Phil. Trans. Royal Soc. 1904) repeated the experiment with a cylindrical condenser made of ebony, rotating in a magnetic field parallel to its own axis. He observed a change proportional to K — 1 and not to K.
Thus the above set of electro-magnetic experiments contradict the Hertz-Heaviside equations, and these must be abandoned.
[P. C. M.]
Note 2.
Lorentz Transformation.
Lorentz. Versuch einer theorie der elektrischen und optischen Erscheinungen im bewegten Körpern.
(Leiden—1895).
Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also notes 73, 86, pages 318, 328.
Lorentz wanted to explain the Michelson-Morley null-effect. In order to do so, it was obviously necessary to explain the Fitzgerald contraction. Lorentz worked on the hypothesis that an electron itself undergoes contraction when moving. He introduced new variables for the moving system defined by the following set of equations.
x¹ = β(x - ut), y¹ = y, z¹ = z, t¹ = β(t - (u/c²)·x)
and for velocities, used
vx¹ = β²vx + u, vy¹ = βvy, vz¹ = βvz and ρ¹ = ρ/β.
With the help of the above set of equations, which is known as the Lorentz transformation, he succeeded in showing how the Fitzgerald contraction results as a consequence of “fortuitous compensation of opposing effects.”
It should be observed that the Lorentz transformation is not identical with the Einstein transformation. The Einsteinian addition of velocities is quite different as also the expression for the “relative” density of electricity.
It is true that the Maxwell-Lorentz field equations remain practically unchanged by the Lorentz transformation, but they are changed to some slight extent. One marked advantage of the Einstein transformation consists in the fact that the field equations of a moving system preserve exactly the same form as those of a stationary system.
It should also be noted that the Fresnelian convection coefficient comes out in the theory of relativity as a direct consequence of Einstein’s addition of velocities and is quite independent of any electrical theory of matter.
[P. C. M.]