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| History of Theories of Aether and Electricity | |||||||||
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Electric and magnetic Science prior to the introduction of the potentials
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of the magnetic body, so as to be incapable of passing from one
element to the next Suppose that an amount m of the positive magnetic fluid is located at a point (x, y, z) ; the components of the magnetic intensity, or force exerted on unit magnetic pole, at a point (\xi, \eta, \zeta) will evidently be
-m \frac{\partial}{\partial\xi}\big(\frac{1}{r}\big), \: -m \frac{\partial}{\partial\eta}\big(\frac{1}{r}\big), \: -m \frac{\partial}{\partial\zeta}\big(\frac{1}{r}\big),
where r denotes \{(\xi-x)^2 +(\eta - y)^2 + (\zeta - z)^2\}^1/2. Hence if we
consider next a magnetic element in which equal quantities of
the two magnetic fluids are displaced from each other parallel
to the x-axis, the components of the magnetic intensity at
(\xi, \eta, \zeta) will be the negative derivates, with respect to \xi, \eta, \zeta
respectively, of the function
A \frac{\partial}{\partial x}\big(\frac{1}{r}\big)
where the quantity A, which does not involve (\xi, \eta, \zeta), may be
called the magnetic moment of the element : it may be measured
by the couple required to maintain the element in equilibrium
at a definite angular distance from the magnetic meridian. If the displacement of the two fluids from each other in the element is not parallel to the axis of x, it is easily seen that the expression corresponding to the last is
A \frac{\partial}{\partial x}\big(\frac{1}{r}\big) + B \frac{\partial}{\partial y}\big(\frac{1}{r}\big) + C \frac{\partial}{\partial z}\big(\frac{1}{r}\big)
where the vector (A, B, C) now denotes the magnetic moment
of the element. Thus the magnetic intensity at an external point (\xi, \eta, \zeta) due to any magnetic body has the components
\big(-\frac{\partial V}{\partial\xi}, \: -\frac{\partial V}{\partial\eta}, \: -\frac{\partial V}{\partial\zeta}\big),
where
V = \int\int\int \big(A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + C \frac{\partial}{\partial z}\big)\big(\frac{1}{r}\big) dx\,dy\,dz
integrated throughout the substance of the magnetic body, and
where the vector (A, B, C) or I represents the magnetic moment per unit- volume, or, as it is generally called, the magnetization. The function V was afterwards named by Green the magnetic potential. Poisson, by integrating by parts the preceding expression for the magnetic potential, obtained it in the form
V = \int\int \big( \mathbf{I} \cdot \mathbf{dS}\big)\big(\frac{1}{r}\big) - \int\int\int \big(\frac{1}{r}\big)\, div \,\mathbf{I}\, dx\,dy\,dz
[1]
the first integral being taken over the surface S of the magnetic
body, and the second integral being taken throughout its volume.
This formula shows that the magnetic intensity produced by the
body in external space is the same as would be produced by a
fictitious distribution of magnetic fluid, consisting of a layer
over its surface, of surface-charge (\mathbf{I} \cdot \mathbf{dS}) per element dS,
together with a volume-distribution of density - div\, \mathbf{I} throughout its substance.
These fictitious magnetizations are generally
known as Poisson's equivalent surface- and volume-distributions
of magnetism. Poisson, moreover, perceived that at a point in a very small cavity excavated within the magnetic body, the magnetic potential has a limiting value which is independent of the shape of the cavity as the dimensions of the cavity tend to zero ; but that this is not true of the magnetic intensity, which in such a small cavity depends on the shape of the cavity. Taking the cavity to be spherical, he showed that the magnetic intensity within it is
grad \, V + 4/3\pi\,\mathbf{I}
[2]
where \mathbf{I} denotes the magnetization at the place.
This memoir also contains a discussion of the magnetism temporarily induced in soft iron and other magnetizable metals by the approach of a permanent magnet. Poisson accounted for the properties of temporary magnets by assuming that they contain embedded in their substance a great number of small spheres, which are perfect conductors for the magnetic fluids ; so that the resultant magnetic intensity in the interior of one of these small spheres must be zero. He showed that such a sphere, when placed in a field of magnetic intensity \mathbf{F}, [3] must acquire a magnetic moment of amount 3/4 \mathbf{F} x the volume of the sphere, in order to counteract within the sphere the force \mathbf{F}. Thus if k_p denote the total volume of these spheres contained within a unit volume of the temporary magnet, the magnetization will be \mathbf{I}, where
4/3\pi \mathbf(I) = k_p \mathbf{F}
and F denotes the magnetic intensity within a spherical cavity
excavated in the body. This is Poisson s law of induced magnetism. It is known that some substances acquire a greater degree of temporary magnetization than others when placed in the same circumstances : Poisson accounted for this by supposing that the quantity k_p varies from one substance to another. But the experimental data show that for soft iron k_p must have a value very near unity, which would obviously be impossible if k_p is to mean the ratio of the volume of spheres contained within a region to the total volume of the region. [4] The physical interpretation assigned by Poisson to his formulae must therefore be rejected, although the formulae themselves retain their value. Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George Green [5] (b. 1793, d. 1841). Green's treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point : to this function Green gave the name potential, by which it has always since been known. [6] Near the beginning of the memoir is established the celebrated formula connecting surface and volume integrals, which is now generally called Green's Theorem, and of which Poisson's result on the equivalent surface- and volume-distributions of magnetization is a particular application. By using this theorem to investigate the properties of the potential, Green arrived at many results of remarkable beauty and interest. We need only mention, as an example of the power of his method, the following : Suppose that there is a hollow conducting shell, bounded by two closed surfaces, and that a number of electrified bodies are placed, some within and some without it ; and let the inner surface and interior bodies be called the interior system, and the outer surface and exterior bodies be called the exterior system. Then all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth ; and all those of the exterior system will be the same as if the interior system did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity equal to the whole of that originally contained in the shell itself and in all the interior bodies. It will be evident that electrostatics had by this time attained a state of development in which further progress could be hoped for only in the mathematical superstructure, unless experiment should unexpectedly bring to light phenomena of an entirely new character. This will therefore be a convenient place to pause and consider the rise of another branch of electrical philosophy. [1] If the components of a vector \mathbf{a} are denoted by (a_x , a_y , a_z ), the quantity a_x b_x + a_y b_y + a_z b_z is called the scalar product of two vectors \mathbf{a} and \mathbf{b}, and is denoted by (a \dot b). The quantity \frac{\partial a_x}{\partial x} + \frac{\partial a_y}{\partial y} + \frac{\partial a_z}{\partial z} is called the divergence of the vector \mathbf{a}, and is is denoted by div \, \mathbf{a}. [2] The vector whose components are -\frac{\partial V}{\partial x} -\frac{\partial V}{\partial y } - \frac{\partial V}{\partial z} is denoted by grad \mathbf{V}. [3] In the present work, vectors will generally be distinguished by heavy type. [4] This objection was advanced by Maxwell in 430 of his Treatise. An attempt to overcome it was made by Betti : cf. p. 377 of his Lessons on the Potential. [5] An essay on the application of mathematical analysis to the theories of electricity and magnetism, Nottingham, 1828 : reprinted in The Mathematical Papers of the late George Green, p. 1. [6] Euler in 1744 (De methodis inveniendi . . .) had spoken of the vis potentialis what would now be called the potential energy possessed by an elastic body when bent. |