Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/261

We may call these the equations of magnetization, and they indicate that in the electromagnetic system the magnetic induction ${\mathfrak {B}}$, considered as a vector, is the sum, in the Hamiltonian sense, of two vectors, the magnetic force ${\mathfrak {H}}$, and the magnetization ${\mathfrak {J}}$, multi plied by 4 π, or

${\mathfrak {B}}={\mathfrak {H}}+4\pi {\mathfrak {J}}.$

In certain substances, the magnetization depends on the magnetic force, and this is expressed by the system of equations of induced magnetism given at Arts. 426 and 435.

606.] Up to this point of our investigation we have deduced everything from purely dynamical considerations, without any reference to quantitative experiments in electricity or magnetism. The only use we have made of experimental knowledge is to recognise, in the abstract quantities deduced from the theory, the concrete quantities discovered by experiment, and to denote them by names which indicate their physical relations rather than their mathematical generation.

In this way we have pointed out the existence of the electromagnetic momentum ${\mathfrak {A}}$ as a vector whose direction and magnitude vary from one part of space to another, and from this we have deduced, by a mathematical process, the magnetic induction, ${\mathfrak {B}}$, as a derived vector. We have not, however, obtained any data for determining either ${\mathfrak {A}}$ or ${\mathfrak {B}}$ from the distribution of currents in the field. For this purpose we must find the mathematical connexion between these quantities and the currents.

We begin by admitting the existence of permanent magnets, the mutual action of which satisfies the principle of the conservation of energy. We make no assumption with respect to the laws of magnetic force except that which follows from this principle, namely, that the force acting on a magnetic pole must be capable of being derived from a potential.

We then observe the action between currents and magnets, and we find that a current acts on a magnet in a manner apparently the same as another magnet would act if its strength, form, and position were properly adjusted, and that the magnet acts on the current in the same way as another current. These observations need not be supposed to be accompanied with actual measurements of the forces. They are not therefore to be considered as furnishing numerical data, but are useful only in suggesting questions for our consideration.

The question these observations suggest is, whether the magnetic field produced by electric currents, as it is similar to that produced