Page:Elementary Text-book of Physics (Anthony, 1897).djvu/287

§ 248] through the rest of the closed surface from within outward. The tubes of force of the magnet are therefore closed tubes, passing through the magnet in the manner already described. The tubes thus considered, in which account is taken of the effect of the distributions on the disk-shaped cavity within the magnet, are called tubes of induction.

If the magnet be a bar placed with its axis along the lines of force in a uniform magnetic field and magnetized by induction, the induction (§ 246) is equal to $$F = R + 4\pi I.$$ If the magnetization of the bar be proportional to the force of the field, so that $$I = kR,$$ we have The number of tubes of induction which pass through unit area in a cross-section of the bar is equal to this, for the total number of tubes that pass through the section of the magnet is $$(R + 4\pi I)a;$$ that is, $$N = \mu R.$$

The coefficient $$k$$ is called the coefficient of induced magnetization; it is assumed to be zero for a vacuum, and may be either positive or negative. The coefiicient $$\mu = 1 + 4\pi k$$ is called the magnetic inductive capacity or the magnetic permeability. It must be noticed that when $$k > 0,$$ so that the induction is greater than the magnetic force of the field, the resultant magnetic force within the body is less than the magnetic force of the field, because the poles induced in the body act in the opposite sense to the force of the field.

248. Energy in a Magnetic Field.—On the view we are now taking, that the actions between magnets are due to a condition of the medium which occupies the field, it is natural to suppose that the energy of a set of magnets is distributed in the field. We will find a law for this distribution, which associates the energy with the tubes of induction.

The energy of the system is manifestly equal to the work that would be required to construct that system. We will first show that this may be expressed, in terms of the magnet poles and of the potentials of the places occupied by them, by the formula $$\textstyle \sum \displaystyle \tfrac{1}{2}mV.$$