DIELECTRIC DISPLACEMENT CURRENT Electromagnetic Induction.—Faraday found that, when a magnet is moved near a circuit so as to change the number of unit tubes of magnetic force passing through the circuit, a current is induced in the circuit. In fig. 20 let ABC be a wire circuit, and let there be a magnetic pole of strength in at a point P. Suppose the pole to be moving towards the circuit along any path QPO, and let the motion be adjusted so that a constant current C is induced in the circuit. If the solid angle subtended by the circuit at P is denoted by w, then the power, or work per unit time, required to keep the pole moving is equal to mCdw/dt, where I denotes the time, because the work required to bring up a unit pole from a great distance to P is equal to Cw, when the current C is constant. The electrical power de veloped in the circuit is equal to EC, where E is the electromotive force induced in the circuit. Hence we have EC = mCdw/dt, or E = rndw/ dt. If the magnetic permeability of the medium is µ, then the field strength F due to the pole is at a distance r from the pole. The number of unit tubes of magnetic force coming out of the pole is equal to 47rm, since the cross section of a unit tube is equal to z/uF and = (1/µF) = = 47rsn. The number of unit tubes from the pole passing through the circuit is therefore w/47r, or mw; so that if N= —mw, then mdw/dt = —dN/dt. The electromotive force induced in the circuit is therefore equal to —dN/dt. We conclude that whenever the number N of unit tubes of magnetic force passing through a circuit is changing, there is an induced electromotive force in the circuit equal to —dN/cit.
When a current C is flowing round a circuit the tubes of force of the field due to the current pass through the circuit. The strength of the field is proportional to the current, so that the number N of unit tubes passing through the circuit due to the current is proportional to the current. If then S denotes the number of unit tubes passing through the circuit due to unit current in it we have N =SC. The constant S is called the self-induction or the inductance of the circuit. When the current C varies, there is an induced electromotive force in the circuit equal to —dN/dt or to —Sdc/dt. If an electromotive force E is applied to the circuit, by means of a battery or otherwise, the current will be given by the equation C= (E—SdC/dt)/R, where R is the resistance of the circuit. The induced electro motive force in a circuit is taken equal to —dN/dt, instead of +dN/dt, because the induced electromotive force of self induction opposes the variations of the current; i.e., when an electromotive force E is applied to a circuit in which the current is zero, the current increases with the time and the induced electro motive force is in the opposite direction to the applied electro motive force E. The solution of the equation when E is constant and C = o at t = o, is C = (I — i,s ) . The current therefore increases with the time at a diminishing rate, Ee Kris and finally becomes equal to E/R. dC/dt is equal to and so at the start is equal to E/S. If, while the steady current E/R is flowing, the electromotive force E is suddenly reduced to zero, we have SdC/dt+CR = o. The solution of this equation is C= /S, where t is reckoned from the instant at which E is reduced to zero. In this case the current dies away and finally becomes zero.
The practical unit of self-induction is called a henry, of ter the American physicist Joseph Henry (q.v.), and is a self induction such that the induced electromotive force is equal to one volt when the current is changing at the rate of one ampere per second. Since the volt is equal to electromagnetic units of potential difference and the ampere to electromagnetic units of cur rent, it follows that the henry is equal to electromagnetic units of self-induction. If one volt is applied to a circuit of one Ohm resistance and one henry inductance, then C= i so that after one second C=0.6321 amperes, and of ter io seconds amperes.