Page:Elementary Text-book of Physics (Anthony, 1897).djvu/361

§ 395] of tubes of induction which pass through circuit 1 in consequence of the current in the other circuit. Then the energy of circuit 1, in consequence of the presence of the other circuit, is $$i_{1}N_{1};$$ the energy of circuit 2, in consequence of the presence of circuit 1, will be similarly $$i_{2}N_{2}.$$ Now since there will be no mutual action between the circuits if either one of them is removed to an infinite distance, the work done in removing one of them is equal to its energy due to the presence of the other; and since, manifestly, the same amount of work is done if either one of the circuits be kept fixed and the other removed to an infinite distance, their energies must be equal, or $$i_{1}N_{1} = i_{2}N_{2}.$$ Now $$N_{1}$$ and $$N_{2}$$ are proportional respectively to the currents in circuits 2 and 1. Let $$M_{1}$$ represent the number of tubes of induction which pass through circuit 1 in consequence of unit current in circuit 2, and $$M_{2}$$ the corresponding number which pass through circuit 2 in consequence of unit current in circuit 1. Then $$i_{1}i_{2}M_{1} = i_{1}i_{2}M_{2}$$ or $$M_{1} = M_{2} = M.$$ The number of tubes of induction which pass through either circuit in consequence of a unit current in the other circuit is the same; the coefficient $$M$$ which expresses this number is called the coefficient of mutual induction.

The energy of two circuits is equal to the energy which they possess due to their own currents, and the energy which each of them possesses due to the current in the other. If $$L$$ and $$N$$ are their coefficients of self-induction and $$M$$ their coefficient of mutual induction, their energy is equal to $$\tfrac{1}{2}Li_{1}^2 + Mi_{1}i_{2} + \tfrac{1}{2}Ni_{2}^2 .$$ This energy may be represented as divided between the two circuits by the equivalent formula $$\tfrac{1}{2}i_{1}(Li_{1} + Mi_{2}) + \tfrac{1}{2}i_{2} (Mi_{1} + Ni_{2}),$$ where the terms represent one half the current in the circuit multiplied by the number of tubes of induction which pass through the circuit.

295. Motion of a Circuit in a Magnetic Field.—The motion of a circuit in a magnetic field, if the current in it be supposed constant, may always be found, from the results of the preceding sections, by the help of the general rule that the motion is such as to make the energy of the circuit as small as possible. In the simple case where the magnetic field is due to a north magnet pole, the