Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/236

to make the instrument very sensible to the action of the force if it exists.

The chief difficulty in the experiments arose from the disturbing action of the earth s magnetic force, which caused the electromagnet to act like a dip-needle. The results obtained were on this account very rough, but no evidence of any change in θ could be obtained even when an iron core was inserted in the coil, so as to make it a powerful electromagnet.

If, therefore, a magnet contains matter in rapid rotation, the angular momentum of this rotation must be very small compared with any quantities which we can measure, and we have as yet no evidence of the existence of the terms T_me derived from their mechanical action.

576.] Let us next consider the forces acting on the currents of electricity, that is, the electromotive forces.

Let Y be the effective electromotive force due to induction, the electromotive force which must act on the circuit from without to balance it is Y' = -Y, and, by Lagrange s equation,

$Y=-Y'=-{\frac {d}{dt}}{\frac {dT}{d{\dot {y}}}}+{\frac {dT}{dy}}.$

Since there are no terms in T involving the coordinate y, the second term is zero, and Y is reduced to its first term. Hence, electromotive force cannot exist in a system at rest, and with constant currents.

Again, if we divide Y into three parts, Y_m, Y_e, Y_me, corresponding to the three parts of T, we find that, since T_m does not contain ${\dot {y}}$, Y_m = 0.

We also find

$Y_{e}=-{\frac {d}{dt}}{\frac {dT_{e}}{d{\dot {y}}}}.$

Here ${\frac {dT_{e}}{d{\dot {y}}}}$ is a linear function of the currents, and this part of the electromotive force is equal to the rate of change of this function. This is the electromotive force of induction discovered by Faraday. We shall consider it more at length afterwards.

577.] From the part of T, depending on velocities multiplied by currents, we find

$Y_{me}=-{\frac {d}{dt}}{\frac {dT_{me}}{d{\dot {y}}}}.$

Now ${\frac {dT_{me}}{d{\dot {y}}}}$ is a linear function of the velocities of the conductors. If, therefore, any terms of T_me have an actual existence, it would be possible to produce an electromotive force independently of all existing currents by simply altering the velocities of the conductors.