Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/256

If we now put

$-\Psi '=F{\frac {\delta x}{\delta t}}+G{\frac {\delta y}{\delta t}}+H{\frac {\delta z}{\delta t}},$

(6)

${\frac {dF'}{dt}}=-{\frac {d\Psi '}{dx}}-c{\frac {\delta y}{\delta t}}+b{\frac {\delta z}{\delta t}}+{\frac {dF}{dt}}.$

(7)

The equation for P, the component of the electromotive force parallel to x, is, by (B),

$P=c{\frac {\delta y}{\delta t}}-b{\frac {\delta z}{\delta t}}-{\frac {dF}{dt}}-{\frac {d\Psi }{dx}},$

(8)

referred to the fixed axes. Substituting the values of the quantities as referred to the moving axes, we have

$P'=c{\frac {\delta y'}{\delta t}}-b{\frac {\delta z'}{\delta t}}-{\frac {dF'}{dt}}-{\frac {d(\Psi +\Psi ')}{dx}},$

(8)

for the value of P referred to the moving* axes.

601.] It appears from this that the electromotive force is expressed by a formula of the same type, whether the motions of the conductors be referred to fixed axes or to axes moving in space, the only difference between the formulae being that in the case of moving axes the electric potential Ψ must be changed into Ψ + Ψ'.

In all cases in which a current is produced in a conducting circuit, the electromotive force is the line-integral

$E=\int {\left(P{\frac {dx}{ds}}+Q{\frac {dy}{ds}}+R{\frac {dz}{ds}}\right)ds}.$

(10)

taken round the curve. The value of Ψ disappears from this integral, so that the introduction of Ψ' has no influence on its value. In all phenomena, therefore, relating to closed circuits and the currents in them, it is indifferent whether the axes to which we refer the system be at rest or in motion. See Art. 668.

On the Electromagnetic Force acting on a Conductor which carries an Electric Current through a Magnetic Field.

602.] We have seen in the general investigation, Art. 583, that if x₁ is one of the variables which determine the position and form of the secondary circuit, and if X₁ is the force acting on the secondary circuit tending to increase this variable, then

$X_{1}={\frac {dM}{dx_{1}}}i_{1}i_{2}.$

(1)

Since $i_{1}$ is independent of $x_{1}$, we may write

$Mi_{1}=p=\int {\left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds},$

(2)