Page:On Faraday's Lines of Force.pdf/72

226 The equivalent currents are $$\frac{1}{2}I_0$$, and $$\frac{1}{2}I_0\frac{R}{R'}\frac{n'}{n}$$, and their duration is $$\tau$$.

When the communication with the source of the current is cut off, there will be a change of $$R$$. This will produce a change in the value of $$\tau$$, so that if $$R$$ be suddenly increased, the strength of the secondary current will be increased, and its duration diminished. This is the case in the ordinary coil-machines. The quantity N depends on the form of the machine, and may be determined by experiment for a machine of any shape.

XII. Spherical shell revolving in magnetic ﬁeld.

Let us next take the case of a revolving shell of conducting matter under the inﬂuence of a uniform ﬁeld of magnetic force. The phenomena are explained by Faraday in his Experimental Researches, Series II., and references are there given to previous experiments.

Let the axis of $$z$$ be the axis of revolution, and let the angular velocity be $$\omega$$. Let the magnetism of the ﬁeld be represented in quantity by $$I$$, inclined at an angle $$\theta$$ to the direction of $$z$$, in the plane of $$zx$$.

Let $$R$$ be the radius of the spherical shell, and $$T$$ the thickness. Let the quantities $$\alpha_0, beta_0, \gamma_0$$ be the electro-tonic functions at any point of space; $$a_1, b_1, c_1, \alpha_1, \beta_1, \gamma_1$$ symbols of magnetic quantity and intensity; $$a_2, b_2, c_2, \alpha_2, \beta_2, \gamma_2$$ of electric quantity and intensity. Let $$p_2$$ be the electric tension at any point,




 * $$\alpha_2=\frac{dp_2}{dx}+ka_2$$
 * rowspan=3| ....................(1),
 * $$\beta_2=\frac{dp_2}{dy}+kb_2$$
 * $$\gamma_2=\frac{dp_2}{dz}+kc_2$$
 * }
 * $$\gamma_2=\frac{dp_2}{dz}+kc_2$$
 * }


 * $$\frac{da_2}{dx}+\frac{db_2}{dy}+\frac{dc_2}{dz}=0$$..............(2);


 * $$\therefore \frac{d\alpha_2}{dx}+\frac{d\beta_2}{dy}+\frac{d\gamma_2}{dz}=\nabla^2p.$$