Page:Thomson1881.djvu/8

 or they may also be written

{{MathForm2|(4)|$$\left.\begin{array}{ll} \alpha & =\frac{\mu e}{R^{3}}(q(z-\zeta)-r(y-\eta)),\\ \\\beta & =\frac{\mu e}{R^{3}}(r(x-\xi)-p(z-\zeta)),\\ \\\gamma & =\frac{\mu e}{R^{3}}(p(y-n)-q(x-\xi)),\end{array}\right\} $$}}

Comparing these expressions with those given by Ampere for the magnetic force produced by a current, we see that the magnetic force due to the moving sphere is the same as that produced per unit length of a current whose intensity is $$\mu e\sqrt{p^{2}+q^{2}+r^{2}}$$, situated at the centre of the sphere, the direction of the positive current coinciding with the direction of motion of the sphere. The resultant magnetic force produced by the sphere at any point is $$\omega\mu e\sin\epsilon/\rho^{2}$$, ω being the velocity of the sphere, and ε the angle between the direction of motion of the sphere and the radius vector ρ drawn from the centre of the sphere to the point; the direction of the force is perpendicular both to the direction of motion of the sphere and the radius vector from the centre of the sphere to the point; and the direction of the force and the direction of motion are related to each other like translation and rotation in a right-handed screw.

It may be useful to form a rough numerical idea of the magnitude of the greatest magnetic force which could be produced by a moving charged sphere. The greatest value of the force $$=\mu KVa\omega/\rho^{2}$$, where a is the radius and V the potential of the sphere. Now if F be the greatest electric force which can exist without discharge, the greatest value of V is Fa. According to Mr. Macfarlane's experiments F is, roughly speaking, about $$3\times10^{12}$$, $$\mu K=\tfrac{1}{9\times10^{20}}$$; substituting these values, the greatest value of the magnetic force becomes $$\tfrac{1a^{2}\omega}{3p^{2}10^{8}}$$. Now $$\frac{a}{\rho}$$ cannot be greater than unity; so the greatest value of the force is $$\omega/3\times10^{8}$$. If the sphere were attached to an arm of such length that it described a metre in each complete revolution of the arm, and if the arm were to make 100 revolutions a second, ω would equal 104, and the greatest magnetic force would be $$1/3\times10^{4}=.000033$$. Prof. Rowland, in his experiments on the magnetic effects of electric convection, measured a magnetic force only about one tenth of this.