TERRESTRIAL CURRENTS. 507 



531. TERRESTRIAL CURRENTS. Let us consider, for example, 

 a sphere magnetised uniformly. Let us suppose that a conducting 

 arc, resting with one end at the pole and the other on the 

 equator, turns about the axis with a uniform motion; this arc 

 will cut the same flow of force as if it were applied on the sur- 

 face along a meridian. An element ds, the velocity of which is v, 

 cuts in each unit of time, a flow of force equal to Zvds, Z 

 being the perpendicular component of the magnetic force on the 

 corresponding parallel. If V be the velocity at the equator, F^ 

 the magnetic force at the pole, a the radius of the sphere, and A 

 the latitude of the element ds, we have 



z> = VcosA, Z = FpsinA, ds = ad\. 



The flow of force cut in each unit of time by the whole arc 

 is equal to the electromotive force of induction *?, which gives 



e VFtf sin A cos \d\ - F v Va. 



Jo 2 



If the arc is insulated, this value of e represents the difference 

 of potentials at the two ends. 



If the ends of the arc were connected to two bodies of the 

 same capacity C, these bodies would acquire, after a longer or 

 shorter time, equal and opposite statical charges, the absolute value 



of which would be -Ce. 

 2 



Finally, if the arc were closed by a fixed conductor on the 

 inside or outside of the sphere, the circuit would be traversed by 

 a continuous uniform current, from the equator to the pole or 

 conversely. 



As the Earth may be compared to a sphere magnetised uni- 

 formly, it will be seen that an external arc, which does not share 

 the rotatory motion, should be traversed by an induction current 

 from the equator to the pole, for the direction and magnitude of 

 the induced currents only depend on the relative motion of the arc 

 and of the magnetic system. It is probable that this induction 

 plays an important part in certain natural phenomena, such as 

 the aurora borealis (which seem to be electrical discharges in the 

 upper regions of the atmosphere), the currents observed on the 

 surface of the Earth in telegraphic wires, and the perturbations of 

 the magnetic elements. 



