Momentum in the Electric Field. 337 
Hence the momentum contained in the space between the 
surface of the moving charged sphere (whose radius we 
represent by a) and a concentric sphere of radius R is equi- 
valent to a momentum through the centre of the sphere 
parallel to z, the direction of motion of the sphere, and equal 
to ew G — =); The whole momentum outside the sphere is 
a 
w 
eS. 
colt 
a 
If the moving electrified body were ellipsoidal instead of 
spherical, then if the velocities parallel to the axes of the 
ellipsoid were w, v, w, the momenta parallel to the axes of 
2, y, z would be Au, Bu, Cw respectively, where A, B, C are 
not equal. Thus in this case the momentum would not be in 
the direction of the velocity unless the velocity were parallel 
to a principal axis of the ellipsoid : except in this case there 
would be a couple acting on the ellipsoid. For let O be 
the centre of the ellipsoid at any time, after a time d¢ let 
the centre come to O’; let I be the momentum at ¢ acting in 
the direction OL; then at ¢+6¢ the momentum is I in the 
direction O’L’, O’L’ being parallel to OL. Thus the momentum 
in the field at the time t+ 6¢ is not the same as it was at the 
time t, the difference is equivalent to the moment of momentum 
due to —I at O and iat O’. The moment of momentum lost 
by the field must be transferred to the moving body ; hence 
this will be acted upon by a couple in the plane LOO’. The 
components of the moment of this couple parallel to the axes 
of x, y, 2 are respectively 
(B—C) ww, (C—A)uw, (A—B)we. 
Charged Sphere rotating about an Awis.—If a uniformly 
charged sphere of radius a is rotating with angular velocity 
w about an axis through its centre parallel to z, we can 
easily prove that the components of magnetic force at a 
point distant r (7> a) from the centre of the sphere are given 
by the equations 
ewa? d? 1 
a= 
3 dadzr’ 
tag da 1 
rae dydz 1” 
pare, I 
i oe ae 
e being the charge on the sphere. 
