Momentum in the Electric Field. 349 
is equal to the component at P of the vector potential due 
to the current. Hence we see that the vector potential at any 
point is the momentum due to the magnetic force produced 
by the system giving the vector potential and the electric 
field due to unit charge at the point. We thus get a physical 
interpretation for the vector potential, and see that instead of 
being merely an analytical device, it represents a most im- 
portant physical property of the system. We shall show 
later on that the principle of the conservation of momentum 
leads at once to the ordinary equations for the induction 
of currents. 
Momentum due to Two Moving Particles.—Let P and Q be 
the moving particles, e and e! their charges, R the distance 
between them, 2, 2}, W1 3 U2, V2, We the components of their 
velocities, then from the results proved on page 344 we see that 
the momentum in the field is equivalent to the following 
distribution. 
A momentum passing through P parallel to the axis of x 
equal to 
2 ep puee’ ae ( 
= a) tm) spa, a U9 og Gg | or Wo 
ee Ry me ON dae” Gasdy, | dx.de, 
aR aR aR ) 
and a momentum through Q parallel to x equal to 
2 pe” pee’ Meni oe Gok aR a?R 
sh Ug+ “a Uy — ghee (v1 da? +, dady, + WwW, aa 5 
a and bare the radii of the spheres whose centres are P and Q 
over which the charges are spread, and 2, 1, 21, 2a, Yay 22 are 
the coordinates of P and Q respectively. We have of course 
#R i d?7R J a? et d?R 
dx,? PERE i dxsdy, fii daydy, 
There are, of course, momenta through P and Q parallel 
to the axes of y and z respectively given by symmetrical 
expressions. These expressions enable us to find the veloci- 
ties acquired when a charged particle P at rest in the presence 
of another, Q, is struck by an impulse. Let us suppose that 
the particle P initially at rest is struck by an impulse I, 
parallel to 2, then the momentum in the charged particles 
and in the field must be equivalent to this impulse. Hence, if 
Uj, Vj, Wi, Us, Vz, W2 are the velocities acquired by P and Q 
