Momentum in the Electric Field. 333 
Case of a Pole and Charged Point.—If we have a pole of 
strength m at A anda charge e at B, the preceding rule shows 
that the momentum at P is at right angles to the plane APB 
and is per unit volume equal to 
1 ee dain APB. 
4ar BP? AP? 
This distribution of momentum can easily be shown to be 
equivalent to a moment of momentum em around AB; the 
direction of rotation being opposite to that for a right-handed 
screw when the direction of translation is from A to B. 
The resultant momentum in any direction vanishes; as 
this is true for all positions of A and B it follows that however 
the pointand pole may move about in the field the momentum 
in any direction gained by the pole must be equal and oppo- 
site to that gained by the point; 7. e., the forces on the pole 
and point must be equal and opposite. 
To find these forces, let us suppose that in a time o¢ the 
point B comes to B’, A remaining at rest; then the moment 
of momentum in the field has changed from em along AB to 
emalong AB’; 2. ¢., the moment of momentum in the field has 
changed by emsin ¢ at right angles to AB in the plane ABA’. 
The material system consisting of the pole and point must 
have gained the moment of momentum lost by the field; the 
moment of momentum emsin din the assigned direction is 
em sin 
AB 
plane BAA’ and an equal and opposite momentum at B. 
Thus we may suppose A to have gained and B lost this 
momentum in time of, the momentum gained by A per unit 
min $ (5. Now 
time, 7. e. the force acting upon A, is - 
, Where @ is the angle between BB’ the 
equivalent to momentum at A at right angles to the 
BB’ sin AB | 
AB 
direction of motion of B and AB;; thus the force on A is 
sin 7 
emsin@ BB’ _ emvsin@ 
wast a ABE 
where v is the velocity of B. 
Thus a moving charge is acted upon by a force at right 
angles to the direction of motion, at right angles also to the 
magnetic force, and equal to Hevsin @: hence the principle of 
momentum gives us in a very simple way the force ona 
moving charged point. 
Exactly the same result will apply if the pole moves instead 
