Force on a Magnetic Particle in a Magnetic Field. 39 
Let PQ, P’Q’ (fig. 2) be two indefinitely close lines of 
6) 
magnetic force, and let H be the magnetic intensity at P 
and H’ that at P’, PP’ and QQ’ being normals to the line 
of force at P and Q. ‘Then, since 
HM. dil 
pe) an 
it easily follows that 
|g (IEA ol 
Py = PQ 
The work done in carrying unit pole round the circuit 
PQQ’P’P is, therefore, zero; from which it follows that the 
field of force is conservative. 
As an example of a non-conservative field, take the case of 
a current of electricity in a straight wire or cylindrical liquid 
column of circular cross section. The intensity of field 
inside, on the supposition that the current is uniformly 
Ae : . 2Cr 
distributed over the area of the cross section, 1s > where 7 
is the distance from the axis and a the radius of the wire. 
: Hs ee ARUP 0 
The force on a particle of soft iron inside is 4 “, but it is 
in the direction in which the magnetic intensity diminishes 
most rapidly. Insidea wire conveying a current of electricity 
a magnetic potential does not exist, and the field is non- 
conservative. 
We may now goa step farther and find the direction of 
the force on a magnetic particle ina plane field of given 
magnetic intensity, assuming that there exists a distribution 
of electric currents at right angles to the plane. Let the 
direction of the current be the positive direction through 
