Alternating Magnetic Force in a Solenoid. 
9 
If the currents in the coil be oscillatory, kept np by an impressed E. 
M. F.= E sin pt where p = 2rcn ( n denoting the frequency), then the 
magnetic force in the core will also be oscillatory; and calling it H , we 
will have also H—H 1 sin pt + H 2 cos pt. t 
If this value of H be substituted in the general differential equation 
for H just above given, we shall have the two equations: 
!||| H'i)=-xH 2 
and 
7-iF rH '* )= + xH ' 
where x is put for From these two equations we infer that the 
common equation which both H x and H & will satisfy is 
Id d 1 
r dr dr r 
d 
dr 
H = — x i H . . . 
. (5) 
This is an equation of exactly the same form as the elastic equation (2). 
If H x be made equal to A a M+A 2 N where M and N are functions of r 
and equal to 
M=l — 
x b r l 
2-ip 2*. . . . 12‘ 2 
xr 2 2H*6' 2 x 5 2 10 
4 cc 3 r* + 2\ .. .10* . 
then H=(AM+BN) sin nt + (AN — BM) cos nt. 
If H 2 =B q M+B 2 N and A 0 = — B 2 while A 2 = — B 0 
This can be shown by writing H x and H 2 in series, viz.: 
H X =A 0 + A x r + A 2 r 2 + etc. 
H 2 =B a + B 1 r+ B 2 r~ + e tc. 
inserting the series in the biquadratic differential equations, as val¬ 
ues of H x or U 2 , differentiating as required and equating to zero the 
coefficients of the different powers of r. All the odd coefficients A 
and B disappear. We need only the A 0 and A 2 B 0 and B 2 To satisfy 
| Fleming calls p the pidsations of the current; in harmonic motion 
p— —jT- = a constant. 
dt 
2 
