Geology Reference
In-Depth Information
In this conditions, the disk works as a battery
supplying an emf given by:
where we have used the initial condition
I
(0)
D
0.
Let us assume now that ¨/¨
0
>1. In this in-
stance, the argument of the exponential is positive
and the current increases until it reaches some
value, say
I
1
, at time
t
D
t
1
. This current flows
through the solenoid in the direction indicated in
Fig.
4.5
, thereby, the induced magnetic field
B
has the same direction of the external field
B
0
.
If at time
t
D
t
1
the field
B
0
is instantaneously
removed, although in reality the presence of self-
induction effectively prevents an instantaneous
removal of the external field, the Eq. (
4.41
)for
t
t
1
becomes a homogeneous equation:
2
¨ŒB.t/
C
B
0
b
2
a
2
1
E
.t/
D
(4.38)
This expression shows that an increase of
B
determines an increase of electromotive force,
which in turn implies an increase of the current
flowing through the solenoid. The increase of cur-
rent determines in turn an increase of
B
and so on.
We say that the system has
positive feedback
.Let
R
and
be, respectively, the electric resistance
(in Ohm) of the circuit and the self-inductance
of the solenoid. Clearly, the total emf (
4.38
)
must be equal to the electromotive force that is
necessary to allow flowing of a current
I
through
a circuit having electric resistance
R
,plusacom-
ponent that compensates the emf (
4.36
) associ-
ated with current variations. The former is given
by
Ohm
'
slaw
:
L
1
I.t/
D
0
dI
dt
C
R
L
¨
¨
0
(4.44)
Assuming the initial condition:
I
(
t
1
)
D
I
1
,we
have the following solution:
I.t/
D
I
1
exp
1
.t
t
1
/
R
L
¨
¨
0
I
t
t
1
E
.t/
D
RI.t/
(4.39)
(4.45)
Therefore, we have:
This is an exact solution only for times
t
much greater than the decay constant of the
external field
B
0
.Itimpliesthateveninabsence
of external field the system sustains a self-excited
magnetic field
B
D
B
(
t
). By (
4.34
)thisfieldis
given by:
B.t/
D
B
1
exp
2
¨ŒB.t/
C
B
0
b
2
a
2
dI
dt
C
RI
D
1
L
1
2
¨ŒB.t/
C
B
0
b
2
.for b>>a/
(4.40)
Š
1
.t
t
1
/
R
L
¨
¨
0
I
t
t
1
Using (
4.34
) for the solenoid field
B
,gives:
1
I.t/
D
(4.46)
¨B
0
b
2
2
dI
dt
C
R
L
¨
¨
0
(4.41)
L
Therefore, we can draw the following conclu-
sions:
where:
1. If ¨/¨
0
< 1, both
I
and
B
decay exponentially;
2. If ¨
D
¨
0
, we have a stationary field
B
(
t
)
D
B
1
;
3. If ¨/¨
0
> 1, both
I
and
B
increase exponen-
tially.
2R
0
nb
2
¨
0
(4.42)
Equation
4.41
is a linear first-order differential
equation with constant coefficients. The solution
is:
Thus, the simple analog model of a self-
excited dynamo composed by a solenoid and
a rotating disk furnishes a theoretical background
for the existence of self-sustained, persistent,
1
exp
1
t
(4.43)
¨B
0
b
2
2R
.
1
R
L
¨
¨
0
I.t/
D
¨
=
¨
0
/
