I have already mentioned Hughes' researches on the form of
electromagnet best adapted for rapid signaling. I have also
incidentally mentioned the fact that where rapidly varying currents
are employed, the strength of the electric current that a given
battery can yield is determined not so much by the resistance of the
electric circuit as by its electric inertia. It is not a very easy
task to explain precisely what happens to an electric circuit when the
current is turned on suddenly. The current does not suddenly rise to
its full value, being retarded by inertia. The ordinary law of Ohm in
its simple form no longer applies; one needs to apply that other law
which bears the name of the law of Helmholtz, the use of which is to
give us an expression, not for the final value of the current, but for
its value at any short time, t, after the current has been turned on.
The strength of the current after a lapse of a short time, t, cannot
be calculated by the simple process of taking the electromotive force
and dividing it by the resistance, as you would calculate steady
currents.
In symbols, Helmholtz's law is:
i_{t} = E/R ( 1 - e^{-(R/L)t} )
In this formula i_{t} means the strength of the current after the
lapse of a short time t; E is the electromotive force; R, the
resistance of the whole circuit; L, its coefficient of self-induction;
and _e_ the number 2.7183, which is the base of the Napierian
logarithms. Let us look at this formula; in its general form it
resembles Ohm's law, but with a new factor, namely, the expression
contained within the brackets. The factor is necessarily a fractional
quantity, for it consists of unity less a certain negative
exponential, which we will presently further consider. If the factor
within brackets is a quantity less than unity, that signifies that
i_{t} will be less than E ÷ R. Now the exponential of negative sign,
and with negative fractional index, is rather a troublesome thing to
deal with in a popular lecture. Our best way is to calculate some
values, and then plot it out as a curve. When once you have got it
into the form of a curve, you can begin to think about it, for the
curve gives you a mental picture of the facts that the long formula
expresses in the abstract. Accordingly we will take the following
case. Let E = 2 volts; R = 1 ohm; and let us take a relatively large
self-induction, so as to exaggerate the effect; say let L = 10 quads.
This gives us the following:
________________________________________
| | | |
| t_{(sec.)} | e^{+(R/L)t} | i_{t} |
--------------+--------------+---------|
| 0 | 1 | 0 |
| 1 | 1.105 | 0.950 |
| 2 | 1.221 | 1.810 |
| 5 | 1.649 | 3.936 |
| 10 | 2.718 | 6.343 |
| 20 | 7.389 | 8.646 |
| 30 | 20.08 | 9.501 |
| 60 | 403.4 | 9.975 |
| 120 | 16200.0 | 9.999 |
----------------------------------------
In this case the value of the steady current as calculated by Ohm's
law is 10 amperes, but Helmholtz's law shows us that with the great
self-induction which we have assumed to be present, the current, even
at the end of 30 seconds, has only risen up to within 5 percent. of
its final value; and only at the end of two minutes has practically
attained full strength. These values are set out in the highest curve
in Fig. 54, in which, however, the further supposition is made that
the number of spirals, S, in the coils of the electromagnet is 100, so
that when the current attains its full value of 10 amperes, the full
magnetizing power will be Si = 1000. It will be noticed that the
curve rises from zero at first steeply and nearly in a straight line,
then bends over, and then becomes nearly straight again, as it
gradually rises to its limiting value. The first part of the
curve--that relating to the strength of the current after _very small_
interval of time--is the period within which the strength of the
current is governed by inertia (i.e., the self-induction) rather than
by resistance. In reality the current is not governed either by the
self-induction or by the resistance alone, but by the ratio of the
two. This ratio is sometimes called the "time constant" of the
circuit, for it represents _the time_ which the current takes in that
circuit to rise to a definite fraction of its final value.
E = 10
r = 1
R = 100
L = 10
Si
1000 + _..-------------------------------
| . _ _---------
| . .----
| . .- 2 IN SERIES
| . .-
| -
| .: - :
| .: . :
500 | . : __- -:---------------------------
| . : _.- - : 2 IN PARALLEL
| . :. - :
| . / : - :
| . / - :
|. / - : :
|./. : :
|/_____:_____________:____________________________ t
10 20 40 60 80 100 120