t 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 |