When they are all put in series, so that the battery has a much
greater resistance than the rest of the circuit, the current rises
much more rapidly, because of the smallness of the time constant,
although it never attains the same ultimate maximum as when grouped in
the other way. That is to say, if there is self-induction as well as
resistance in the circuit, the amateur rule does not tell you the best
way of arranging the battery. There is another mode of regarding the
matter which is helpful. Self-induction, while the current is growing,
acts as if there were a sort of spurious addition to the resistance of
the circuit; and while the current is dying away it acts of course in
the other way, as if there were a subtraction from the resistance.
Therefore you ought to arrange the battery so that the internal
resistance is equal to the real resistance of the circuit, plus the
spurious resistance during that time. But how much is the spurious
resistance during that time? It is a resistance proportional to the
time that has elapsed since the current was turned on. So then it
comes to a question of the length of time for which you want to work
it. What fraction of a second do you require your signal to be given
in? What is the rate of the vibrator of your electric bell? Suppose
you have settled that point, and that the short time during which the
current is required to rise is called t; then the apparent resistance
at time t after the current is turned on is given by the formula: