Page:A Dynamical Theory of the Electromagnetic Field.pdf/14

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PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

Case of a single Circuit.

(35) The equation of the current ${\displaystyle x}$ in a circuit whose resistance is ${\displaystyle R}$, and whose coefficient of self-induction is ${\displaystyle L}$, acted on by an external electromotive force ${\displaystyle \xi }$, is

 ${\displaystyle \xi -Rx={\frac {d}{dt}}Lx.}$ (13)

When ${\displaystyle \xi }$ is constant, the solution is of the form

${\displaystyle x=b+(a-b)e^{-{\frac {R}{L}}t},}$

where ${\displaystyle a}$ is the value of the current at the commencement, and ${\displaystyle b}$ is its final value.

The total quantity of electricity which passes in time ${\displaystyle t}$, where ${\displaystyle t}$ is great, is

 ${\displaystyle \int _{0}^{t}{xdt}=bt+(a-b){\frac {L}{R}}}$ (14)

The value of the integral of ${\displaystyle x^{2}}$ with respect to the time is

 ${\displaystyle \int _{0}^{t}x^{2}dt=b^{2}t+(a-b){\frac {L}{R}}({\frac {3b+a}{2}}).}$ (15)

The actual current changes gradually from the initial value ${\displaystyle a}$ to the final value ${\displaystyle b}$, but the values of the integrals of ${\displaystyle x}$ and ${\displaystyle x^{2}}$ are the same as if a steady current of intensity ${\displaystyle {\tfrac {1}{2}}(a+b)}$ were to flow for a time ${\displaystyle 2{\tfrac {L}{R}}}$, and were then succeeded by the steady current ${\displaystyle b}$. The time ${\displaystyle 2{\tfrac {L}{R}}}$ is generally so minute a fraction of a second, that the effects on the galvonometer and dynamometer may be calculated as if the impulse were instantaneous.

If the circuit consists of a battery and a coil, then, when the circuit is first complete, the effects are the same as if the current had only half its final strength during the time ${\displaystyle 2{\tfrac {L}{R}}}$. This diminution of the current, due to induction, is sometimes called the counter-current.

(36) If an additional resistance ${\displaystyle r}$ is suddenly thrown into the circuit, as by breaking contact, so as to force the current to pass through a thin wire of resistance ${\displaystyle r}$, then the original current is ${\displaystyle a={\tfrac {\xi }{R}}}$, and the final current is ${\displaystyle b={\tfrac {\xi }{R+r}}}$.

The current of induction is then ${\displaystyle {\tfrac {1}{2}}\xi {\tfrac {2R+r}{R(R+r)}}}$, and continues for a time 2${\displaystyle {\tfrac {L}{(R+r)}}}$. The current is greater than that which the battery can maintain in the two wires ${\displaystyle R}$ and ${\displaystyle r}$, and may be sufficient to ignite the thin wire ${\displaystyle r}$.

When contact is broken by separating the wires in air, this additional resistance is given by the interposed air, and since the electromotive force across the new resistance is very great, a spark will be formed across.