Page:EB1911 - Volume 09.djvu/232

|the subsequent motion of electricity in the wire. If ${\text{Q}}$ is the quantity of electricity in the condenser initially, and $g$ that at any time $t$ after completing the circuit, then the energy stored up in the condenser at that instant is ${\tfrac {1}{2}}q^{2}/{\text{C,}}$ and the energy associated with the circuit is ${\tfrac {1}{2}}{\text{L}}(dq/dt)^{2},$ and the rate of dissipation of energy by resistance is ${\text{R}}(dq/dt)^{2},$ since $dq/dt{=}i$ is the discharge current. Hence we can construct an equation of energy which expresses the fact that at any instant the power given out by the condenser is partly stored in the circuit and partly dissipated, as heat in it. Mathematically this is expressed as follows:—

$-{\frac {d}{dt}}\left[{\tfrac {1}{2}}{\frac {q^{2}}{\text{C}}}\right]{=}{\frac {d}{dt}}\left[{\tfrac {1}{2}}{\text{L}}\left({\frac {dq}{dt}}\right)^{2}\right]+{\text{R}}\left({\frac {dq}{dt}}\right)^{2}$

or

${\frac {d^{2}q}{dt^{2}}}+{\frac {\text{R}}{\text{L}}}{\frac {dq}{dt}}+{\frac {1}{\text{LC}}}q{=}0.$

The above equation has two solutions according as ${\text{R}}^{2}/4{\text{L}}^{2}$ is greater or less than $1/{\text{LC}}$ In the first case the current $i$ in the circuit can be expressed by the equation

$i{=}{\text{Q}}{\frac {\alpha ^{2}+\beta ^{2}}{2\beta }}e^{-\alpha }(e^{\beta t}-e^{-\beta t}),$

where $\alpha {=}{\text{R}}/2{\text{L}},\,\,\beta {=}{\sqrt {\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}-{\frac {1}{\text{LC}}}$, ${\text{Q}}$ is the value of $q$ when $t=0,$ and $e$ is the base of Napierian logarithms; and in the second case by the equation

$i{=}{\text{Q}}{\frac {\alpha ^{2}+\beta ^{2}}{\beta }}e^{-\alpha t}\sin \beta t$

where

$\alpha {=}{\text{R}}/2{\text{L}},$⁠and⁠$\beta {=}{\sqrt {{\frac {1}{\text{LC}}}-{\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}}\cdot$

These expressions show that in the first case the discharge current of the jar is always in the same direction and is a transient unidirectional current. In the second case, however, the current is an oscillatory current gradually decreasing in amplitude, the frequency $n$ of the oscillation being given by the expression

$n{=}{\frac {1}{2\pi }}{\sqrt {{\frac {1}{\text{LC}}}-{\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}}\cdot$

In those cases in which the resistance of the discharge circuit is very small, the expression for the frequency $n$ and for the time period of oscillation ${\text{R}}$ take the simple forms $n{=}1,\,\,2\pi {\sqrt {\text{LC}}}$ or ${\text{T}}{=}1/n{=}2\pi {\sqrt {\text{LC}}}$}}

The above investigation shows that if we construct a circuit consisting of a condenser and inductance placed in series with one another, such circuit has a natural electrical time period of its own in which the electrical charge in it oscillates if disturbed. It may therefore be compared with a pendulum of any kind which when displaced oscillates with a time period depending on its inertia and on its restoring force.

The study of these electrical oscillations received a great impetus after H. R. Hertz showed that when taking place in electric circuits of a certain kind they create electromagnetic waves (see Electric Waves) in the dielectric surrounding the oscillator, and an additional interest was given to them by their application to telegraphy. If a Leyden jar and a circuit of low resistance but some inductance in series with it are connected across the secondary spark gap of an induction coil, then when the coil is set in action we have a series of bright noisy sparks, each of which consists of a train of oscillatory electric discharges from the jar. The condenser becomes charged as the secondary electromotive force of the coil is created at each break of the primary current, and when the potential difference of the condenser coatings reaches a certain value determined by the spark-ball distance a discharge happens. This discharge, however, is not a single movement of electricity in one direction but an oscillatory motion with gradually decreasing amplitude. If the oscillatory spark is photographed on a revolving plate or a rapidly moving film, we have evidence in the photograph that such a spark consists of numerous intermittent sparks gradually becoming feebler. As the coil continues to operate, these trains of electric discharges take place at regular intervals. We can cause a train of electric oscillations in one circuit to induce similar oscillations in a neighbouring circuit, and thus construct an oscillation transformer or high frequency induction coil.

Alternating Currents.—The study of alternating currents of electricity began to attract great attention towards the end of the 19th century by reason of their application in electrotechnics and especially to the transmission of power. A circuit in which a simple periodic alternating current flows is called a single phase circuit. The important difference between such a form of current flow and steady current flow arises from the fact that if the circuit has inductance then the periodic electric current in it is not in step with the terminal potential difference or electromotive force acting in the circuit, but the current lags behind the electromotive force by a certain fraction of the periodic time called the “phase difference.” If two alternating currents having a fixed difference in phase flow in two connected separate but related circuits, the two are called a two-phase current. If three or more single-phase currents preserving a fixed difference of phase flow in various parts of a connected circuit, the whole taken together is called a polyphase current. Since an electric current is a vector quantity, that is, has direction as well as magnitude, it can most conveniently be represented by a line denoting its maximum value, and if the alternating current is a simple periodic current then the root-mean-square or effective value of the current is obtained by dividing the maximum value by ${\sqrt {2}}.$ Accordingly when we have an electric circuit or circuits in which there are simple periodic currents we can draw a vector diagram, the lines of which represent the relative magnitudes and phase differences of these currents.

A vector can most conveniently be represented by a symbol such as $a+\iota b$ where $a$ stands for any length of $a$ units measured horizontally and $b$ for a length $b$ units measured vertically, and the smybol $\iota$ is a sign of perpendicularity, and equivalent analytically^[1] to ${\sqrt {-1}}.$ Accordingly if ${\text{E}}$ represents the periodic electromotive force (maximum value) acting in a circuit of resistance ${\text{R}}$ and inductance ${\text{L}}$ and frequency $n,$ and if the current considered as a vector is represented by ${\text{I,}}$ it is easy to show that a vector equation exists between these quantities as follows:—

${\text{E}}{=}{\text{RI}}+\iota 2\pi n{\text{LI}}.$

Since the absolute magnitude of a vector $a+\iota b$ is ${\sqrt {(a^{2}+b^{2})}},$ it follows that considering merely magnitudes of current and electromotive force and denoting them by symbols ${\text{(E) (I)}}$ we have the following equation connecting ${\text{(E)}}$ and ${\text{(I)}}$:—

$({\text{I}}){=}{\text{(E)}}/{\sqrt {{\text{R}}^{2}+p^{2}{\text{L}}^{2}}},$

where $p$ stands for $2\pi n.$ If the above equation is compared with the symbolic expression of Ohm's law, it will be seen that the quantity ${\sqrt {({\text{R}}^{2}+p^{2}{\text{L}}^{2})}}$ takes the place of resistance ${\text{R}}$ in the expression of Ohm. This quantity ${\sqrt {({\text{R}}^{2}+p^{2}{\text{L}}^{2})}}$ is called the “impedance” of the alternating circuit. The quantity $p{\text{L}}$ is called the “reactance” of the alternating circuit, and it is therefore obvious that the current in such a circuit lags behind the electromotive force by an angle, called the angle of lag, the tangent of which is $p{\text{L}}/{\text{R}}.$

Currents in Networks of Conductors.—In dealing with problems connected with electric currents we have to consider the laws which govern the flow of currents in linear conductors (wires), in plane conductors (sheets). and throughout the mass of a material conductor.^[2] In the first case consider the collocation of a number of linear conductors, such as rods or wires of metal, joined at their ends to form a network of conductors, The network consists of a number of conductors joining certain points and forming meshes. In each conductor a current may exist, and along each conductor there is a fall of potential, or an active electromotive force may be acting in it. Each conductor has a certain resistance. To find the current in each conductor when the individual resistances and electromotive forces are given, proceed as follows:—Consider any one mesh. The sum of all the electromotive forces which exist in the branches bounding that mesh must be equal to the sum of all the products of the resistances into the currents flowing along them, or $\sum {\text{(E)}}=\sum {\text{(C.R.)}}$ Hence if we consider each mesh as traversed by imaginary currents all circulating in the same direction, the real currents are the sums or differences of these imaginary cyclic currents in each branch. Hence we may assign to each mesh a cycle symbol ${x,y,z,}$ &c., and form a cycle equation. Write down the cycle symbol for a mesh and prefix as coefficient the sum of all the resistances which bound that cycle, then subtract the cycle symbols of each adjacent cycle, each multiplied by the value of the bounding or common resistances, and equate this sum to the total electromotive force acting round the cycle. Thus if ${xyz}$ are the cycle currents, and ${abc}$ the resistances bounding the mesh $x,$ and $b$ and $c$ those separating it from the meshes $y$ and $z,$ and ${\text{E}}$ an electromotive force in the branch $a,$ then

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1. See W. G. Rhodes, An Elementary Treatise on Alternating Currents (London, 1902), chap. vii.
2. See J. A. Fleming, “Problems on the Distribution of Electric Currents in Networks of Conductors,” Phil. Mag. (1885), or Proc. Phys. Soc. Lond. (1885), 7; also Maxwell, Electricity and Magnetism (2nd ed.), vol. i. p. 374, §280, 282b.