Page:EB1911 - Volume 09.djvu/230

forms a “solenoid,” and if the solenoid is bent round so that its two ends come together we have an endless solenoid. Consider such a solenoid of mean length $l$ and ${\text{N}}$ turns of wire. If it is made endless, the magnetic force ${\text{H}}$ is the same everywhere along the central axis and the line integral along the axis is ${\text{H}}l.$ If the current is denoted by ${\text{I}},$ then ${\text{NI}}$ is the total current, and accordingly $4\pi {\text{NI}}={\text{H}}l,$ or ${\text{H}}=4\pi {\text{NI}}/l,$ For a thin endless solenoid the axial magnetic force is therefore $4\pi$ times the current-turns per unit of length. This holds good also for a long straight solenoid provided its length is large compared with its diameter. It can be shown that if insulated wire is wound round a sphere, the turns being all parallel to lines of latitude, the magnetic force in the interior is constant and the lines of force therefore parallel. The magnetic force at a point outside a conductor conveying a current can by various means be measured or compared with some other standard magnetic forces, and it becomes then a means of measuring the current. Instruments called galvanometers and ammeters for the most part operate on this principle.

Thermal Effects of Currents.—J. P. Joule proved that the heat produced by a constant current in a given time in a wire having a constant resistance is proportional to the square of the strength of the current. This is known as Joule’s law, and it follows, as already shown, as an immediate consequence of Ohm’s law and the fact that the power dissipated electrically in a conductor, when an electromotive force ${\text{E}}$ is applied to its extremities, producing thereby a current ${\text{I}}$ in it, is equal to ${\text{EI.}}$

If the current is alternating or periodic, the heat produced in any time ${\text{T}}$ is obtained by taking the sum at equidistant intervals of time of all the values of the quantities ${\text{R}}i^{2}dt,$ where $dt$ represents a small interval of time and $i$ is the current at that instant. The quantity ${\text{T}}^{-1}\int _{0}^{\text{T}}i^{2}dt$ is called the mean-square-value of the variable current, $i$ being the instantaneous value of the current, that is, its value at a particular instant or during a very small interval of time $dt.$ The square root of the above quantity, or

$\left[{\text{T}}^{-1}\int _{0}^{\text{T}}i^{2}dt\right]^{{\frac {1}{2}},}$

is called the root-mean-square-value, or the effective value of the current, and is denoted by the letters R.M.S.

Currents have equal heat-producing power in conductors of identical resistance when they have the same R.M.S. values. Hence periodic or alternating currents can be measured as regards their R.M.S. value by ascertaining the continuous current which produces in the same time the same heat in the same conductor as the periodic current considered. Current measuring instruments depending on this fact, called hot-wire ammeters, are in common use, especially for measuring alternating currents. The maximum value of the periodic current can only be determined from the R.M.S. value when we know the wave form of the current. The thermal effects of electric currents in conductors are dependent upon the production of a state of equilibrium between the heat produced electrically in the wire and the causes operative in removing it. If an ordinary round wire is heated by a current it loses heat, (1) by radiation, (2) by air convection or cooling, and (3) by conduction of heat out of the ends of the wire. Generally speaking, the greater part of the heat removal is effected by radiation and convection.

If a round sectioned metallic wire of uniform diameter $d$ and length $l$ made of a material of resistivity $p$ has a current of ${\text{A}}$ amperes passed through it, the heat in watts produced in any time $t$ seconds is represented by the value of $4{\text{A}}^{2}\rho lt/10^{9}\pi d^{2},$ where $d$ and $l$ must be measured in centimetres and $\rho$ in absolute C.G.S. electromagnetic units. The factor $10^{9}$ enters because one ohm is 10⁹ absolute electromagnetic C.G.S. units (see Units, Physical). If the wire has an emissivity $e,$ by which is meant that $e$ units of heat reckoned in joules or watt-seconds are radiated per second from unit of surface, then the power removed by radiation in the time $t$ is expressed by $\pi dlet.$ Hence when thermal equilibrium is established we have $4{\text{A}}^{2}plt/10^{9}\pi d^{2}=\pi dlet,$ or ${\text{A}}^{2}=10^{9}\pi ^{2}ed^{3}/4\rho .$ If the diameter of the wire is reckoned in mils (1 mil = .001 in.), and if we take $e$ to have a value 0.1, an emissivity which will generally bring the wire to about 60° C., we can put the above formula in the following forms for circular sectioned copper, iron or platinoid wires, viz.

${\text{A}}={\sqrt {d^{2}/500}}$ for copper wires

${\text{A}}={\sqrt {d^{2}/4000}}$ for iron wires

${\text{A}}={\sqrt {d^{2}/5000}}$ for platinoid wires.

These expressions give the ampere value of the current which will bring bare, straight or loosely coiled wires of $d$ mils in diameter to about 60° C. when the steady state of temperature is reached. Thus, for instance, a bare straight copper wire 50 mils in diameter (= 0.05 in.) will be brought to a steady temperature of about 60° C. if a current of ${\sqrt {50^{3}/500}}={\sqrt {250}}=16$ amperes (nearly) is passed through it, whilst a current of ${\sqrt {25}}=5$ amperes would bring a platinoid wire to about the same temperature.

A wire has therefore a certain safe current-carrying capacity which is determined by its specific resistance and emissivity, the latter being fixed by its form, surface and surroundings. The emissivity increases with the temperature, else no state of thermal equilibrium could be reached. It has been found experimentally that whilst for fairly thick wires from 8 to 60 mils in diameter the safe current varies approximately as the 1.5th power of the diameter, for fine wires of 1 to 3 mils it varies more nearly as the diameter.

Action of one Current on Another.—The investigations of Ampère in connexion with electric currents are of fundamental importance in electrokinetics. Starting from the discovery of Oersted, Ampère made known the correlative fact that not only is there a mechanical action between a current and a magnet, but that two conductors conveying electric currents exert mechanical forces on each other. Ampère devised ingenious methods of making one portion of a circuit movable so that he might observe effects of attraction or repulsion between this circuit and some other fixed current. He employed for this purpose an astatic circuit B, consisting of a wire bent into a double rectangle round which a current flowed first in one and then in the opposite direction (fig. 5). In this way the circuit was removed from the action of the earth’s magnetic field, and yet one portion of it could be submitted to the action of any other circuit C. The astatic circuit was pivoted by suspending it in mercury cups q, p, one of which was in electrical connexion with the tubular support A, and the other with a strong insulated wire passing up it.

Fig. 5. Ampère devised certain crucial experiments, and the theory deduced from them is based upon four facts and one assumption.^[1] He showed (1) that wire conveying a current bent back on itself produced no action upon a proximate portion of a movable astatic circuit; (2) that if the return wire was bent zig-zag but close to the outgoing straight wire the circuit produced no action on the movable one, showing that the effect of an element of the circuit was proportional to its projected length; (3) that a closed circuit cannot cause motion in an element of another circuit free to move in the direction of its length; and (4) that the action of two circuits on one and the same movable circuit was null if one of the two fixed circuits was $n$ times greater than the other but $n$ times further removed from the movable circuit. From this last experiment by an ingenious line of reasoning he proved that the action of an element of current on another element of current varies inversely as a square of their distance. These experiments enabled him to construct a mathematical expression of the law of action between two elements of conductors conveying currents. They also enabled him to prove that an element of current may be resolved like a force into components in different directions, also that the force produced by any element of the circuit on an element of any other circuit was perpendicular to the line joining the elements and inversely as the square of their distance. Also he showed that this force was an attraction if the currents in the elements were in the same direction, but a repulsion if they were in opposite directions. From these experiments and deductions from them he built up a complete formula for the action of one element of a current of length $d{\text{S}}$

1. See Maxwell, Electricity and Magnetism, vol. ii. chap. ii.