Page:EB1911 - Volume 09.djvu/229

Rh of the electromotive force acting in the circuit reckoned in volts by the resistance in ohms, or $$\text{A} = \text{V} / \text{R}.$$ Ohm established his law by a course of reasoning which was similar to that on which J. B. J. Fourier based his investigations on the uniform motion of heat in a conductor. As a matter of fact, however, Ohm’s law merely states the direct proportionality of steady current to steady electromotive force in a circuit, and asserts that this ratio is governed by the numerical value of a quality of the conductor, called its resistance, which is independent of the current, provided that a correction is made for the change of temperature produced by the current. Our belief, however, in its universality and accuracy rests upon the close agreement between deductions made from it and observational results, and although it is not derivable from any more fundamental principle, it is yet one of the most certainly ascertained laws of electrokinetics.

Ohm’s law not only applies to the circuit as a whole but to any part of it, and provided the part selected does not contain a source of electromotive force it may be expressed as follows:—The difference of potential (P.D.) between any two points of a circuit including a resistance $$\text{R},$$ but not including any source of electromotive force, is proportional to the product of the resistance and the current $$i$$ in the element, provided the conductor remains at the same temperature and the current is constant and unidirectional. If the current is varying we have, however, to take into account the electromotive force (E.M.F.) produced by this variation, and the product $$\text{R}i$$ is then equal to the difference between the observed P.D. and induced E.M.F.

We may otherwise define the resistance of a circuit by saying that it is that physical quality of it in virtue of which energy is dissipated as heat in the circuit when a current flows through it. The power communicated to any electric circuit when a current is created in it by a continuous unidirectional electromotive force $$\text{E}$$ is equal to $$\text{E}i,$$ and the energy dissipated as heat in that circuit by the conductor in a small interval of time $$dt$$ is measured by $$\text{E}idt.$$ Since by Ohm’s law $$\text{E} = \text{R}i,$$ where $$\text{R}$$ is the resistance of the circuit, it follows that the energy dissipated as heat per unit of time in any circuit is numerically represented by $$\text{R}i^2,$$ and therefore the resistance is measured by the heat produced per unit of current, provided the current is unvarying.

Inductance.—As soon as we turn our attention, however, to alternating or periodic currents we find ourselves compelled to take into account another quality of the circuit, called its “inductance.” This may be defined as that quality in virtue of which energy is stored up in connexion with the circuit in a magnetic form. It can be experimentally shown that a current cannot be created instantaneously in a circuit by any finite electromotive force, and that when once created it cannot be annihilated instantaneously. The circuit possesses a quality analogous to the inertia of matter. If a current $$i$$ is flowing in a circuit at any moment, the energy stored up in connexion with the circuit is measured by $$\tfrac{1}{2}\text{L}i^2,$$ where $$\text{L},$$ the inductance of the circuit, is related to the current in the same manner as the quantity called the mass of a body is related to its velocity in the expression for the ordinary kinetic energy, viz. $$\tfrac{1}{2}\text{M}v^2.$$ The rate at which this conserved energy varies with the current is called the “electrokinetic momentum” of this circuit ($$= \text{L}i$$). Physically interpreted this quantity signifies the number of lines of magnetic flux due to the current itself which are self-linked with its own circuit.

Magnetic Force and Electric Currents.—In the case of every circuit conveying a current there is a certain magnetic force (see ) at external points which can in some instances be calculated. Laplace proved that the magnetic force due to an element of length $$d\text{S}$$ of a circuit conveying a current $$\text{I}$$ at a point $$\text{P}$$ at a distance $$\text{r}$$ from the element is expressed by $$\text{I}d\text{S} \sin \theta / r^2,$$ where $$\theta$$ is the angle between the direction of the current element and that drawn between the element and the point. This force is in a direction perpendicular to the radius vector and to the plane containing it and the element of current. Hence the determination of the magnetic force due to any circuit is reduced to a summation of the effects due to all the elements of length. For instance, the magnetic force at the centre of a circular circuit of radius $$r$$ carrying a steady current $$\text{I}$$ is $$2\pi \text{I} / r,$$ since all elements are at the same distance from the centre. In the same manner, if we take a point in a line at right angles to the plane of the circle through its centre and at a distance $$d,$$ the magnetic force along this line is expressed by $$2\pi r^2 \text{I} / (r^2 + d^2)^{\frac{3}{2}}.$$ Another important case is that of an infinitely long straight current. By summing up the magnetic force due to each element at any point $$\text{P}$$ outside the continuous straight current $$\text{I}$$ and at a distance $$d$$ from it, we can show that it is equal to $$2\text{I} / d$$ or is inversely proportional to the distance of the point from the wire. In the above formula the current $$\text{I}$$ is measured in absolute electromagnetic units. If we reckon the current in amperes $$\text{A},$$ then $$\text{I} = \text{A} / 10.$$

It is possible to make use of this last formula, coupled with an experimental fact, to prove that the magnetic force due to an element of current varies inversely as the square of the distance. If a flat circular disk is suspended so as to be free to rotate round a straight current which passes through its centre, and two bar magnets are placed on it with their axes in line with the current, it is found that the disk has no tendency to rotate round the current. This proves that the force on each magnetic pole is inversely as its distance from the current. But it can be shown that this law of action of the whole infinitely long straight current is a mathematical consequence of the fact that each element of the current exerts a magnetic force which varies inversely as the square of the distance. If the current flows $$\text{N}$$ times, round the circuit instead of once, we have to insert $$\text{NA} / 10$$ in place of $$\text{I}$$ in all the above formulae. The quantity $$\text{NA}$$ is called the “ampere-turns” on the circuit, and it is seen that the magnetic field at any point outside a circuit is proportional to the ampere-turns on it and to a function of its geometrical form and the distance of the point.

There is therefore a distribution of magnetic force in the field of every current-carrying conductor which can be delineated by lines of magnetic force and rendered visible to the eye by iron filings (see ). If a copper wire is passed vertically through a hole in a card on which iron filings are sprinkled, and a strong electric current is sent through the circuit, the filings arrange themselves in concentric circular lines making visible the paths of the lines of magnetic force (fig. 3). In the same manner, by passing a circular wire through a card and sending a strong current through the wire we can employ iron filings to delineate for us the form of the lines of magnetic force (fig. 4).

In all cases a magnetic pole of strength $$\text{M},$$ placed in the field of an electric current, is urged along the lines of force with a mechanical force equal to $$\text{MH},$$ where $$\text{H}$$ is the magnetic force. If then we carry a unit magnetic pole against the direction in which it would naturally move we do work. The lines of magnetic force embracing a current-carrying conductor are always loops or endless lines.

The work done in carrying a unit magnetic pole once round a circuit conveying a current is called the “line integral of magnetic force” along that path. If, for instance, we carry a unit pole in a circular path of radius $$r$$ once round an infinitely long straight filamentary current $$\text{I},$$ the line integral is $$4\pi \text{I}.$$ It is easy to prove that this is a general law, and that if we have any currents flowing in a conductor the line integral of magnetic force taken once round a path linked with the current circuit is $$4\pi$$ times the total current flowing through the circuit. Let us apply this to the case of an endless solenoid. If a copper wire insulated or covered with cotton or silk is twisted round a thin rod so as to make a close spiral, this