less will be the reluctance and the more lines of force will be established by a given number of ampère turns.

The reluctance of an iron ring may be calculated from the formula

where l is the mean length of the ring, A is its cross section, and is the permeability of the iron. If a circuit is made up of parts that have different permeabilities their reluctances must be calculated separately and added together when the parts are in series.

When the armature is not across the poles the reluctance is greatly increased because of the small permeability of the air through which the lines of force must pass. Therefore the number of lines of force established in that case is very much less than with the armature across the poles.

The force with which such a magnet holds its armature is proportional to the area of its poles, and is expressed by the approximate formula

where B represents the induction or number of lines of force per square centimeter of the surface between pole and armature, and A is the combined areas of the two poles.

Problems.

1. How much current is flowing in one rail of an electric railway which runs in a north and south direction and causes a deflection of 45° in a compass needle held 30 cm. above the center of the rail; taking the strength of the horizontal component of the earth's magnetic field as 0.20?

2. Find the strength of the magnetic field at the center of a circular coil of 7 turns of wire 18 cm. in diameter when carrying a current of 3 ampères.

3. What will be the deflection of a magnetic needle at the center of the coil in the last problem if the coil is placed with its plane vertical and in the magnetic meridian at a point where the earth's horizontal force is 0.16?

4. How many lines of force will be set up in a horseshoe magnet with iron armature, the iron circuit having an average cross section of

36 sq. cm., each leg being 15 cm. long and the two legs 12 cm. apart between centers? On each leg is a coil of 400 turns of wire carrying a current of 5 ampères. The permeability of the iron may be taken as 100.

5. Find the force in Kilograms which an electromagnet can sustain when it is magnetized so that there are 6000 lines of force per sq. cm. in the core, each pole piece having an area of 36 sq. cms.

INTERACTION OF CURRENTS AND MAGNETS.

685. Mutual Action of Parallel Currents.-Two parallel conductors carrying currents in the same direction attract each other, while if the currents are in opposite directions they repel. This may be shown by means of Ampère's frame, a light rectangular frame of wire connected to a battery through two mercury cups so that it can freely revolve, as shown in figure 390.

FIG. 390.

If a second frame having a number of turns of wire through which a current passes is brought up so that one of its edges is parallel and near to one of the vertical wires of the pivoted frame, the attraction or repulsion of parallel currents is easily demonstrated.

Also if the frame B is held under the pivoted frame so that its upper edge is at right angles to the lower wire of the movable

frame the latter will then turn until the two are parallel and with the adjacent currents in the same direction.

686. Magnetic Field Around Parallel Currents.-If the lines of force of two parallel currents are studied by means of iron filings or a compass needle, they will be found as in figure 391 when the currents are both in the same direction. While if the currents are in opposite directions the resultant lines of magnetic force are as shown in figure 392.

According to Faraday's conception, the attraction in the first case may be explained by a tension in the magnetized medium or a tendency for it to shorten up in the direction of lines of force; on the other hand, the repulsion in the second case

is also in accordance with Faraday's idea that there is a pressure or tendency for a magnetized medium to expand at right angles to the lines of force.

It is also to be noticed that in the first case the field of force is stronger just outside of the conductors than it is between them, for between the two the magnetic effect of the one is opposed by the other; while in the second case the two act together to produce a strong magnetic field between them and a weaker field outside.

687. Action Between Current and Magnetic Field.-In the case just considered each conductor may be thought of as acted on by the magnetic field due to the other. That there is such a reaction between a magnetic field and a conductor carrying a current may be demonstrated by presenting one pole of a bar magnet to one of the vertical branches of Ampère's frame (§685) when the wire will move across the lines of force of the magnet. Or if a current of electricity is established in a light flexible conductor of tinsel cord hanging between the poles of a horseshoe magnet the cord is repelled outward from between the

poles when the current is downward and the poles are situated as shown in figure 393. If the current is reversed or if the magnet is turned over so that the poles are interchanged. the cord is drawn inward. The field of force due to a current flowing across a uniform magnetic field is shown in figure 394, where the current is supposed to flow downward in a wire which intersects the paper perpendicularly at O. The dotted lines are the lines of force of the uniform field, the circles are those of the current, and the full lines are the resultant lines of magnetic force. Clearly a tension in the medium along lines of force and pressure at right angles will urge the conductor in the

FIG. 393. Current in magnetic field.

FIG. 394.

direction shown by the arrow. At points nearer the top of the diagram than O, the force due to the current acts with the original field, while below 0, the two are in opposition, hence the field above O is strengthened by the current while it is weakened below the conductor, there being a neutral point P where there is no magnetic force at all.

These experiments lead to the following general rule:

When the magnetic field immediately adjoining a conductor carrying a current is strengthened on one side and weakened on the other by the effect of the current, the conductor is urged toward that side where the field is weakened.

If the current is not at right angles to the lines of force of the field, only that component of the magnetic force which is perpendicular to the conductor is effective, so that the effective magnetic force, the current, and the force acting on the conductor to

move it are in three directions mutually at right angles to each other, and their relation can always be determined by the rule just given or by the following mnemonic rule due to Fleming.

Hold the left hand, with forefinger extended and middle finger at right angles to it, so that the forefinger points in the direction of the magnetic force and the middle finger in the direction of the current; the thumb, at right angles to both, will then point in the direction in which the conductor is urged.

The amount of the force F experienced by the conductor is
F=1HI

where is the length of the conductor in the field, H is the strength of the component of the magnetic field at right angles to the conductor, and I is the current strength, all being measured in C. G. S. units.

688. Magnet and Current in a Coil.-The mutual action of a magnet and a current in a coil may be studied by a little light circular coil of wire connected to zinc and copper terminals which dip into a test-tube containing dilute sulphuric acid, the

FIG. 395.-Magnet and floating current.

whole system being floated in a tank by means of a cork.* On presenting the pole of a bar magnet the coil will set itself so that the lines of force of the magnet are in the same direction through the coil as the lines of force. of the coil itself, thus strengthening the field within the coil. It will then approach the pole and slip over it to the middle of the magnet.

If the magnet is now pulled out and quickly thrust through the coil in the opposite direction, the coil will slip off from the magnet, revolve so as to present the opposite face, and then again approach it. It may be seen that these actions result from the general rule of paragraph 687. For in each portion of the circular circuit the magnetic field is strengthened on one side of the conductor and weakened on the other and each part strives to move from

* In this experiment the test-tube should be weighted with shot until the cork is entirely submerged, only the upper part of the test-tube projecting above the surface, otherwise the motions will be greatly impeded by the surface viscosity of the water.