earth in these latitudes k is about 30, so that the term

1.6, and so is small compared to the term k. The cylinder of iron thus tends to set itself parallel to the direction of the field.

In the case of a diamagnetic body the value of k is so small that 1 +2πk is practically unity, and the term

k is very nearly equal to k, 1+2πk so that in a uniform field there is no measurable directive force exerted upon even a cylinder of bismuth (= -0.6 10-6). When a diamagnetic cylinder is placed between the poles of a strong electro-magnet it is, however, found that the cylinder tends to turn so as to set itself with its length at right angles to the lines of force. The reason is, that the field between the poles of such a magnet is not uniform, being stronger near the centre than at the edge, and the diamagnetic body turns so that as much of itself as possible is in the weaker part of the field. In a diamagnetic body the permeability is less than unity, while the permeability of air is unity and the medium with the higher permeability tends to force the other away from the stronger parts of the field, so that the greatest number of tubes of induction may crowd into a given space.

Solids are not the only bodies which exhibit magnetic properties; thus oxygen and some solutions of iron salts are paramagnetic, while water and alcohol are diamagnetic.

By means of these liquids it can be shown that the direction in which a cylindrical tube filled with, say, a paramagnetic liquid tends to set itself depends on the susceptibility of the surrounding medium. Thus a tube containing a weak solution of ferric chloride will in air or water set itself parallel to the direction of the field, since its susceptibility is greater than that of either air or water. If, however, it is surrounded by a stronger solution of ferric chloride, it will behave like a diamagnetic body and set itself with its length perpendicular to the direction of the lines of force of the field. This effect is at once explainable if we consider that when the tube containing the weak solution is placed in the stronger solution, since the permeability of the contents of the tube is less than that of the surrounding medium, the induction through the tube will be less than that which would exist if the tube were removed, and the tube is practically diamagnetic with respect to the surrounding stronger solution.

It is therefore evident that in order to account for diamagnetism it is not necessary to assume that these bodies have a negative susceptibility, but only that their susceptibility is less than that of air, or, since the susceptibility of air and of a vacuum are very nearly the same, less than that of a vacuum. Since the susceptibility and the permeability are related by the equation

μ=1+4πή,

and that for the most diamagnetic body known the susceptibility is less than 1/4, the permeability will in all cases be greater than zero.

PART VI.-ELECTRO-MAGNETISM

CHAPTER XII

FORCES ACTING ON CONDUCTORS CONVEYING

CURRENTS

509. Force acting on a Straight Conductor conveying a Current when placed in a Magnetic Field.-If a straight conductor, in which a current is flowing, is placed in a magnetic field, so that it is at right angles to the lines of force of the field, then, owing to the magnetic field due to the current, the distribution of the lines of force of the field will be altered. In Fig. 490 are shown the lines of force due to a conductor which is perpendicular to the plane of the paper and passes through the point A when placed in a uniform magnetic field in which the lines of force ran parallel to the line CD. Remembering that we have every reason to suppose that there exists a tension along the lines of force, and a pressure at right angles, while the lines of force act as if they were connected with the body by which they are produced, it is evident that, as a result of the crowding of the lines of force on one side of the conductor, and their separation on the other, the conductor conveying the current will be acted upon by a force in the direction of the arrow.

If the current flows downwards, the lines of force are circles which run in the clockwise direction, and at the upper part of the diagram they strengthen the magnetic field, since they run in the same direction as the lines of force of the field. In the lower part of the diagram the lines of force due to the current and to the field are in opposite directions, and therefore the resultant magnetic field is the difference of the fields due to the two causes. The direction in which the conductor tends to move is therefore at right angles to the direction of the lines of the field, and towards the part of the field where the lines of force due to the current are in the opposite direction to the lines of force of the field. Since the direction of the lines of force of the current can at once be remembered by one of the rules given in § 472, the direction of the force acting on a conductor in a magnetic field can at once be remembered. Fleming has given a convenient rule for remembering the direction in which a conductor conveying a current in a magnetic field will tend to move. If the index finger of the left hand is held pointing in the direction of the lines

of force of the field, and the middle finger in the direction of the current, the conductor will tend to move in the direction of the outstretched

thumb, and at right angles to the lines of force of the field. A study of Fig. 491 will make the matter clear. Thus a vertical wire in which a current was flowing downwards would, on account of the earth's hori

zontal component, be acted upon by a force tending to move it in an easterly direction. In this case, according to our rule, the left hand must be held with the index finger pointing towards the north, since the lines of force of the horizontal component

of the earth's field run from south to north, and with the middle finger pointing downwards. The outstretched thumb will then point towards the east.

Ampère, who made a lengthy series of experiments on the forces acting on conductors in which currents are flowing, showed that if a conductor of length is traversed by a current of C c.g.s. units, and is placed at right angles to the lines of force of a uniform magnetic field of strength H, the force acting on the conductor will be equal to ICH. If the current is measured in amperes, then, since the ampere is one-tenth of a c.g.s. unit, the force will be one-tenth of the above.

If the conductor is not at right angles to the direction of the lines of force of the field, in calculating the force we must resolve the field into two components, one perpendicular to the direction of the current, and the other parallel. Then the component parallel to the direction of the current will produce no force on the conductor, and the force due to the other component is calculated by the formula given above.

510. Force acting on a Rectangular Coil conveying a Current when in a Magnetic Field.-As an example of the application of the formula given in the last section, we may calculate the force acting on a rectangular coil when placed in a uniform magnetic field. Suppose that the field is of strength H, and that the lines of force are horizontal, and run from south to north. Let the coil, ABCD (Fig. 492), consist of a single turn of wire in the form of a rectangle of length and breadth 6, and let it be placed with its plane in the vertical plane parallel to the direction of the field. Since the top and bottom of the rectangle, AB and CD, are parallel to the direction of the lines of force of the field, they will experience no force. If a current of C c.g.s. units is flowing round the rectangle,

FIG. 492.

H

so that its direction in AB is from A to B, the vertical side AD will be acted upon by a force ICH in the direction of EF. In the same way the vertical side BC will be acted upon by an equal force in the direction GK. The resultant of these forces, since they are equal and opposite parallel forces, is a couple, of which the magnitude is b.ICH, tending to turn the rectangle round in the anticlockwise direction, when looked at from above.

Next suppose that the rectangle is allowed to turn round under the influence of this couple into the position A'B'C'D', in which its plane is perpendicular to the direction of the lines of force of the field. In this position the top A'B' is now perpendicular to the lines of force, and therefore experiences a force. By the rule we see that this will be an upward force of bCH. Since the direction of the current in the bottom of the rectangle is opposite to that in A'B', this portion of the circuit will be acted upon by a downward force bCH, which will produce equilibrium with the force exerted upon A'B'. Since the current in the vertical side

A'D' is upwards, the force on this side is still ICH in the direction EF, while the force on the vertical side B'C' is ICH in the direction GK. Since these two forces are now not only equal and opposite, but act in the same straight line, they are in equilibrium. Thus, in this position, the rectangle is in a state of stable equilibrium, and there is no force, due to the magnetic field, either tending to move it bodily or to turn it about any axis.

From the above investigations we see that the rectangle tends to set itself in such a position that the number of tubes of force which pass through it is a maximum, and that the direction of the lines of force due to the circuit is, inside the coil, the same as that of the lines of force of the field.

If from analogy with a magnet we call the face of a coil at which the lines of force leave the space included by the coil the north surface of the coil, then we may express the results to which we have been led as follows: A circuit in which a current is flowing tends to turn so that the number of tubes of force due to the field entering its south face is a maximum. Since the direction of the lines of force due to the coil in the position A'B' run inside the coil in the same direction as the lines of force due to the field, we may also summarise the result as follows: A coil in which a current is flowing will tend to set itself so that the total number of lines of force which pass through it are a maximum; that is, so that the total induction through the coil is a maximum.

511*. Magnetic Shell.-A thin plate of magnetic material which is magnetised so that the direction of magnetisation is everywhere perpendicular to the surfaces of the sheet is called a magnetic shell. The product of the thickness of the shell into the intensity of magnetisation is called the strength of the shell.

It can be shown that the magnetic force exerted by a closed circuit in which a current C is flowing is the same as that which would be exerted by a magnetic shell which occupied the space bounded by the circuit, and of which the strength was equal to C. The side of the shell which is a north pole must, of course, correspond to what was called the north side of the circuit in the last section. If S is the area of the space enclosed by the circuit, so that S is the area of the pole of the shell, the strength of each pole will be IS, for, as we have seen in § 501, the surface density of the free magnetism on the pole is numerically equal to the intensity of magnetisation. Hence the number of tubes of force which leave the north face of the shell and enter the south face, completing their course through the substance of the shell, is

4πSI.

The number of tubes of force which pass through any given area is called the induction through that area; or, to distinguish between this use of the term and that in § 502, we may call the total number of lines