§ 6. Field due to a Circular Current. We will now complete what was said about a circular current in Chapter XVII. § 2. Near the wire the lines of force will practically be small circles about the wire, as in Chapter XVII. § 1, since the rest of the circuit will be relatively unimportant. But in mean positions the lines will be the resultants due to the whole circuit, and will not be circles, but curves of some other form situated symmetrically with respect to the central line and to the plane of the circular current.

Field at the centre of the circular current.- Now consider the field at the centre of the circle. Referring to §§ 2 and 3, we see that the following considerations will determine the field.

(i.) All the elements of this circular current are the same distance ✓ from the centre, and therefore the field-strength at the centre is directly proportional to the length 2 r of the circumference.

(ii.) The field will also be proportional to the inverse square of the common distance r, or to where r is the radius of the circle.

I

(iii.) If the current be given in electro-magnetic measure, it will be easy to find the magnetic field-strength at the centre, or the number of dynes with which unit pole would there be urged in a direction perpendicular to the plane of the circle.

From the above it follows that if c be the current in absolute measure, and I the magnetic field-strength at the centre, then.

The force with which a pole of strength μ is urged will be .

(i.)

The couple acting on a small needle of magnetic moment m, placed at the centre and lying in the plane of the circuit, will be.

If the current C be given in ampères, we must divide each of the above results by 10.

§ 7. Magnetic Shells; and the Fields due to Them. If we regard only the external magnetic field produced by an electric circuit, we find that we can replace the circuit by a thin steel sheet

whose faces are of opposite polarities; such a sheet magnet, or magnetic shell, giving exactly the same magnetic field as did the current, provided that certain conditions as to its thickness and its degree of magnetisation be observed.

But it will be necessary first to give some account of the nature of a magnetic shell, or, as it is otherwise expressed, of lamellar distributions of magnetism. We return to 'currents'

in § 9.

Consider an enormous number of small straight magnetic needles placed side by side, until they form a sheet whose area is very large as compared with the length of each little needle; that is, a sheet very large as compared with its thickness. If the needles be all turned one way, so that the one side of the sheet be formed of their north poles and the other side of their south poles, then we have what is called a lamellar distribution of magnetism ; or we have a magnetic shell.

In the case of magnetic needles, whose length was very great as compared with their thickness, it was convenient to speak of their poles and of their length; the product of the length and polestrength giving us that magnetic moment of the needle (called μl or m) upon which so much was found to depend. The reader will see from Chapter III. § 16, and from Chapter XVII., that upon the magnetic moment of the needle depended the external field that it gave, and also the couple with which a magnetic field acted upon it in turn.

Now in the case of a magnetic shell we must express ourselves somewhat differently. We must here consider the intensity p of the surface magnetisation, and the thickness t of the shell.

To take the case with which we shall be concerned, viz. that of a uniform shell, we may suppose all the little needles to have been equal both in length t, and in pole-strength. Then, if the sum of all the little poles that form 1 square centimètre of the surface of the shell would, if concentrated at one point, form a single pole of strength measured by p units, we may very naturally say that the density of magnetism on the surface of the shell is measured by p; the word 'density' being here used in the same sense as in electrostatics; viz. quantity [in this case magnetic quantity] per unit area.

If the thickness of the shell be called t, the product .p is

called the strength of the shell. It is the same as the magnetic moment of a needle of length and of pole-strength, representing the concentrated magnetism of 1 square centimètre of the surface. We will use the single symbol j for the strength of a shell.

Now let us consider any point P external to the shell. Let the boundary of the shell subtend a solid angle measured by at the point P; and let the strength of the shell be j. Then it can be shown that the field at the point P depends upon j, upon

, and upon the way in which alters in value as we move from point to point. As long as the thickness is relatively very small, it is found (by mathematical methods belonging properly to the integral calculus) that we are concerned with the strength j; not with and separately.

The reader must notice that (according to the mathematical investigation not given here) the field at P does not depend upon the shape of the shell, so long as its edge remains the same. If the shell be bent into a surface nearly closed, but having a hole of a certain shape and size left unclosed, it will give the same external field as would a shell that just fitted this hole; since the two would have the same edge. And a completely closed shell gives no external field.

§ 8. Magnetic Potentials due to Magnetic Shells.-The reader is now referred to what was said of potential, and of force

Note on solid angles.-Let us consider a point P and a sphere of unit radius described about P as centre. Any portion of the surface is said to subtend a solid angle at the centre, this angle being measured by the fraction portion of surface in square centimètres Since the radius is one centimètre, (distance from centre in centimètres)?

we may say that the solid angle is measured by the area of the portion of the surface considered, this area being expressed in square centimètres. Now let lines be drawn from P to the edge of the magnetic shell. These will intercept a portion of the surface of our unit sphere. The solid angle ♫ subtended by the boundary of the magnetic shell at the point P is measured by the portion of the surface, expressed in square centimètres, which is thus intercepted on the surface of the unit sphere described about P as centre.

Thus, since the total surface of unit sphere is 4 square centimètres, the total solid angle about any point is 4 π.

If a point is relatively very close to the surface, the surface subtends (very nearly) a solid angle of 2 π at the point; and subtends exactly 2 when the point is actually on the surface.

as measured by rate of change of potential, in Chapter X. §§ 9 14 and elsewhere. If + unit pole' and ' unit pole' be substituted for unit electricity' and '-unit electricity,' all that was '+ there said applies to magnetic fields. (Thus, for example, a pole μ gives out 4 μ marked lines of magnetic force.) It is then readily understood that if we know the magnetic potential at a point P due to a magnetic shell or due to an electric circuit, and if we know at what rate this potential changes in different directions, then we know all about the magnetic field at P.

In the case of the magnetic shell, it can be shown that the magnetic potential on unit pole at P due to this shell is measured by the product j. The force at P in any direction will, for the general reasons given in Chapter X., be measured by the rate at which this potential changes its value in this direction; and this, since j does not alter for the same shell, depends upon the rate at which changes in that direction. Thus a small shell close by and a large one far off may give the same value of at P, or the same potential at P; but in general the rate at which 2 changes, or the force at P, will be different in the two cases.

=

Inside a completely closed surface formed of a magnetic shell, the potential is everywhere 4π.j, as explained; since now 4 π. It does not, therefore, change from point to point; or there is a constant potential and no force in the hollow of such a closed surface. Outside it the potential and force are both zero, since, as explained, now equals zero.

As was seen in Chapter X., the direction in which the potential changes most rapidly is the direction of the resultant force, or is the direction of 'the' lines of force of the field.

Of course the potential on a pole of μ units will be measured by μjQ.

It follows also that the work done between two positions of the pole μ, at which the shell subtends solid angles 2, and 2, respectively, will be measured by

The mathematical investigation of the potentials and fields due to a magnetic shell demands a knowledge of the infinitesimal calculus. But an investigation in which infinitesimal notation, at any rate, is not used, can be found in Cumming's 'Theory of Electricity' (Macmillan).

X

§ 9. Magnetic Equivalent of an Electric Circuit.--We now return to that which was the main reason for the digression of §§ 7 and 8.

If we have on the one hand a circuit carrying a current measured by C in absolute electro-magnetic units, and on the other hand a magnetic shell of the same boundary and of a 'strength' numerically equal to C; and if we examine the potentials and fields at any external point, due to the two re spectively, then it is found that the two give identical results.

The proof of this equivalence demands a knowledge of the infinitesimal calculus; and in the present Course the reader must be content to accept as a result of mathematical analysis confirmed by experiment the statement that .

'As regards the external magnetic field produced, a circuit of current-strength C (as measured in absolute electro-magnetic units) is exactly equivalent to a magnetic shell having the same boundary and having an uniform strength j numerically equal to C.'

So the formulæ of the last section become, for an electric circuit, μ C and μ C (1

2).

Experiment.-De la Rive's floating battery.-A small battery-cell is constructed, composed of two strips the one of zinc and the other copper immersed

in dilute acid in a short wide test tube. The circuit is completed by an insulated wire wound into a small circular coil that has its plane vertical when the test tube is vertical. This small cell and circular current is floated on the surface of a large vessel of water, by means of a cork or other float through which the test tube passes.

We thus have a vertical circular current capable both of rotation on its vertical axis and of horizontal translation.

If we test it by means of a large bar magnet we shall find that it acts as would a thin circular magnetic shell. As could be deduced from Chapter