410.]

SHELLS.

33

direction everywhere normal to its surface, the intensity of the magnetization at any place multiplied by the thickness of the sheet at that place is called the Strength of the magnetic shell at that place.

If the strength of a shell is everywhere equal, it is called a Simple magnetic shell; if it varies from point to point it may be conceived to be made up of a number of simple shells superposed and overlapping each other. It is therefore called a Complex magnetic shell.

Let d8 be an element of the surface of the shell at Q, and the strength of the shell, then the potential at any point, P, due to the element of the shell, is

where is the angle between the vector QP, or r and the normal drawn from the positive side of the shell.

But if do is the solid angle subtended by dS at the point P

and therefore in the case of a simple magnetic shell

V = Φω,

or, the potential due to a magnetic shell at any point is the product of its strength into the solid angle subtended by its edge at the given point*.

410.] The same result may be obtained in a different way by supposing the magnetic shell placed in any field of magnetic force, and determining the potential energy due to the position of the shell.

If V is the potential at the element dS, then the energy due to this element is

dv

dx

dV

dV

+ n + z)ds,

or, the product of the strength of the shell into the part of the surface-integral of V due to the element dS of the shell.

Hence, integrating with respect to all such elements, the energy due to the position of the shell in the field is equal to the product of the strength of the shell and the surface-integral of the magnetic induction taken over the surface of the shell.

Since this surface-integral is the same for any two surfaces which

This theorem is due to Gauss, General Theory of Terrestrial Magnetism, § 38.

have the same bounding edge and do not include between them any centre of force, the action of the magnetic shell depends only on the form of its edge.

Now suppose the field of force to be that due to a magnetic pole of strength m. We have seen (Art. 76, Cor.) that the surfaceintegral over a surface bounded by a given edge is the product of the strength of the pole and the solid angle subtended by the edge at the pole. Hence the energy due to the mutual action of the pole and the shell is

Φηω,

and this (by Green's theorem, Art. 100) is equal to the product of the strength of the pole into the potential due to the shell at the pole. The potential due to the shell is therefore Þ w.

411.] If a magnetic pole m starts from a point on the negative surface of a magnetic shell, and travels along any path in space so as to come round the edge to a point close to where it started but on the positive side of the shell, the solid angle will vary continuously, and will increase by 4 during the process. The work done by the pole will be 4πÞm, and the potential at any point on the positive side of the shell will exceed that at the neighbouring point on the negative side by 4.

If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4, being positive when the positive side of the shell is inward. Hence such a shell exerts no action on any magnet placed either outside or inside the shell.

412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, the distribution of magnetism is called Lamellar. If is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point xyz by a line drawn within the magnet, then the conditions of lamellar magnetization are

The quantity, 4, which thus completely determines the magnetization at any point may be called the Potential of Magnetization. It must be carefully distinguished from the Magnetic Potential. 413.] A magnet which can be divided into complex magnetic shells is said to have a complex lamellar distribution of magThe condition of such a distribution is that the lines of

415.]

POTENTIAL DUE TO A LAMELLAR MAGNET.

35

magnetization must be such that a system of surfaces can be drawn This condition is expressed by the

cutting them at right angles.

well-known equation

is

Forms of the Potentials of Solenoidal and Lamellar Magnets.

414.] The general expression for the scalar potential of a magnet

where p denotes the potential at (x, y, z) due to a unit magnetic pole placed at , n, S, or in other words, the reciprocal of the distance between (§, 1, 5), the point at which the potential is measured, and (x, y, z), the position of the element of the magnet to which it is due.

This quantity may be integrated by parts, as in Arts. 96, 386.

dA dB dC

dx

V = [[p(At + Bm + Cw) ds = [[[p4+ 5+ d) de dy dz, [[p(Al+Bm+Cn) -SSS da dy dz

where l, m, n are the direction-cosines of the normal drawn outwards from dS, an element of the surface of the magnet.

When the magnet is solenoidal the expression under the integral sign in the second term is zero for every point within the magnet, so that the triple integral is zero, and the scalar potential at any point, whether outside or inside the magnet, is given by the surfaceintegral in the first term.

The scalar potential of a solenoidal magnet is therefore completely determined when the normal component of the magnetization at every point of the surface is known, and it is independent of the form of the solenoids within the magnet.

415.] In the case of a lamellar magnet the magnetization is determined by 4, the potential of magnetization, so that

аф B = dy

dy dz

d2p d2p
dx2 dy2

Π

The second term is zero unless the point (, 7, 5) is included in the magnet, in which case it becomes 4 π (4) where (4) is the value of at the point έ, n, C. The surface-integral may be expressed in terms of r, the line drawn from (x, y, z) to (§, n, ), and @ the angle which this line makes with the normal drawn outwards from dS, so that the potential may be written.

ф cos 0 dS+4π (†),

where the second term is of course zero when the point (§, 7, 5) is not included in the substance of the magnet.

The potential, V, expressed by this equation, is continuous even at the surface of the magnet, where becomes suddenly zero, for

if we write

and if

is the value of £ at a point just within the surface, and

, that at a point close to the first but outside the surface,

The quantity 2 is not continuous at the surface of the magnet. The components of magnetic induction are related to 2 by the equations

416.] In the case of a lamellar distribution of magnetism we may also simplify the vector-potential of magnetic induction. Its x-component may be written

F

афар = SSS Cay d=

By integration by parts we may put this in the form of the surface-integral

dp dz dy

Tz

dojas.

аф dz

-n

dy

The other components of the vector-potential may be written down from these expressions by making the proper substitutions.

On Solid Angles.

417.] We have already proved that at any point P the potential

418.]

SOLID ANGLES.

37.

due to a magnetic shell is equal to the solid angle subtended by: the edge of the shell multiplied by the strength of the shell. As we shall have occasion to refer to solid angles in the theory of electric currents, we shall now explain how they may be measured.

Definition. The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. This area is to be reckoned positive or negative according as it lies on the left or the righthand of the path of the radius vector as seen from the given point.

Let (,, ) be the given point, and let (x, y, z) be a point on the closed curve. The coordinates x, y, z are functions of s, the length of the curve reckoned from a given point. They are periodic functions of 8, recurring whenever s is increased by the whole length of the closed curve.

We may calculate the solid angle o directly from the definition thus. Using spherical coordinates with centre at (§, n, 5), and putting x − §

= r sin 0 cos p, y-n= r sin 0 sin 4, z-= r cos 0, we find the area of any curve on the sphere by integrating

the integration being extended round the curve s.

If the axis of z passes once through the closed curve the first term is 2. If the axis of z does not pass through it this term is zero.

418.] This method of calculating a solid angle involves a choice of axes which is to some extent arbitrary, and it does not depend solely on the closed curve. Hence the following method, in which no surface is supposed to be constructed, may be stated for the sake of geometrical propriety.

As the radius vector from the given point traces out the closed curve, let a plane passing through the given point roll on the closed curve so as to be a tangent plane at each point of the curve in succession. Let a line of unit-length be drawn from the given, point perpendicular to this plane. As the plane rolls round the