A Treatise on Electricity and Magnetism/Part III/Chapter III
MAGNETIC SOLENOIDS AND SHELLS.
On Particular Forms of Magnets.
407.] If a long narrow filament of magnetic matter like a wire is magnetized everywhere in a longitudinal direction, then the product of any transverse section of the filament into the mean intensity of the magnetization across it is called the strength of the magnet at that section. If the filament were cut in two at the section without altering the magnetization, the two surfaces, when separated, would be found to have equal and opposite quantities of superficial magnetization, each of which is numerically equal to the strength of the magnet at the section.
A filament of magnetic matter, so magnetized that its strength is the same at every section, at whatever part of its length the section be made, is called a Magnetic Solenoid.
If m is the strength of the solenoid, ds an element of its length, r the distance of that element from a given point, and ε the angle which r makes with the axis of magnetization of the element, the potential at the given point due to the element is
Integrating this expression with respect to s, so as to take into account all the elements of the solenoid, the potential is found to be
r1 being the distance of the positive end of the solenoid, and r2 that of the negative end from the point where V exists.
Hence the potential due to a solenoid, and consequently all its magnetic effects, depend only on its strength and the position of its ends, and not at all on its form, whether straight or curved, between these points.
Hence the ends of a solenoid may be called in a strict sense its poles.
If a solenoid forms a closed curve the potential due to it is zero at every point, so that such a solenoid can exert no magnetic action, nor can its magnetization be discovered without breaking it at some point and separating the ends.
If a magnet can be divided into solenoids, all of which either form closed curves or have their extremities in the outer surface of the magnet, the magnetization is said to be solenoidal, and, since the action of the magnet depends entirely upon that of the ends of the solenoids, the distribution of imaginary magnetic matter will be entirely superficial.
Hence the condition of the magnetization being solenoidal is
where A, B, C are the components of the magnetization at any point of the magnet.
408.] A longitudinally magnetized filament, of which the strength varies at different parts of its length, may be conceived to be made up of a bundle of solenoids of different lengths, the sum of the strengths of all the solenoids which pass through a given section being the magnetic strength of the filament at that section. Hence any longitudinally magnetized filament may be called a Complex Solenoid.
If the strength of a complex solenoid at any section is m, then the potential due to its action is
This shews that besides the action of the two ends, which may in this case be of different strengths, there is an action due to the distribution of imaginary magnetic matter along the filament with a linear density
409.] If a thin shell of magnetic matter is magnetized in a 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 dS be an element of the surface of the shell at Q, and Φ the strength of the shell, then the potential at any point, P, due to the element of the shell, is
where ε is the angle between the vector QP, or r and the normal drawn from the positive side of the shell.
But if dω is the solid angle subtended by dS at the point P
and therefore in the case of a simple magnetic shell
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
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 magnetiinduction taken over the surface of the shell.
Since this surface-integral is the same for any two surfaces 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 surface-integral 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
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 4π during the process. The work done by the pole will be 4πΦm, 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 4πΦ.
If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4πΦ, 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 φ is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point x y z by a line drawn within the magnet, then the conditions of lamellar magnetization are
The quantity, φ, 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 magnetism. The condition of such a distribution is that the lines of magnetization must be such that a system of surfaces can be drawn cutting them at right angles. This condition is expressed by the well-known equation
Forms of the Potentials of Solenoidal and Lamellar Magnets.
414.] The general expression for the scalar potential of a magnet is
where p denotes the potential at (x, y, z) due to a unit magnetic pole placed at ξ, η, ζ, or in other words, the reciprocal of the distance between (ξ, η, ζ), 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.
where l, m, n are the direction-cosines of the normal drawn out wards 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 surface-integral 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 φ, the potential of magnetization, so that
The expression for V may therefore be written
Integrating this expression by parts, we find
The second term is zero unless the point (ξ, η, ζ) is included in the magnet, in which case it becomes 4 π (φ) where (φ) is the value of φ at the point ξ, η, ζ. The surface-integral may be expressed in terms of r, the line drawn from (x, y, z) to (ξ, η, ζ), and the angle which this line makes with the normal drawn outwards from dS, so that the potential may be written
where the second term is of course zero when the point (ξ, η, ζ) 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 φ becomes suddenly zero, for if we write
and if Ω1 is the value of Ω at a point just within the surface, and Ω2 that at a point close to the first but outside the surface,
The quantity Ω is not continuous at the surface of the magnet.
The components of magnetic induction are related to Ω 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
By integration by parts we may put this in the form of the surface-integral
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 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 right-hand of the path of the radius vector as seen from the given point.
Let (ξ, η, ζ) 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 s, recurring whenever s is increased by the whole length of the closed curve.
We may calculate the solid angle ω directly from the definition thus. Using spherical coordinates with centre at (ξ, η, ζ), and putting
we find the area of any curve on the sphere by integrating
or, using the rectangular coordinates,
the integration being extended round the curve s.
If the axis of z passes once through the closed curve the first term is 2π. 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 closed curve the extremity of the perpendicular will trace a second closed curve. Let the length of the second closed curve be σ, then the solid angle subtended by the first closed curve is
This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circumference of the polar curve, is numerically equal to the circumference of a great circle of the sphere.
This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem.
419.] A closed curve s is given in space, and we have to find the solid angle subtended by s at a given point P.
If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if σ is the path of the pole as it approaches the point P, the potential must be the result of a line- integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and σ.
When P is at an infinite distance, the solid angle is evidently zero. As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point approaches.
As the point P moves from P to P' over the element dσ, the element QQ' of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ' will sweep over an area on the spherical surface, which we may write
To find Π let us suppose P fixed while the closed curve is moved parallel to itself through a distance dσ equal to PP' but in the opposite direction. The relative motion of the point P will be the same as in the real case.
During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to QQ' and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment dω which we are in search of.
To determine the value of this solid angle, let θ and θ' be the angles which ds and dσ make with PQ respect ively, and let φ be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds . dσ on a plane perpendicular to PQ or r will be
and since this is equal to r2dω, we find
420.] We may express the angles θ, θ', and φ in terms of r, and its differential coefficients with respect to s and σ, for
We thus find the following value for Π2,
A third expression for Π in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is dω and whose axis is r is
But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and dσ on the axis of x, y and z, as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of Π,
This expression gives the value of Π free from the ambiguity of sign introduced by equation (5).
421.] The value of ω, the solid angle subtended by the closed curve at the point P, may now be written
where the integration with respect to s is to be extended completely round the closed curve, and that with respect to σ from A a fixed point on the curve to the point P. The constant ω0 is the value of the solid angle at the point A. It is zero if A is at an infinite distance from the closed curve.
The value of ω at any point P is independent of the form of the curve between A and P provided that it does not pass through the magnetic shell itself. If the shell be supposed infinitely thin, and if P and P' are two points close together, but P on the positive and P' on the negative surface of the shell, then the curves AP and AP' must lie on opposite sides of the edge of the shell, so that PAP' is a line which with the infinitely short line P'P forms a closed circuit embracing the edge. The value of ω at P exceeds that at P' by 4π, that is, by the surface of a sphere of radius unity.
Hence, if a closed curve be drawn so as to pass once through the shell, or in other words, if it be linked once with the edge of the shell, the value of the integral extended round both curves will be 4π.
This integral therefore, considered as depending only on the closed curve s and the arbitrary curve AP, is an instance of a function of multiple values, since, if we pass from A to P along different paths the integral will have different values according to the number of times which the curve AP is twined round the curve s.
If one form of the curve between A and P can be transformed into another by continuous motion without intersecting the curve s, the integral will have the same value for both curves, but if during the transformation it intersects the closed curve n times the values of the integral will differ by 4πn.
If s and σ are any two closed curves in space, then, if they are not linked together, the integral extended once round both is zero.
If they are intertwined n times in the same direction, the value of the integral is 4πn. It is possible, however, for two curves to be intertwined alternately in opposite directions, so that they are inseparably linked together though the value of the integral is zero. See Fig. 4.
It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while describing a closed curve in presence of a closed electric current, and indicating the geometrical connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibnitz, Euler and Vandermonde. We have now, however, some progress to report, chiefly due to Riemann, Helmholtz and Listing.
422.] Let us now investigate the result of integrating with respect to s round the closed curve.
One of the terms of Π in equation (7) is
If we now write for brevity
the integrals being taken once round the closed curve s, this term of Π may be written
and the corresponding term of will be
Collecting all the terms of Π, we may now write
By assuming dσ successively in the direction of the axes of x, y and z, we obtain for the values of the components of the magnetic force
This quantity is evidently the rate of decrement of ω, the magnetic potential, in passing along the curve σ, or in other words, it is the magnetic force in the direction of dσ.
The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential ω, functions having a series of values, but are perfectly determinate for every point in space.
The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction:
Let a point Q travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When Q has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P.
Potential Energy of a Magnetic Shell placed in a Magnetic Field.
423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength φ placed in a magnetic field whose potential is V, is
where l, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell.
Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we may write
where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell.
If we now suppose that the magnetic field is that due to a second magnetic shell whose strength is φ', the values of F, G, H will be
where the integrations are extended once round the curve s', which forms the edge of this shell.
Substituting these values in the expression for M we find
where the integration is extended once round s and once round s'. This expression gives the potential energy due to the mutual action of the two shells, and is, as it ought to be, the same when s and s' are interchanged. This expression with its sign reversed, when the strength of each shell is unity, is called the potential of the two closed curves s and s'. It is a quantity of great importance in the theory of electric currents. If we write ε for the angle between the directions of the elements ds and ds', the potential of s and s' may be written
It is evidently a quantity of the dimension of a line.