the Leyden Jar,” ib. 167 [ii.], p. 599, containing many valuable observations on the residual charge of Leyden jars; W. E. Ayrton and J. Perry, “A Preliminary Account of the Reduction of Observations on Strained Material, Leyden Jars and Voltameters,” Proc. Roy. Soc., 1880, 30, p. 411, showing experiments on residual charge of condensers and a comparison between the behaviour of dielectrics and glass fibres under torsion. In connexion with this paper the reader may also be referred to one by L. Boltzmann, “Zur Theorie der elastischen Nachwirkung,” Wien. Acad. Sitz.-Ber., 1874, 70.
Distribution of Electricity on Conductors.—We now proceed to consider in more detail the laws which govern the distribution of electricity at rest upon conductors. It has been shown above that the potential due to a charge of q units placed on a very small sphere, commonly called a point-charge, at any distance x is q/x. The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges q1, q2, q3, &c., distributed in any manner, is the sum of them separately, or
where x1, x2, x3, &c., are the distances of the respective point charges from the point in question at which the total potential is required. The resultant electric force E at that point is then obtained by differentiating V, since E = −dV /dx, and E is in the direction in which V diminishes fastest. In any case, therefore, in which we can sum up the elementary potentials at any point we can calculate the resultant electric force at the same point.
We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force. Let us assume the field divided up into tubes of electric force as already explained, and these cut normally by equipotential surfaces. We can then establish some important properties of these tubes and surfaces. At each point in the field the electric force can have but one resultant value. Hence the equipotential surfaces cannot cut each other. Let us suppose any other surface described in the electric field so as to cut the closely compacted tubes. At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle θ to the normal to the selected surface at that point. Let dS be an element of the surface. Then the quantity E cos θdS is the product of the normal component of the force and an element of the surface, and if this is summed up all over the surface we have the total electric flux or induction through the surface, or the surface integral of the normal force mathematically expressed by ∫E cos θdS, provided that the dielectric constant of the medium is unity.
 |
| Fig. 3. |
We have then a very important theorem as follows:—If any closed surface be described in an electric field which wholly encloses or wholly excludes electrified bodies, then the total flux through this surface is equal to 4π- times the total quantity of electricity within it.[1] This is commonly called Stokes’s theorem. The proof is as follows:—Consider any point-charge E of electricity included in any surface S, S, S (see fig. 3), and describe through it as centre a cone of small solid angle dω cutting out of the enclosing surface in two small areas dS and dS′ at distances x and x′. Then the electric force due to the point charge q at distance x is q/x, and the resolved part normal to the element of surface dS is q cosθ / x2. The normal section of the cone at that point is equal to dS cosθ, and the solid angle dω is equal to dS cosθ / x2. Hence the flux through dS is qdω. Accordingly, since the total solid angle round a point is 4π, it follows that the total flux through the closed surface due to the single point charge q is 4πq, and what is true for one point charge is true for any collection forming a total charge Q of any form. Hence the total electric flux due to a charge Q through an enclosing surface is 4πQ, and therefore is zero through one enclosing no electricity.
Stokes’s theorem becomes an obvious truism if applied to an incompressible fluid. Let a source of fluid be a point from which an incompressible fluid is emitted in all directions. Close to the source the stream lines will be radial lines. Let a very small sphere be described round the source, and let the strength of the source be defined as the total flow per second through the surface of this small sphere. Then if we have any number of sources enclosed by any surface, the total flow per second through this surface is equal to the total strengths of all the sources. If, however, we defined the strength of the source by the statement that the strength divided by the square of the distance gives the velocity of the liquid at that point, then the total flux through any enclosing surface would be 4π times the strengths of all the sources enclosed. To every proposition in electrostatics there is thus a corresponding one in the hydrokinetic theory of incompressible liquids.
Let us apply the above theorem to the case of a small parallel-epipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z). Its angular points have then co-ordinates (x ± 12dx, y ± 12dy, z ± 12dz). Let this rectangular prism be supposed to be wholly filled up with electricity of density ρ; then the total quantity in it is ρ dx dy dz. Consider the two faces perpendicular to the x-axis. Let V be the potential at the centre of the prism, then the normal forces on the two faces of area dy·dx are respectively
| − ( | dV | + 12 | d2V | dx) and ( | dV | − 12 | d2V | dx), |
| dx | dx2 | dx | dx2 |
and similar expressions for the normal forces to the other pairs of faces dx·dy, dz·dx. Hence, multiplying these normal forces by the areas of the corresponding faces, we have the total flux parallel to the x-axis given by −(d2V / dx2) dx dy dz, and similar expressions for the other sides. Hence the total flux is
| − ( | d2V | + | d2V | + | d2V | ) dx dy dz, |
| dx2 | dy2 | dz2 |
and by the previous theorem this must be equal to 4πρdx dy dz.
Hence
| d2V | + | d2V | + | d2V | + 4πρ = 0 |
| dx2 | dy2 | dz2 |
This celebrated equation was first given by S. D. Poisson, although previously demonstrated by Laplace for the case when ρ = 0. It defines the condition which must be fulfilled by the potential at any and every point in an electric field, through which ρ is finite and the electric force continuous. It may be looked upon as an equation to determine ρ when V is given or vice versa. An exactly similar expression holds good in hydrokinetics, provided that for the electric potential we substitute velocity potential, and for the electric force the velocity of the liquid.
The Poisson equation cannot, however, be applied in the above form to a region which is partly within and partly without an electrified conductor, because then the electric force undergoes a sudden change in value from zero to a finite value, in passing outwards through the bounding surface of the conductor. We can, however, obtain another equation called the “surface characteristic equation” as follows:—Suppose a very small area dS described on a conductor having a surface density of electrification σ. Then let a small, very short cylinder be described of which dS is a section, and the generating lines are normal to the surface. Let V1 and V2 be the potentials at points just outside and inside the surface dS, and let n1 and n2 be the normals to the surface dS drawn outwards and inwards; then −dV1 / dn1 and −dV2 / dn2 are the normal components of the force over the ends of the imaginary small cylinder. But the force perpendicular to the curved surface of this cylinder is everywhere zero. Hence the total flux through the surface considered is −{(dV1 / dn1) + (dV2 / dn2)} dS, and this by a previous theorem must be equal to 4πσdS, or the total included electric quantity. Hence we have the surface characteristic equation,[2]
Let us apply these theorems to a portion of a tube of electric force. Let the part selected not include any charged surface. Then since the generating lines of the tube are lines of force, the component of the electric force perpendicular to the curved surface of the tube is everywhere zero. But the electric force is normal to the ends of the tube. Hence if dS and dS′ are the areas of the ends, and +E and -E′ the oppositely directed electric forces at the ends of the tube, the surface integral of normal force on the flux over the tube is
and this by the theorem already given is equal to zero, since the tube includes no electricity. Hence the characteristic quality of a tube of electric force is that its section is everywhere inversely as the electric force at that point. A tube so chosen that EdS for one section has a value unity, is called a unit tube, since the product of force and section is then everywhere unity for the same tube.
In the next place apply the surface characteristic equation to any point on a charged conductor at which the surface density is σ. The electric force outward from that point is −dV/dn, where dn is a distance measured along the outwardly drawn normal, and the force within the surface is zero. Hence we have
The above is a statement of Coulomb’s law, that the electric force at the surface of a conductor is proportional to the surface density of the charge at that point and equal to 4π times the density.[3]
- ↑ The beginner is often puzzled by the constant appearance of the factor 4π in electrical theorems. It arises from the manner in which the unit quantity of electricity is defined. The electric force due to a point-charge q at a distance r is defined to be q/r2, and the total flux or induction through the sphere of radius r is therefore 4πq. If, however, the unit point charge were defined to be that which produces a unit of electric flux through a circumscribing spherical surface or the electric force at distance r defined to be 14πr2, many theorems would be enunciated in simpler forms.
- ↑ See Maxwell, Electricity and Magnetism, vol. i. § 78b (2nd ed.).
- ↑ Id. ib. vol. i. § 80. Coulomb proved the proportionality of electric surface force to density, but the above numerical relation E = 4πσ was first established by Poisson.
