the ellipsoid a similar and slightly larger one, that distribution will be in equilibrium and will produce a constant potential throughout the interior. Thus if σ is the surface density, δ the thickness of the shell at any point, and ρ the assumed volume density of the matter of the shell, we have σ = Aδρ. Then the quantity of electricity on any element of surface dS is A times the mass of the corresponding element of the shell; and if Q is the whole quantity of electricity on the ellipsoid, Q = A times the whole mass of the shell. This mass is equal to 4πabcρμ; therefore Q = A4πabcρμ and δ = μp, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane. Hence
Accordingly for a given ellipsoid the surface density of free distribution of electricity on it is everywhere proportional to the length of the perpendicular let fall from the centre on Capacity of an ellipsoid. the tangent plane at that point. From this we can determine the capacity of the ellipsoid as follows: Let p be the length of the perpendicular from the centre of the ellipsoid, whose equation is x2/a2 + y2/b2 + z2/c2 = 1 to the tangent plane at x, y, z. Then it can be shown that 1/p2 = x2/a4 + y2/b4 + z2/c4 (see Frost’s Solid Geometry, p. 172). Hence the density σ is given by
| σ = | Q | 1 | . | |
| 4πabc | √(x2 / a4 + y2 / b4 + z2 / c4) |
and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression,
| V = ∫ | σdS | = | Q | ∫ | dS |
| r | 4πabc | r √(x2 / a4 + y2 / b4 + z2 / c4) |
Accordingly the capacity C of the ellipsoid is given by the equation
| 1 | = | 1 | ∫ | dS |
| C | 4πabc | √(x2 + y2 + z2) √(x2 / a4 + y2 / b4 + z2 / c4) |
It has been shown by Professor Chrystal that the above integral may also be presented in the form,[1]
| 1 | = 12 ∫∞0 | dλ |
| C | √{(a2 + λ) (b2 + λ) (c2 + λ)} |
The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.
Thus if the ellipsoid is one of revolution, and ds is an element of arc which sweeps out the element of surface dS, we have
| dS = 2πyds = 2πydx / ( | dx | ) = 2πydx / ( | py | ) = | 2πb2 | dx. |
| ds | b | p |
Hence, since σ = Qp / 4πab2, σdS = Qdx / 2a.
Accordingly the distribution of electricity is such that equal parallel slices of the ellipsoid of revolution taken normal to the axis of revolution carry equal charges on their curved surface.
The capacity C of the ellipsoid of revolution is therefore given by the expression
| 1 | = | 1 | ∫ | dx |
| C | 2a | √(x2 + y2) |
If the ellipsoid is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to
| 1 | = | 1 | logε ( | 1 + e | ) |
| C1 | 2ae | 1 − e |
Whereas if it is an ellipsoid of revolution round the minor axis b (oblate), we have
| 1 | = | sin−1ae |
| C2 | ae |
In each case we have C = a when e = 0, and the ellipsoid thus becomes a sphere.
In the extreme case when e = 1, the prolate ellipsoid becomes a long thin rod, and then the capacity is given by
which is identical with the formula (2) already obtained. In the other extreme case the oblate spheroid becomes a circular disk when e = 1, and then the capacity C2 = 2a/π. This last result shows that the capacity of a thin disk is 2/π = 1/1.571 of that of a sphere of the same radius. Cavendish (Elec. Res. pp. 137 and 347) determined in 1773 experimentally that the capacity of a sphere was 1.541 times that of a disk of the same radius, a truly remarkable result for that date.
Three other cases of practical interest present themselves, viz. the capacity of two concentric spheres, of two coaxial cylinders and of two parallel planes.
Consider the case of two concentric spheres, a solid one enclosed in a hollow one. Let R1 be the radius of the inner sphere, R2 the inside radius of the outer sphere, and R2 the outside radius of the outer spherical shell. Let a charge +Q be Capacity of two concentric spheres. given to the inner sphere. Then this produces a charge −Q on the inside of the enclosing spherical shell, and a charge +Q on the outside of the shell. Hence the potential V at the centre of the inner sphere is given by V = Q/R1 − Q/R2 + Q/R3. If the outer shell is connected to the earth, the charge +Q on it disappears, and we have the capacity C of the inner sphere given by
Such a pair of concentric spheres constitute a condenser (see Leyden Jar), and it is obvious that by making R2 nearly equal to R1, we may enormously increase the capacity of the inner sphere. Hence the name condenser.
The other case of importance is that of two coaxial cylinders. Let a solid circular sectioned cylinder of radius R1 be enclosed in a coaxial tube of inner radius R2. Then when the inner Capacity of two coaxial cylinders. cylinder is at potential V1 and the outer one kept at potential V2 the lines of electric force between the cylinders are radial. Hence the electric force E in the interspace varies inversely as the distance from the axis. Accordingly the potential V at any point in the interspace is given by
where R is the distance of the point in the interspace from the axis, and A is a constant. Hence V2 − V1 = −A log R2/R1. If we consider a length l of the cylinder, the charge Q on the inner cylinder is Q = 2πR1lσ, where σ is the surface density, and by Coulomb’s law σ = E1/4π, where E1 = A/R1 is the force at the surface of the inner cylinder.
Accordingly Q = 2πR1lA / 4πR1 = Al/2. If then the outer cylinder be at zero potential the potential V of the inner one is
This formula is important in connexion with the capacity of electric cables, which consist of a cylindrical conductor (a wire) enclosed in a conducting sheath. If the dielectric or separating insulator has a constant K, then the capacity becomes K times as great.
The capacity of two parallel planes can be calculated at once if we neglect the distribution of the lines of force near the edges of the plates, and assume that the only field is the uniform field Capacity of two parallel planes. between the plates. Let V1 and V2 be the potentials of the plates, and let a charge Q be given to one of them. If S is the surface of each plate, and d their distance, then the electric force E in the space between them is E = (V1 − V2)/d. But if σ is the surface density, E = 4πσ, and σ = Q/S. Hence we have
In this calculation we neglect altogether the fact that electric force distributed on curved lines exists outside the interspace between the plates, and these lines in fact extend from the back of one “Edge effect.” plate to that of the other. G. R. Kirchhoff (Gesammelte Abhandl. p. 112) has given a full expression for the capacity C of two circular plates of thickness t and radius r placed at any distance d apart in air from which the edge effect can be calculated. Kirchhoff’s expression is as follows:—
| C = | πr2 | + | r | { d logε | 16πr (d + t) | + t logε | d + t | } |
| 4πd | 4πd | εd2 | t |
In the above formula ε is the base of the Napierian logarithms. The first term on the right-hand side of the equation is the expression for the capacity, neglecting the curved edge distribution of electric force, and the other terms take into account, not only the uniform field between the plates, but also the non-uniform field round the edges and beyond the plates.
In practice we can avoid the difficulty due to irregular distribution of electric force at the edges of the plate by the use of a guard plate as first suggested by Lord Kelvin.[2] If a large plate has a circular hole cut in it, and this is nearly filled up by a Guard plates. circular plate lying in the same plane, and if we place another large plate parallel to the first, then the electric field between this second plate and the small circular plate is nearly uniform; and if S is the area of the small plate and d its distance from the opposed plate, its capacity may be calculated by the simple formula C = S / 4πd. The outer larger plate in which the hole is cut is called the “guard plate,” and must be kept at the same potential as the smaller inner or “trap-door plate.” The same arrangement can be supplied to a pair of coaxial cylinders. By placing metal plates on either side of a larger sheet of dielectric or insulator we can construct a condenser of relatively large capacity. The instrument known as a Leyden jar (q.v.) consists of a glass bottle coated within and without for three parts of the way up with tinfoil.
- ↑ See article “Electricity,” Encyclopaedia Britannica (9th edition), vol. viii. p. 30. The reader is also referred to an article by Lord Kelvin (Reprint of Papers on Electrostatics and Magnetism, p. 178), entitled “Determination of the Distribution of Electricity on a Circular Segment of a Plane, or Spherical Conducting Surface under any given Influence,” where another equivalent expression is given for the capacity of an ellipsoid.
- ↑ See Maxwell, Electricity and Magnetism, vol. i. pp. 284–305 (3rd ed., 1892).
