Thus, consider a sphere uniformly charged with Q units of positive electricity. It is a fundamental theorem in attractions that a thin spherical shell of matter which attracts according to the law of the inverse square acts on all external points asPotential of a sphere.if it were concentrated at its centre. Hence a sphere having a charge Q repels a unit charge placed at a distance x from its centre with a force Q/x² dynes, and therefore the work W in ergs expended in bringing the unit up to that point from an infinite distance is given by the integralW =∫x∞Qx−2dx = Q/x(1).Hence the potential at the surface of the sphere, and therefore the potential of the sphere, is Q/R, where R is the radius of the sphere in centimetres. The quantity of electricity which must be given to the sphere to raise it to unit potential is therefore R electrostatic units. The capacity of a conductor is defined to be the charge required to raise its potential to unity, all other charged conductors being at an infinite distance. This capacity is then a function of the geometrical dimensions of the conductor, and can be mathematically determined in certain cases. Since the potential of a small charge of electricity dQ at a distance r is equal to dQ/r, and since the potential of all parts of a conductor is the same in those cases in which the distribution of surface density of electrification is uniform or symmetrical with respect to some point or axis in the conductor, we can calculate the potential by simply summing up terms like σdS/r, where dS is an element of surface, σ the surface density of electricity on it, and r the distance from the symmetrical centre. The capacity is then obtained as the quotient of the whole charge by this potential. Thus the distribution of electricity on a sphere in free space must be uniform, and all parts of the charge are at anCapacity of a sphere.equal distance R from the centre. Accordingly the potentialatthe centre is Q/R. But this must be the potentialofthe sphere, since all parts are at the same potential V. Since the capacity C is the ratio of charge to potential, the capacity of the sphere in free space is Q/V = R, or is numerically the same as its radius reckoned in centimetres.We can thus easily calculate the capacity of a long thin wire like a telegraph wire far removed from the earth, as follows: Let 2r be the diameter of the wire, l its length, and σ the uniformCapacity of a thin rod.surface electric density. Then consider a thin annulus of the wire of width dx; the charge on it is equal to 2πrσ/dx units, and the potential V at a point on the axis at a distance x from the annulus due to this elementary charge isV = 2∫l/202πrσdx = 4πrσ{loge(½l + √r² + ¼l²) − loger}.√(r² + x²)If, then, r is small compared with l, we have V = 4πrσlogel/r. But the charge is Q = 2πrσ, and therefore the capacity of the thin wire is given byC = ½ logel/r(2).A more difficult case is presented by the ellipsoid5. We have first to determine the mode in which electricity distributes itself on a conducting ellipsoid in free space. It must be such a distribution that the potential in the interior will bePotential of an ellipsoid.constant, since the electric force must be zero. It is a well-known theorem in attractions that if a shell is made of gravitative matter whose inner and outer surfaces are similar ellipsoids, it exercises no attraction on a particle of matter in its interior6. Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a + da), (b + db), (c + dc), where da/a = db/b = dc/c = μ. The potential of such a shell at any internal point is constant, and the equipotential surfaces for external space are ellipsoids confocal with the ellipsoidal shell. Hence if we distribute electricity over an ellipsoid, so that its density is everywhere proportional to the thickness of a shell formed by describing roundthe 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. Henceσ = Qp / 4πabc(3).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 onCapacity 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’sSolid Geometry, p. 172). Hence the density σ is given byσ =Q1.4πabc√(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∫dSr4πabcr √(x² / a4+ y² / b4+ z² / c4)(4).Accordingly the capacity C of the ellipsoid is given by the equation1=1∫dSC4πabc√(x² + y² + z²) √(x² / a4+ y² / b4+ z² / c4)(5).It has been shown by Professor Chrystal that the above integral may also be presented in the form,71= ½∫∞0dλC√{(a² + λ) (b² + λ) (c² + λ)}(6).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 havedS = 2πyds = 2πydx /(dx)= 2πydx /(py)=2πb²dx.dsbpHence, 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 expression1=1∫dxC2a√(x² + y²)(7).If the ellipsoid is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to1=1logε(1 + e)C12ae1 − e(8).Whereas if it is an ellipsoid of revolution round the minor axis b (oblate), we have1=sin−1aeC²ae(9).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 byC1= a / logε2a/b(10),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 R1be the radius of the inner sphere, R2the inside radius of the outer sphere, and R2the outside radius of the outer spherical shell. Let a charge +Q beCapacity 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 byC = 1/R1− 1/R2= (R2− R1) / R1R2(11).Such a pair of concentric spheres constitute a condenser (seeLeyden Jar), and it is obvious that by making R2nearly equal to R1, we may enormously increase the capacity of the inner sphere. Hence the namecondenser.The other case of importance is that of two coaxial cylinders. Let a solid circular sectioned cylinder of radius R1be enclosed in a coaxial tube of inner radius R2. Then when the innerCapacity of two coaxial cylinders.cylinder is at potential V1and the outer one kept at potential V2the 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 byE = −dV/dR = A/R or V = −A ∫ R−1dR,(12),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/R1is 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 isV = A log (R2/R1), and its capacity C = l/2 log R2/R1.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 fieldCapacity of two parallel planes.between the plates. Let V1and V2be 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(V1− V2) d = 4πQ / S or C = Q / (V1− V2) = S / 4πd(13).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 logε16πr (d + t)+ t logεd + t}4πd4πdεd²t(14).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.8If a large plate has a circular hole cut in it, and this is nearly filled up by aGuard 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.If we have a number of such condensers we can combine them in “parallel” or in “series.” If all the plates on one side are connected together and also those on the other, the condensers are joined in parallel. If C1, C2, C3, &c., are the separateSystems of condensers.capacities, then Σ(C) = C1+ C2+ C3+ &c., is the total capacity in parallel. If the condensers are so joined that the inner coating of one is connected to the outer coating of the next, they are said to be in series. Since then they are all charged with the same quantity of electricity, and the total over all potential difference V is the sum of each of the individual potential differences V1, V2, V3, &c., we haveQ = C1V1= C2V2= C3V3= &c., and V = V1+ V2+ V3+ &c.The resultant capacity is C = Q/V, andC = 1 / (1/C1+ 1/C2+ 1/C3+ &c.) = 1 / Σ(1/C)(15).These rules provide means for calculating the resultant capacity when any number of condensers are joined up in any way.If one condenser is charged, and then joined in parallel with another uncharged condenser, the charge is divided between them in the ratio of their capacities. For if C1and C2are the capacities and Q1and Q2are the charges after contact, then Q1/C1and Q2/C2are the potential differences of the coatings and must be equal. Hence Q1/C1= Q2/C2or Q1/Q2= C1/C2. It is worth noting that if we have a charged sphere we can perfectly discharge it by introducing it into the interior of another hollow insulated conductor and making contact. The small sphere then becomes part of the interior of the other and loses all charge.Measurement of Capacity.—Numerous methods have been devised for the measurement of the electrical capacity of conductors in those cases in which it cannot be determined by calculation. Such a measurement may be anabsolutedetermination or arelativeone. The dimensions of a capacity in electrostatic measure is a length (seeUnits, Physical). Thus the capacity of a sphere in electrostatic units (E.S.U.) is the same as the number denoting its radius in centimetres. The unit of electrostatic capacity is therefore that of a sphere of 1 cm. radius.9This unit is too small for practical purposes, and hence a unit of capacity 900,000 greater, called a microfarad, is generally employed. Thus for instance the capacity in free space of a sphere 2 metres in diameter would be 100/900,000 = 1/9000 of a microfarad. The electrical capacity of the whole earth considered as a sphere is about 800 microfarads. An absolute measurement of capacity means, therefore, a determination in E.S. units made directly without reference to any other condenser. On the other hand there are numerous methods by which the capacities of condensers may be compared and a relative measurement made in terms of some standard.One well-known comparison method is that of C.V. de Sauty. The two condensers to be compared are connected in the branches of a Wheatstone’s Bridge (q.v.) and the other two arms completed with variable resistance boxes. These armsRelative determinations.are then altered until on raising or depressing the battery key there is no sudden deflection either way of the galvanometer. If R1and R2are the arms’ resistances and C1and C2the condenser capacities, then when the bridge is balanced we have R1: R2= C1: C2.Another comparison method much used in submarine cable work is the method of mixtures, originally due to Lord Kelvin and usually called Thomson and Gott’s method. It depends on the principle that if two condensers of capacity C1and C2are respectively charged to potentials V1and V2, and then joined in parallel with terminals of opposite charge together, the resulting potential difference of the two condensers will be V, such thatV =(C1V1− C2V2)(C + C)(16);and hence if V is zero we have C1: C2= V2: V1.The method is carried out by charging the two condensers to be compared at the two sections of a high resistance joining the ends of a battery which is divided into two parts by a movable contact.10This contact is shifted until such a point is found by trial that the two condensers charged at the different sections and then joined as above described and tested on a galvanometer show no charge. Various special keys have been invented for performing the electrical operations expeditiously.A simple method for condenser comparison is to charge the two condensers to the same voltage by a battery and then discharge them successively through a ballistic galvanometer (q.v.) and observe the respective “throws” or deflections of the coil or needle. These are proportional to the capacities. For the various precautions necessary in conducting the above tests special treatises on electrical testing must be consulted.Fig. 2.In the absolute determination of capacity we have to measure the ratio of the charge of a condenser to its plate potential difference. One of the best methods for doing this is to charge the condenser by the known voltage of a battery, and thenAbsolute determinations.discharge it through a galvanometer and repeat this process rapidly and successively. If a condenser of capacity C is charged to potential V, and discharged n times per second through a galvanometer, this series of intermittent discharges is equivalent to a current nCV. Hence if the galvanometer is calibrated by a potentiometer (q.v.) we can determine the value of this current in amperes, and knowing the value of n and V thus determine C. Various forms of commutator have been devised for effecting this charge and discharge rapidly by J.J. Thomson, R.T. Glazebrook, J.A. Fleming and W.C. Clinton and others.11One form consists of a tuning-fork electrically maintained in vibration of known period, which closes an electric contact at every vibration and sets another electromagnet in operation, which reverses a switch and moves over one terminal of the condenser from a battery to a galvanometer contact. In another form, a revolving contact is used driven by an electric motor, which consists of an insulating disk having on its surface slips of metal and three wire brushes a, b, c (see fig. 2) pressing against them. The metal slips are so placed that, as the disk revolves, the middle brush, connected to one terminal of the condenser C, is alternately put in conductive connexion with first one and then the other outside brush, which are joined respectively to the battery B and galvanometer G terminals. From the speed of this motor the number of commutations per second can be determined. The above method is especially useful for the determinations of very small capacities of the order of 100 electrostatic units or so and upwards.
Thus, consider a sphere uniformly charged with Q units of positive electricity. It is a fundamental theorem in attractions that a thin spherical shell of matter which attracts according to the law of the inverse square acts on all external points asPotential of a sphere.if it were concentrated at its centre. Hence a sphere having a charge Q repels a unit charge placed at a distance x from its centre with a force Q/x² dynes, and therefore the work W in ergs expended in bringing the unit up to that point from an infinite distance is given by the integral
W =∫x∞Qx−2dx = Q/x
(1).
Hence the potential at the surface of the sphere, and therefore the potential of the sphere, is Q/R, where R is the radius of the sphere in centimetres. The quantity of electricity which must be given to the sphere to raise it to unit potential is therefore R electrostatic units. The capacity of a conductor is defined to be the charge required to raise its potential to unity, all other charged conductors being at an infinite distance. This capacity is then a function of the geometrical dimensions of the conductor, and can be mathematically determined in certain cases. Since the potential of a small charge of electricity dQ at a distance r is equal to dQ/r, and since the potential of all parts of a conductor is the same in those cases in which the distribution of surface density of electrification is uniform or symmetrical with respect to some point or axis in the conductor, we can calculate the potential by simply summing up terms like σdS/r, where dS is an element of surface, σ the surface density of electricity on it, and r the distance from the symmetrical centre. The capacity is then obtained as the quotient of the whole charge by this potential. Thus the distribution of electricity on a sphere in free space must be uniform, and all parts of the charge are at anCapacity of a sphere.equal distance R from the centre. Accordingly the potentialatthe centre is Q/R. But this must be the potentialofthe sphere, since all parts are at the same potential V. Since the capacity C is the ratio of charge to potential, the capacity of the sphere in free space is Q/V = R, or is numerically the same as its radius reckoned in centimetres.
We can thus easily calculate the capacity of a long thin wire like a telegraph wire far removed from the earth, as follows: Let 2r be the diameter of the wire, l its length, and σ the uniformCapacity of a thin rod.surface electric density. Then consider a thin annulus of the wire of width dx; the charge on it is equal to 2πrσ/dx units, and the potential V at a point on the axis at a distance x from the annulus due to this elementary charge is
If, then, r is small compared with l, we have V = 4πrσlogel/r. But the charge is Q = 2πrσ, and therefore the capacity of the thin wire is given by
C = ½ logel/r
(2).
A more difficult case is presented by the ellipsoid5. We have first to determine the mode in which electricity distributes itself on a conducting ellipsoid in free space. It must be such a distribution that the potential in the interior will bePotential of an ellipsoid.constant, since the electric force must be zero. It is a well-known theorem in attractions that if a shell is made of gravitative matter whose inner and outer surfaces are similar ellipsoids, it exercises no attraction on a particle of matter in its interior6. Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a + da), (b + db), (c + dc), where da/a = db/b = dc/c = μ. The potential of such a shell at any internal point is constant, and the equipotential surfaces for external space are ellipsoids confocal with the ellipsoidal shell. Hence if we distribute electricity over an ellipsoid, so that its density is everywhere proportional to the thickness of a shell formed by describing roundthe 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
σ = Qp / 4πabc
(3).
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 onCapacity 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’sSolid Geometry, p. 172). Hence the density σ is given by
and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression,
(4).
Accordingly the capacity C of the ellipsoid is given by the equation
(5).
It has been shown by Professor Chrystal that the above integral may also be presented in the form,7
(6).
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
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
(7).
If the ellipsoid is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to
(8).
Whereas if it is an ellipsoid of revolution round the minor axis b (oblate), we have
(9).
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
C1= a / logε2a/b
(10),
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 R1be the radius of the inner sphere, R2the inside radius of the outer sphere, and R2the outside radius of the outer spherical shell. Let a charge +Q beCapacity 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
C = 1/R1− 1/R2= (R2− R1) / R1R2
(11).
Such a pair of concentric spheres constitute a condenser (seeLeyden Jar), and it is obvious that by making R2nearly equal to R1, we may enormously increase the capacity of the inner sphere. Hence the namecondenser.
The other case of importance is that of two coaxial cylinders. Let a solid circular sectioned cylinder of radius R1be enclosed in a coaxial tube of inner radius R2. Then when the innerCapacity of two coaxial cylinders.cylinder is at potential V1and the outer one kept at potential V2the 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
E = −dV/dR = A/R or V = −A ∫ R−1dR,
(12),
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/R1is 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
V = A log (R2/R1), and its capacity C = l/2 log R2/R1.
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 fieldCapacity of two parallel planes.between the plates. Let V1and V2be 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
(V1− V2) d = 4πQ / S or C = Q / (V1− V2) = S / 4πd
(13).
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:—
(14).
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.8If a large plate has a circular hole cut in it, and this is nearly filled up by aGuard 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.
If we have a number of such condensers we can combine them in “parallel” or in “series.” If all the plates on one side are connected together and also those on the other, the condensers are joined in parallel. If C1, C2, C3, &c., are the separateSystems of condensers.capacities, then Σ(C) = C1+ C2+ C3+ &c., is the total capacity in parallel. If the condensers are so joined that the inner coating of one is connected to the outer coating of the next, they are said to be in series. Since then they are all charged with the same quantity of electricity, and the total over all potential difference V is the sum of each of the individual potential differences V1, V2, V3, &c., we have
Q = C1V1= C2V2= C3V3= &c., and V = V1+ V2+ V3+ &c.
The resultant capacity is C = Q/V, and
C = 1 / (1/C1+ 1/C2+ 1/C3+ &c.) = 1 / Σ(1/C)
(15).
These rules provide means for calculating the resultant capacity when any number of condensers are joined up in any way.
If one condenser is charged, and then joined in parallel with another uncharged condenser, the charge is divided between them in the ratio of their capacities. For if C1and C2are the capacities and Q1and Q2are the charges after contact, then Q1/C1and Q2/C2are the potential differences of the coatings and must be equal. Hence Q1/C1= Q2/C2or Q1/Q2= C1/C2. It is worth noting that if we have a charged sphere we can perfectly discharge it by introducing it into the interior of another hollow insulated conductor and making contact. The small sphere then becomes part of the interior of the other and loses all charge.
Measurement of Capacity.—Numerous methods have been devised for the measurement of the electrical capacity of conductors in those cases in which it cannot be determined by calculation. Such a measurement may be anabsolutedetermination or arelativeone. The dimensions of a capacity in electrostatic measure is a length (seeUnits, Physical). Thus the capacity of a sphere in electrostatic units (E.S.U.) is the same as the number denoting its radius in centimetres. The unit of electrostatic capacity is therefore that of a sphere of 1 cm. radius.9This unit is too small for practical purposes, and hence a unit of capacity 900,000 greater, called a microfarad, is generally employed. Thus for instance the capacity in free space of a sphere 2 metres in diameter would be 100/900,000 = 1/9000 of a microfarad. The electrical capacity of the whole earth considered as a sphere is about 800 microfarads. An absolute measurement of capacity means, therefore, a determination in E.S. units made directly without reference to any other condenser. On the other hand there are numerous methods by which the capacities of condensers may be compared and a relative measurement made in terms of some standard.
One well-known comparison method is that of C.V. de Sauty. The two condensers to be compared are connected in the branches of a Wheatstone’s Bridge (q.v.) and the other two arms completed with variable resistance boxes. These armsRelative determinations.are then altered until on raising or depressing the battery key there is no sudden deflection either way of the galvanometer. If R1and R2are the arms’ resistances and C1and C2the condenser capacities, then when the bridge is balanced we have R1: R2= C1: C2.
Another comparison method much used in submarine cable work is the method of mixtures, originally due to Lord Kelvin and usually called Thomson and Gott’s method. It depends on the principle that if two condensers of capacity C1and C2are respectively charged to potentials V1and V2, and then joined in parallel with terminals of opposite charge together, the resulting potential difference of the two condensers will be V, such that
(16);
and hence if V is zero we have C1: C2= V2: V1.
The method is carried out by charging the two condensers to be compared at the two sections of a high resistance joining the ends of a battery which is divided into two parts by a movable contact.10This contact is shifted until such a point is found by trial that the two condensers charged at the different sections and then joined as above described and tested on a galvanometer show no charge. Various special keys have been invented for performing the electrical operations expeditiously.
A simple method for condenser comparison is to charge the two condensers to the same voltage by a battery and then discharge them successively through a ballistic galvanometer (q.v.) and observe the respective “throws” or deflections of the coil or needle. These are proportional to the capacities. For the various precautions necessary in conducting the above tests special treatises on electrical testing must be consulted.
In the absolute determination of capacity we have to measure the ratio of the charge of a condenser to its plate potential difference. One of the best methods for doing this is to charge the condenser by the known voltage of a battery, and thenAbsolute determinations.discharge it through a galvanometer and repeat this process rapidly and successively. If a condenser of capacity C is charged to potential V, and discharged n times per second through a galvanometer, this series of intermittent discharges is equivalent to a current nCV. Hence if the galvanometer is calibrated by a potentiometer (q.v.) we can determine the value of this current in amperes, and knowing the value of n and V thus determine C. Various forms of commutator have been devised for effecting this charge and discharge rapidly by J.J. Thomson, R.T. Glazebrook, J.A. Fleming and W.C. Clinton and others.11One form consists of a tuning-fork electrically maintained in vibration of known period, which closes an electric contact at every vibration and sets another electromagnet in operation, which reverses a switch and moves over one terminal of the condenser from a battery to a galvanometer contact. In another form, a revolving contact is used driven by an electric motor, which consists of an insulating disk having on its surface slips of metal and three wire brushes a, b, c (see fig. 2) pressing against them. The metal slips are so placed that, as the disk revolves, the middle brush, connected to one terminal of the condenser C, is alternately put in conductive connexion with first one and then the other outside brush, which are joined respectively to the battery B and galvanometer G terminals. From the speed of this motor the number of commutations per second can be determined. The above method is especially useful for the determinations of very small capacities of the order of 100 electrostatic units or so and upwards.
Dielectric constant.—Since all electric charge consists in a state of strain or polarization of the dielectric, it is evident that the physical state and chemical composition of the insulator must be of great importance in determining electrical phenomena. Cavendish and subsequently Faraday discovered this fact, and the latter gave the name “specific inductive capacity,” or “dielectric constant,” to that quality of an insulator which determines the charge taken by a conductor embedded in it when charged to a given potential. The simplest method of determining it numerically is, therefore, that adopted by Faraday.12He constructed two equal condensers, each consisting of a metal ball enclosed in a hollow metal sphere, and he provided also certain hemispherical shells of shellac, sulphur, glass, resin, &c., which he could so place in one condenser between the ball and enclosing sphere that it formed a condenser with solid dielectric. He then determined the ratio of the capacities of the two condensers, one with air and the other with the solid dielectric. This gave the dielectric constant K of the material. Taking the dielectric constant of air as unity he obtained the following values, for shellac K = 2.0, glass K = 1.76, and sulphur K = 2.24.
Table I.—Dielectric Constants(K)of Solids(Kfor Air= 1).
Since Faraday’s time, by improved methods, but depending essentially upon the same principles, an enormous number of determinations of the dielectric constants of various insulators, solid, liquid and gaseous, have been made (see tables I., II., III. and IV.). There are very considerable differences between the values assigned by different observers, sometimes no doubt due to differences in method, but in most cases unquestionably depending on variations in the quality of the specimens examined. The value of the dielectric constant is greatly affected by the temperature and the frequency of the applied electric force.
Table II.—Dielectric Constant(K)of Liquids.
Table III.—Dielectric Constant of some Bodies at a very low Temperature(−185° C.) (Fleming and Dewar).
The above determinations at low temperature were made with either a steady or a slowly alternating electric force applied a hundred times a second. They show that the dielectric constant of a liquid generally undergoes great reduction in value when the liquid is frozen and reduced to a low temperature.13
The dielectric constants of gases have been determined by L. Boltzmann and I. Klemenčič as follows:—
Table IV.—Dielectric Constants(K)of Gases at15°C. and760mm. Vacuum= 1.
In general the dielectric constant is reduced with decrease of temperature towards a certain limiting value it would attain at the absolute zero. This variation, however, is not always linear. In some cases there is a very sudden drop at or below a certain temperature to a much lower value, and above and below the point the temperature variation is small. There is also a large difference in most cases between the value for a steadily applied electric force and a rapidly reversed or intermittent force—in the last case a decrease with increase of frequency. Maxwell (Elec. and Magn.vol. ii. § 788) showed that the square root of the dielectric constant should be the same number as the refractive index for waves of the same frequency (seeElectric Waves). There are very few substances, however, for which the optical refractive index has the same value as K for steady or slowly varying electric force, on account of the great variation of the value of K with frequency.
There is a close analogy between the variation of dielectric constant of an insulator with electric force frequency and that of the rigidity or stiffness of an elastic body with the frequency of applied mechanical stress. Thus pitch is a soft and yielding body under steady stress, but a bar of pitch if struck gives a musical note, which shows that it vibrates and is therefore stiff or elastic for high frequency stress.
Residual Charges in Dielectrics.—In close connexion with this lies the phenomenon of residual charge in dielectrics.14If a glass Leyden jar is charged and then discharged and allowed to stand awhile, a second discharge can be obtained from it, and in like manner a third, and so on. The reappearance of the residual charge is promoted by tapping the glass. It has been shown that this behaviour of dielectrics can be imitated by a mechanical model consisting of a series of perforated pistons placed in a tube of oil with spiral springs between each piston.15If the pistons are depressed and then released, and then the upper piston fixed awhile, a second discharge can be obtained from it, and the mechanical stress-strain diagram of the model is closely similar to the discharge curve of a dielectric. R.H.A. Kohlrausch called attention to the close analogy between residual charge and the elastic recovery of strained bodies such as twisted wire or glass threads. If a charged condenser is suddenly discharged and then insulated, the reappearance of a potential difference between its coatings is analogous to the reappearance of a torque in the case of a glass fibre which has been twisted, released suddenly, and then gripped again at the ends.
For further information on the qualities of dielectrics the reader is referred to the following sources:—J. Hopkinson, “On the Residual Charge of the Leyden Jar,”Phil. Trans., 1876, 166 [ii.], p. 489, where it is shown that tapping the glass of a Leyden jar permits the reappearance of the residual charge; “On the Residual Charge ofthe 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 ofqunits 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, orq1/x1+ q2/x2+ q3/x3+ &c. = Σ (q/x) = V(17),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.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.16This 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 asourceof 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 ± ½dx, y ± ½dy, z ± ½dz). 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+ ½d²Vdx)and(dV− ½d²Vdx),dxdx²dxdx²and 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,dx²dy²dz²and by the previous theorem this must be equal to 4πρdx dy dz.Henced²V+d²V+d²V+ 4πρ = 0dx²dy²dz²(18).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 V1and V2be the potentials at points just outside and inside the surface dS, and let n1and n2be the normals to the surface dS drawn outwards and inwards; then −dV1/ dn1and −dV2/ dn2are 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,17(dV1/ dn1) + (dV2/ dn2) + 4πσ = 0(19).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 isEdS − E′dS′(20),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−dV/dn = 4.0πσ or σ = −(¼π) dV/dn = E/4π.The above is a statement of Coulomb’s law, thatthe 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.18If we define the positive direction along a tube of electric force as the direction in which a small body charged with positive electricity would tend to move, we can summarize the above facts in a simple form by saying that,if we have any closed surface described in any manner in an electric field, the excess of the number of unit tubes which leave the surface over those which enter it is equal to 4π-times the algebraic sum of all the electricity included within the surface.Every tube of electric force must therefore begin and end on electrified surfaces of opposite sign, and the quantities of positive and negative electricity on its two ends are equal, since the force E just outside an electrified surface is normal to it and equal to σ/4π, where σ is the surface density; and since we have just proved that for the ends of a tube of force EdS = E′dS′, it follows that σdS = σ′dS′, or Q = Q′, where Q and Q′ are the quantities of electricity on the ends of the tube of force. Accordingly, since every tube sent out from a charged conductor must end somewhere on another charge of opposite sign, it follows that the two electricities always exist in equal quantity, and that it is impossible to create any quantity of one kind without creating an equal quantity of the opposite sign.Fig. 4.We have next to consider the energy storage which takes place when electric charge is created,i.e.when the dielectric is strained or polarized. Since the potential of a conductor is defined to be the work required to move a unit of positive electricity from the surface of the earth or from an infinite distance from all electricity to the surface of the conductor, it follows that the work done in putting a small charge dq into a conductor at a potential v is v dq. Let us then suppose that a conductor originally at zero potential has its potential raised by administering to it small successive doses of electricity dq. The first raises its potential to v, the second to v′ and so on, and the nth to V. Take any horizontal line and divide it into small elements of length each representing dq, and draw vertical lines representing the potentials v, v′, &c., and after each dose. Since the potential rises proportionately to the quantity in the conductor, the ends of these ordinates will lie on a straight line and define a triangle whose base line is a length equal to the total quantity Q and height a length equal to the final potential V. The element of work done in introducing the quantity of electricity dq at a potential v is represented by the element of area of this triangle (see fig. 4), and hence the work done in charging the conductor with quantity Q to final potential V is ½QV, or since Q = CV, where C is its capacity, the work done is represented by ½CV² or by ½Q² / C.If σ is the surface density and dS an element of surface, then ∫σdS is the whole charge, and hence ½ ∫ VσdS is the expression for the energy of charge of a conductor.We can deduce a remarkable expression for the energy stored up in an electric field containing electrified bodies as follows:19Let V denote the potential at any point in the field. Consider the integralW =1∫ ∫ ∫ {(dV)²+(dV)²+(dV)²}dx dy dz.8πdxdydz(21)where the integration extends throughout the whole space unoccupied by conductors. We have by partial integration∫ ∫ ∫ (dV)²dx dy dz =∫ ∫VdVdy dz −∫ ∫ ∫Vd²Vdx dy dz,dxdxdx²and two similar equations in y and z. Hence1∫ ∫ ∫ {(dV)²+(dV)²+(dV)²}dx dy dz =8πdxdydz1∫ ∫VdVdS −1∫ ∫ ∫V∇V dx dy dz8πdn8π(22)where dV/dn means differentiation along the normal, and ∇ stands for the operator d²/dx² + d²/dy² + d²/dz². Let E be the resultant electric force at any point in the field. Then bearing in mind that σ = (1/4π) dV/dn, and ρ = −(1/4π) ∇V, we have finally1∫ ∫ ∫E²dv =1∫ ∫Vσ dS +1∫ ∫ ∫Vρ dv.8π22The first term on the right hand side expresses the energy of the surface electrification of the conductors in the field, and the second the energy of volume density (if any). Accordingly the term on the left hand side gives us the whole energy in the field.Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to ½DE where D is the displacement or polarization in the dielectric.Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that ∇V = 0. Hence that distribution of potential which is necessary to satisfy Laplace’s equation is also one which makes the potential energy a minimum and therefore the energy stable. Thus the actual distribution of electricity on the conductor in the field is not merelyastable distribution, it isthe onlypossible stable distribution.Fig. 5.Method of Electrical Images.—A very powerful method of attacking problems in electrical distribution was first made known by Lord Kelvin in 1845 and is described as the method of electrical images.20By older mathematical methods it had only been possible to predict in a few simple cases the distribution of electricity at rest on conductors of various forms. The notion of an electrical image may be easily grasped by the following illustration: Let there be at A (see fig. 5) a point-charge of positive electricity +q and an infinite conducting plate PO, shown in section, connected to earth and therefore at zero potential. Then the charge at A together with the induced surface charge on the plate makes a certain field of electric force on the left of the plate PO, which is a zero equipotential surface. If we remove the plate, and yet by any means can keep the identical surface occupied by it a plane of zero potential, the boundary conditions will remain the same, and therefore the field of force to the left of PO will remain unaltered. This can be done by placing at B an equal negative point-charge −q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let −q be placed at B, so that the distance BO is equal to the distance AO, whilst AOB is at right angles to PO. Then the potential at any point P in this ideal plane PO is equal to q/AP − q/BP = O, whilst the resultant force at P due to the two point charges is 2qAO/AP³, and is parallel to AB or normal to PO. Hence if we remove the charge −q at B and distribute electricity over the surface PO with a surface density σ, according to the Coulomb-Poisson law, σ = qAO / 2πAP³, the field of force to the left of PD will fulfil the required boundary conditions, and hence will be the law of distribution of the induced electricity in the case of the actual plate. The point-charge −q at B is called the “electrical image” of the point-charge +q at A.We find a precisely analogous effect in optics which justifies the term “electrical image.” Suppose a room lit by a single candle. There is everywhere a certain illumination due to it. Place across the room a plane mirror. All the space behind the mirror will become dark, and all the space in front of the mirror will acquire an exalted illumination. Whatever this increased illumination may be, it can be precisely imitated by removing the mirror and placing a second lighted candle at the place occupied by the optical image of the first candle in the mirror, that is, as far behind the plane as the first candle was in front. So the potential distribution in the space due to the electric point-charge +q as A together with −q at B is the same as that due to +q at A and the negative induced charge erected on the infinite plane (earthed) metal sheet placed half-way between A and B.Fig. 6.The same reasoning can be applied to determine the electrical image of a point-charge of positive electricity in a spherical surface, and therefore the distribution of induced electricity over a metal sphere connected to earth produced by a point-charge near it. Let +q be any positive point-charge placed at a point A outside a sphere (fig. 6) of radius r, and centre at C, and let P be any point on it. Let CA = d. Take a point B in CA such that CB·CA = r², or CB = r²/d. It is easy then to show that PA : PB = d : r. If then we put a negative point-charge −qr/d at B, it follows that the spherical surface will be a zero potential surface, forq−rq·1= 0PAdPB(24).Another equipotential surface is evidently a very small sphere described round A. The resultant force due to these two point-charges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges. It is not difficult to show thatE = − (d² − r²) q / rAP³(25),in other words, the force at P is inversely as the cube of the distance from A. Suppose then we remove the negative point-charge, and let the sphere be supposed to become conductive and be connected to earth. If we make a distribution of negative electricity over it, which has a density σ varying according to the lawσ = −(d² − r²) q / 4πrAP³(26),that distribution, together with the point-charge +q at A, will make a distribution of electric force at all points outside the sphereexactly similar to that which would exist if the sphere were removed and a negative point charge −qr/d were placed at B. Hence this charge is the electrical image of the charge +q at A in the spherical surface.We may generalize these statements in the following theorem, which is an important deduction from a wider theorem due to G. Green. Suppose that we have any distribution of electricity at rest over conductors, and that we know the potential at all points and consequently the level or equipotential surfaces. Take any equipotential surface enclosing the whole of the electricity, and suppose this to become an actual sheet of metal connected to the earth. It is then a zero potential surface, and every point outside is at zero potential as far as concerns the electric charge on the conductors inside. Then if U is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have U + V = 0 or U = −V at all points outside the earthed metal surface. Therefore, whatever may be the distribution of electric force produced by the charges inside taken alone, it can be exactly imitated for all space outside the metal surface if we suppose the inside charge removed and a distribution of electricity of the same sign made over the metal surface such that its density follows the lawσ = −(¼π) dU / dn(27),where dU/dn is the electric force at that point on the closed equipotential surface considered, due to the original charge alone.Bibliography.—For further developments of the subject we must refer the reader to the numerous excellent treatises on electrostatics now available. The student will find it to be a great advantage to read through Faraday’s three volumes entitledExperimental Researches on Electricity, as soon as he has mastered some modern elementary book giving in compact form a general account of electrical phenomena. For this purpose he may select from the following books: J. Clerk Maxwell,Elementary Treatise on Electricity(Oxford, 1881); J.J. Thomson,Elements of the Mathematical Theory of Electricity and Magnetism(Cambridge, 1895); J.D. Everett,Electricity, founded on part iii. of Deschanel’sNatural Philosophy(London, 1901); G.C. Foster and A.W. Porter,Elementary Treatise on Electricity and Magnetism(London, 1903); S.P. Thompson,Elementary Lessons on Electricity and Magnetism(London, 1903)·When these elementary books have been digested, the advanced student may proceed to study the following: J. Clerk Maxwell,A Treatise on Electricity and Magnetism(1st ed., Oxford, 1873; 2nd ed. by W.D. Niven, 1881; 3rd ed. by J.J. Thomson, 1892); Joubert and Mascart,Electricity and Magnetism, English translation by E. Atkinson (London, 1883); Watson and Burbury,The Mathematical Theory of Electricity and Magnetism(Oxford, 1885); A. Gray,A Treatise on Magnetism and Electricity(London, 1898). In the collectedScientific Papersof Lord Kelvin (3 vols., Cambridge, 1882), of James Clerk Maxwell (2 vols., Cambridge, 1890), and of Lord Rayleigh (4 vols., Cambridge, 1903), the advanced student will find the means for studying the historical development of electrical knowledge as it has been evolved from the minds of some of the master workers of the 19th century.
For further information on the qualities of dielectrics the reader is referred to the following sources:—J. Hopkinson, “On the Residual Charge of the Leyden Jar,”Phil. Trans., 1876, 166 [ii.], p. 489, where it is shown that tapping the glass of a Leyden jar permits the reappearance of the residual charge; “On the Residual Charge ofthe 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 ofqunits 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
q1/x1+ q2/x2+ q3/x3+ &c. = Σ (q/x) = V
(17),
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.
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.16This 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 asourceof 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 ± ½dx, y ± ½dy, z ± ½dz). 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
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
and by the previous theorem this must be equal to 4πρdx dy dz.
Hence
(18).
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 V1and V2be the potentials at points just outside and inside the surface dS, and let n1and n2be the normals to the surface dS drawn outwards and inwards; then −dV1/ dn1and −dV2/ dn2are 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,17
(dV1/ dn1) + (dV2/ dn2) + 4πσ = 0
(19).
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
EdS − E′dS′
(20),
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
−dV/dn = 4.0πσ or σ = −(¼π) dV/dn = E/4π.
The above is a statement of Coulomb’s law, thatthe 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.18
If we define the positive direction along a tube of electric force as the direction in which a small body charged with positive electricity would tend to move, we can summarize the above facts in a simple form by saying that,if we have any closed surface described in any manner in an electric field, the excess of the number of unit tubes which leave the surface over those which enter it is equal to 4π-times the algebraic sum of all the electricity included within the surface.
Every tube of electric force must therefore begin and end on electrified surfaces of opposite sign, and the quantities of positive and negative electricity on its two ends are equal, since the force E just outside an electrified surface is normal to it and equal to σ/4π, where σ is the surface density; and since we have just proved that for the ends of a tube of force EdS = E′dS′, it follows that σdS = σ′dS′, or Q = Q′, where Q and Q′ are the quantities of electricity on the ends of the tube of force. Accordingly, since every tube sent out from a charged conductor must end somewhere on another charge of opposite sign, it follows that the two electricities always exist in equal quantity, and that it is impossible to create any quantity of one kind without creating an equal quantity of the opposite sign.
We have next to consider the energy storage which takes place when electric charge is created,i.e.when the dielectric is strained or polarized. Since the potential of a conductor is defined to be the work required to move a unit of positive electricity from the surface of the earth or from an infinite distance from all electricity to the surface of the conductor, it follows that the work done in putting a small charge dq into a conductor at a potential v is v dq. Let us then suppose that a conductor originally at zero potential has its potential raised by administering to it small successive doses of electricity dq. The first raises its potential to v, the second to v′ and so on, and the nth to V. Take any horizontal line and divide it into small elements of length each representing dq, and draw vertical lines representing the potentials v, v′, &c., and after each dose. Since the potential rises proportionately to the quantity in the conductor, the ends of these ordinates will lie on a straight line and define a triangle whose base line is a length equal to the total quantity Q and height a length equal to the final potential V. The element of work done in introducing the quantity of electricity dq at a potential v is represented by the element of area of this triangle (see fig. 4), and hence the work done in charging the conductor with quantity Q to final potential V is ½QV, or since Q = CV, where C is its capacity, the work done is represented by ½CV² or by ½Q² / C.
If σ is the surface density and dS an element of surface, then ∫σdS is the whole charge, and hence ½ ∫ VσdS is the expression for the energy of charge of a conductor.
We can deduce a remarkable expression for the energy stored up in an electric field containing electrified bodies as follows:19Let V denote the potential at any point in the field. Consider the integral
(21)
where the integration extends throughout the whole space unoccupied by conductors. We have by partial integration
and two similar equations in y and z. Hence
(22)
where dV/dn means differentiation along the normal, and ∇ stands for the operator d²/dx² + d²/dy² + d²/dz². Let E be the resultant electric force at any point in the field. Then bearing in mind that σ = (1/4π) dV/dn, and ρ = −(1/4π) ∇V, we have finally
The first term on the right hand side expresses the energy of the surface electrification of the conductors in the field, and the second the energy of volume density (if any). Accordingly the term on the left hand side gives us the whole energy in the field.
Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to ½DE where D is the displacement or polarization in the dielectric.
Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that ∇V = 0. Hence that distribution of potential which is necessary to satisfy Laplace’s equation is also one which makes the potential energy a minimum and therefore the energy stable. Thus the actual distribution of electricity on the conductor in the field is not merelyastable distribution, it isthe onlypossible stable distribution.
Method of Electrical Images.—A very powerful method of attacking problems in electrical distribution was first made known by Lord Kelvin in 1845 and is described as the method of electrical images.20By older mathematical methods it had only been possible to predict in a few simple cases the distribution of electricity at rest on conductors of various forms. The notion of an electrical image may be easily grasped by the following illustration: Let there be at A (see fig. 5) a point-charge of positive electricity +q and an infinite conducting plate PO, shown in section, connected to earth and therefore at zero potential. Then the charge at A together with the induced surface charge on the plate makes a certain field of electric force on the left of the plate PO, which is a zero equipotential surface. If we remove the plate, and yet by any means can keep the identical surface occupied by it a plane of zero potential, the boundary conditions will remain the same, and therefore the field of force to the left of PO will remain unaltered. This can be done by placing at B an equal negative point-charge −q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let −q be placed at B, so that the distance BO is equal to the distance AO, whilst AOB is at right angles to PO. Then the potential at any point P in this ideal plane PO is equal to q/AP − q/BP = O, whilst the resultant force at P due to the two point charges is 2qAO/AP³, and is parallel to AB or normal to PO. Hence if we remove the charge −q at B and distribute electricity over the surface PO with a surface density σ, according to the Coulomb-Poisson law, σ = qAO / 2πAP³, the field of force to the left of PD will fulfil the required boundary conditions, and hence will be the law of distribution of the induced electricity in the case of the actual plate. The point-charge −q at B is called the “electrical image” of the point-charge +q at A.
We find a precisely analogous effect in optics which justifies the term “electrical image.” Suppose a room lit by a single candle. There is everywhere a certain illumination due to it. Place across the room a plane mirror. All the space behind the mirror will become dark, and all the space in front of the mirror will acquire an exalted illumination. Whatever this increased illumination may be, it can be precisely imitated by removing the mirror and placing a second lighted candle at the place occupied by the optical image of the first candle in the mirror, that is, as far behind the plane as the first candle was in front. So the potential distribution in the space due to the electric point-charge +q as A together with −q at B is the same as that due to +q at A and the negative induced charge erected on the infinite plane (earthed) metal sheet placed half-way between A and B.
The same reasoning can be applied to determine the electrical image of a point-charge of positive electricity in a spherical surface, and therefore the distribution of induced electricity over a metal sphere connected to earth produced by a point-charge near it. Let +q be any positive point-charge placed at a point A outside a sphere (fig. 6) of radius r, and centre at C, and let P be any point on it. Let CA = d. Take a point B in CA such that CB·CA = r², or CB = r²/d. It is easy then to show that PA : PB = d : r. If then we put a negative point-charge −qr/d at B, it follows that the spherical surface will be a zero potential surface, for
(24).
Another equipotential surface is evidently a very small sphere described round A. The resultant force due to these two point-charges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges. It is not difficult to show that
E = − (d² − r²) q / rAP³
(25),
in other words, the force at P is inversely as the cube of the distance from A. Suppose then we remove the negative point-charge, and let the sphere be supposed to become conductive and be connected to earth. If we make a distribution of negative electricity over it, which has a density σ varying according to the law
σ = −(d² − r²) q / 4πrAP³
(26),
that distribution, together with the point-charge +q at A, will make a distribution of electric force at all points outside the sphereexactly similar to that which would exist if the sphere were removed and a negative point charge −qr/d were placed at B. Hence this charge is the electrical image of the charge +q at A in the spherical surface.
We may generalize these statements in the following theorem, which is an important deduction from a wider theorem due to G. Green. Suppose that we have any distribution of electricity at rest over conductors, and that we know the potential at all points and consequently the level or equipotential surfaces. Take any equipotential surface enclosing the whole of the electricity, and suppose this to become an actual sheet of metal connected to the earth. It is then a zero potential surface, and every point outside is at zero potential as far as concerns the electric charge on the conductors inside. Then if U is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have U + V = 0 or U = −V at all points outside the earthed metal surface. Therefore, whatever may be the distribution of electric force produced by the charges inside taken alone, it can be exactly imitated for all space outside the metal surface if we suppose the inside charge removed and a distribution of electricity of the same sign made over the metal surface such that its density follows the law
σ = −(¼π) dU / dn
(27),
where dU/dn is the electric force at that point on the closed equipotential surface considered, due to the original charge alone.
Bibliography.—For further developments of the subject we must refer the reader to the numerous excellent treatises on electrostatics now available. The student will find it to be a great advantage to read through Faraday’s three volumes entitledExperimental Researches on Electricity, as soon as he has mastered some modern elementary book giving in compact form a general account of electrical phenomena. For this purpose he may select from the following books: J. Clerk Maxwell,Elementary Treatise on Electricity(Oxford, 1881); J.J. Thomson,Elements of the Mathematical Theory of Electricity and Magnetism(Cambridge, 1895); J.D. Everett,Electricity, founded on part iii. of Deschanel’sNatural Philosophy(London, 1901); G.C. Foster and A.W. Porter,Elementary Treatise on Electricity and Magnetism(London, 1903); S.P. Thompson,Elementary Lessons on Electricity and Magnetism(London, 1903)·
When these elementary books have been digested, the advanced student may proceed to study the following: J. Clerk Maxwell,A Treatise on Electricity and Magnetism(1st ed., Oxford, 1873; 2nd ed. by W.D. Niven, 1881; 3rd ed. by J.J. Thomson, 1892); Joubert and Mascart,Electricity and Magnetism, English translation by E. Atkinson (London, 1883); Watson and Burbury,The Mathematical Theory of Electricity and Magnetism(Oxford, 1885); A. Gray,A Treatise on Magnetism and Electricity(London, 1898). In the collectedScientific Papersof Lord Kelvin (3 vols., Cambridge, 1882), of James Clerk Maxwell (2 vols., Cambridge, 1890), and of Lord Rayleigh (4 vols., Cambridge, 1903), the advanced student will find the means for studying the historical development of electrical knowledge as it has been evolved from the minds of some of the master workers of the 19th century.