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ELECTROSTATICS, the name given to that department of electrical science
in which the phenomena of electricity at rest are considered. Besides
their ordinary condition all bodies are capable of being thrown into a
physical state in which they are said to be electrified or charged with
electricity. When in this condition they become sources of electric
force, and the space round them in which this force is manifested is
called an "electric field" (see ELECTRICITY). Electrified bodies exert
mechanical forces on each other, creating or tending to create motion,
and also induce electric charges on neighbouring surfaces.
The reader possessed of no previous knowledge of electrical phenomena
will best appreciate the meaning of the terms employed by the aid of a
few simple experiments. For this purpose the following apparatus should
be provided:--(1) two small metal tea-trays and some clean dry tumblers,
the latter preferably varnished with shellac varnish made with alcohol
free from water; (2) two sheets of ebonite rather larger than the
tea-trays; (3) a rod of sealing-wax or ebonite and a glass tube, also
some pieces of silk and flannel; (4) a few small gilt pith balls
suspended by dry silk threads; (5) a gold-leaf electroscope, and, if
possible, a simple form of quadrant electrometer (see ELECTROSCOPE and
ELECTROMETER); (6) some brass balls mounted on the ends of ebonite
penholders, and a few tin canisters. With the aid of this apparatus, the
principal facts of electrostatics can be experimentally verified, as
follows:--
_Experiment I._--Place one tea-tray bottom side uppermost upon three
warm tumblers as legs. Rub the sheet of ebonite vigorously with warm
flannel and lay it rubbed side downwards on the top of the tray. Touch
the tray with the finger for an instant, and lift up the ebonite without
letting the hand touch the tray a second time. The tray is then found to
be electrified. If a suspended gilt pith ball is held near it, the ball
will first be attracted and then repelled. If small fragments of paper
are scattered on the tray and then the other tray held in the hand over
them, they will fly up and down rapidly. If the knuckle is approached to
the electrified tray, a small spark will be seen, and afterwards the
tray will be found to be discharged or unelectrified. If the electrified
tray is touched with the sealing-wax or ebonite rod, it will not be
discharged, but if touched with a metal wire, the hand, or a damp
thread, it is discharged at once. This shows that some bodies are
_conductors_ and others _non-conductors_ or _insulators_ of electricity,
and that bodies can be electrified by friction and impart their electric
charge to other bodies. A charged conductor supported on a non-conductor
retains its charge. It is then said to be insulated.
_Experiment II._--Arrange two tea-trays, each on dry tumblers as before.
Rub the sheet of ebonite with flannel, lay it face downwards on one
tray, touch that tray with the finger for a moment and lift up the
ebonite sheet, rub it again, and lay it face downwards on the second
tray and leave it there. Then take two suspended gilt pith balls and
touch them (a) both against one tray; they will be found to repel each
other; (b) touch one against one tray and the other against the other
tray, and they will be found to attract each other. This proves the
existence of two kinds of electricity, called _positive_ and _negative_.
The first tea-tray is positively electrified, and the second
negatively. If an insulated brass ball is touched against the first tray
and then against the knob or plate of the electroscope, the gold leaves
will diverge. If the ball is discharged and touched against the other
tray, and then afterwards against the previously charged electroscope,
the leaves will collapse. This shows that the two electricities
neutralize each other's effect when imparted equally to the same
conductor.
_Experiment III._--Let one tray be insulated as before, and the
electrified sheet of ebonite held over it, but not allowed to touch the
tray. If the ebonite is withdrawn without touching the tray, the latter
will be found to be unelectrified. If whilst holding the ebonite sheet
over the tray the latter is also touched with an insulated brass ball,
then this ball when removed and tested with the electroscope will be
found to be negatively electrified. The sign of the electrification
imparted to the electroscope when so charged--that is, whether positive
or negative--can be determined by rubbing the sealing-wax rod with
flannel and the glass rod with silk, and approaching them gently to the
electroscope one at a time. The sealing-wax so treated is electrified
negatively or _resinously_, and the glass with positive or _vitreous_
electricity. Hence if the electrified sealing-wax rod makes the leaves
collapse, the electroscopic charge is positive, but if the glass rod
does the same, the electroscopic charge is negative. Again, if, whilst
holding the electrified ebonite over the tray, we touch the latter for a
moment and then withdraw the ebonite sheet, the tray will be found to be
positively electrified. The electrified ebonite is said to act by
"electrostatic induction" on the tray, and creates on it two induced
charges, one of positive and the other of negative electricity. The last
goes to earth when the tray is touched, and the first remains when the
tray is insulated and the ebonite withdrawn.
_Experiment IV._--Place a tin canister on a warm tumbler and connect it
by a wire with the gold-leaf electroscope. Charge positively a brass
ball held on an ebonite stem, and introduce it, without touching, into
the canister. The leaves of the electroscope will diverge with positive
electricity. Withdraw the ball and the leaves will collapse. Replace the
ball again and touch the outside of the canister; the leaves will
collapse. If then the ball be withdrawn, the leaves will diverge a
second time with negative electrification. If, before withdrawing the
ball, after touching the outside of the canister for a moment the ball
is touched against the inside of the canister, then on withdrawing it
the ball and canister are found to be discharged. This experiment proves
that when a charged body acts by induction on an insulated conductor it
causes an electrical separation to take place; electricity of opposite
sign is drawn to the side nearest the inducing body, and that of like
sign is repelled to the remote side, and these quantities are equal in
amount.
_Seat of the Electric Charge._--So far we have spoken of electric charge
as if it resided on the conductors which are electrified. The work of
Benjamin Franklin, Henry Cavendish, Michael Faraday and J. Clerk Maxwell
demonstrated, however, that all electric charge or electrification of
conductors consists simply in the establishment of a physical state in
the surrounding insulator or dielectric, which state is variously called
_electric strain_, _electric displacement_ or _electric polarization_.
Under the action of the same or identical electric forces the intensity
of this state in various insulators is determined by a quality of them
called their _dielectric constant_, _specific inductive capacity_ or
_inductivity_. In the next place we must notice that electrification is
a measurable magnitude and in electrostatics is estimated in terms of a
unit called the _electrostatic unit_ of electric quantity. In the
absolute C.G.S. system this unit quantity is defined as follows:--If we
consider a very small electrified spherical conductor, experiment shows
that it exerts a repulsive force upon another similar and similarly
electrified body. Cavendish and C.A. Coulomb proved that this mechanical
force varies inversely as the square of the distance between the centres
of the spheres. The unit of mechanical force in the "centimetre, gramme,
second" (C.G.S.) system of units is the _dyne_, which is approximately
equal to 1/981 part of the weight of one gramme. A very small sphere is
said then to possess a charge of one electrostatic unit of quantity,
when it repels another similar and similarly electrified body with a
force of one dyne, the centres being at a distance of one centimetre,
provided that the spheres are _in vacuo_ or immersed in some insulator,
the dielectric constant of which is taken as unity. If the two small
conducting spheres are placed with centres at a distance d centimetres,
and immersed in an insulator of dielectric constant K, and carry charges
of Q and Q' electrostatic units respectively, measured as above
described, then the mechanical force between them is equal to QQ'/Kd^2
dynes. For constant charges and distances the mechanical force is
inversely as the dielectric constant.
_Electric Force._--If a small conducting body is charged with Q
electrostatic units of electricity, and placed in any electric field at
a point where the electric force has a value E, it will be subject to a
mechanical force equal to QE dynes, tending to move it in the direction
of the resultant electric force. This provides us with a definition of a
unit of electric force, for it is the strength of an electric field at
that point where a small conductor carrying a unit charge is acted upon
by unit mechanical force, assuming the dielectric constant of the
surrounding medium to be unity. To avoid unnecessary complications we
shall assume this latter condition in all the following discussion,
which is equivalent simply to assuming that all our electrical
measurements are made in air or _in vacuo_.
Owing to the confusion introduced by the employment of the term force,
Maxwell and other writers sometimes use the words _electromotive
intensity_ instead of electric force. The reader should, however, notice
that what is generally called electric force is the analogue in
electricity of the so-called acceleration of gravity in mechanics,
whilst electrification or quantity of electricity is analogous to mass.
If a mass of M grammes be placed in the earth's field at a place where
the acceleration of gravity has a value g centimetres per second, then
the mechanical force acting on it and pulling it downwards is Mg dynes.
In the same manner, if an electrified body carries a positive charge Q
electrostatic units and is placed in an electric field at a place where
the electric force or electromotive intensity has a value E units, it is
urged in the direction of the electric force with a mechanical force
equal to QE dynes. We must, however, assume that the charge Q is so
small that it does not sensibly disturb the original electric field, and
that the dielectric constant of the insulator is unity.
Faraday introduced the important and useful conception of _lines_ and
_tubes_ of electric force. If we consider a very small conductor charged
with a unit of positive electricity to be placed in an electric field,
it will move or tend to move under the action of the electric force in a
certain direction. The path described by it when removed from the action
of gravity and all other physical forces is called a line of electric
force. We may otherwise define it by saying that a line of electric
force is a line so drawn in a field of electric force that its direction
coincides at every point with the resultant electric force at that
point. Let _any_ line drawn in an electric field be divided up into
small elements of length. We can take the sum of all the products of the
length of each element by the resolved part of the electric force in its
direction. This sum, or integral, is called the "line integral of
electric force" or the _electromotive force_ (E.M.F.) along this line.
In some cases the value of this electromotive force between two points
or conductors is independent of the precise path selected, and it is
then called the _potential difference_ (P.D.) of the two points or
conductors. We may define the term potential difference otherwise by
saying that it is the work done in carrying a small conductor charged
with one unit of electricity from one point to the other in a direction
opposite to that in which it would move under the electric forces if
left to itself.
_Electric Potential._--Suppose then that we have a conductor charged
with electricity; we may imagine its surface to be divided up into small
unequal areas, each of which carries a unit charge of electricity. If we
consider lines of electric force to be drawn from the boundaries of
these areas, they will cut up the space round the conductor into tubular
surfaces called tubes of electric force, and each tube will spring from
an area of the conductor carrying a unit electric charge. Hence the
charge on the conductor can be measured by the number of unit electric
tubes springing from it. In the next place we may consider the charged
body to be surrounded by a number of closed surfaces, such that the
potential difference between any point on one surface and the earth is
the same. These surfaces are called "equipotential" or "level surfaces,"
and we may so locate them that the potential difference between two
adjacent surfaces is one unit of potential; that is, it requires one
absolute unit of work (1 erg) to move a small body charged with one unit
of electricity from one surface to the next. These enclosing surfaces,
therefore, cut up the space into shells of potential, and divide up the
tubes of force into electric cells. The surface of a charged conductor
is an equipotential surface, because when the electric charge is in
equilibrium there is no tendency for electricity to move from one part
to the other.
We arbitrarily call the potential of the earth zero, since all potential
difference is relative and there is no absolute potential any more than
absolute level. We call the difference of potential between a charged
conductor and the earth the potential of the conductor. Hence when a
body is charged positively its potential is raised above that of the
earth, and when negatively it is lowered beneath that of the earth.
Potential in a certain sense is to electricity as difference of level is
to liquids or difference of temperature to heat. It must be noted,
however, that potential is a mere mathematical concept, and has no
objective existence like difference of level, nor is it capable per se
of producing physical changes in bodies, such as those which are brought
about by rise of temperature, apart from any question of difference of
temperature. There is, however, this similarity between them.
Electricity tends to flow from places of high to places of low
potential, water to flow down hill, and heat to move from places of high
to places of low temperature. Returning to the case of the charged body
with the space around it cut up into electric cells by the tubes of
force and shells of potential, it is obvious that the number of these
cells is represented by the product QV, where Q is the charge and V the
potential of the body in electrostatic units. An electrified conductor
is a store of energy, and from the definition of potential it is clear
that the work done in increasing the charge q of a conductor whose
potential is v by a small amount dq, is vdq, and since this added charge
increases in turn the potential, it is easy to prove that the work done
in charging a conductor with Q units to a potential V units is 1/2 QV
units of work. Accordingly the number of electric cells into which the
space round is cut up is equal to twice the energy stored up, or each
cell contains half a unit of energy. This harmonizes with the fact that
the real seat of the energy of electrification is the dielectric or
insulator surrounding the charged conductor.[1]
We have next to notice three important facts in electrostatics and some
consequences flowing therefrom.
(i) _Electrical Equilibrium and Potential._--If there be any number of
charged conductors in a field, the electrification on them being in
equilibrium or at rest, the surface of each conductor is an
equipotential surface. For since electricity tends to move between
points or conductors at different potentials, if the electricity is at
rest on them the potential must be everywhere the same. It follows from
this that the electric force at the surface of the conductor has no
component along the surface, in other words, the electric force at the
bounding surface of the conductor and insulator is everywhere at right
angles to it.
By the _surface density_ of electrification on a conductor is meant the
charge per unit of area, or the number of tubes of electric force which
spring from unit area of its surface. Coulomb proved experimentally that
the electric force just outside a conductor at any point is proportional
to the electric density at that point. It can be shown that the
resultant electric force normal to the surface at a point just outside a
conductor is equal to 4[pi][sigma], where [sigma] is the surface
density at that point. This is usually called Coulomb's Law.[2]
(ii) _Seat of Charge._--The charge on an electrified conductor is wholly
on the surface, and there is no electric force in the interior of a
closed electrified conducting surface which does not contain any other
electrified bodies. Faraday proved this experimentally (see
_Experimental Researches_, series xi. S 1173) by constructing a large
chamber or box of paper covered with tinfoil or thin metal. This was
insulated and highly electrified. In the interior no trace of electric
charge could be found when tested by electroscopes or other means.
Cavendish proved it by enclosing a metal sphere in two hemispheres of
thin metal held on insulating supports. If the sphere is charged and
then the jacketing hemispheres fitted on it and removed, the sphere is
found to be perfectly discharged.[3] Numerous other demonstrations of
this fact were given by Faraday. The thinnest possible spherical shell
of metal, such as a sphere of insulator coated with gold-leaf, behaves
as a conductor for static charge just as if it were a sphere of solid
metal. The fact that there is no electric force in the interior of such
a closed electrified shell is one of the most certainly ascertained
facts in the science of electrostatics, and it enables us to demonstrate
at once that particles of electricity attract and repel each other with
a force which is inversely as the square of their distance.
We may give in the first place an elementary proof of the converse
proposition by the aid of a simple lemma:--
_Lemma._--If particles of matter attract one another according to the
law of the inverse square the attraction of all sections of a cone for a
particle at the vertex is the same. _Definition._--The solid angle
subtended by any surface at a point is measured by the quotient of its
apparent surface by the square of its distance from that point. Hence
the total solid angle round any point is 4[pi]. The solid angles
subtended by all normal sections of a cone at the vertex are therefore
equal, and since the attractions of these sections on a particle at the
vertex are proportional to their distances from the vertex, they are
numerically equal to one another and to the solid angle of the cone.
Let us then suppose a spherical shell O to be electrified. Select any
point P in the interior and let a line drawn through it sweep out a
small double cone (see fig. 1). Each cone cuts out an area on the
surface equally inclined to the cone axis. The electric density on the
sphere being uniform, the quantities of electricity on these areas are
proportional to the areas, and if the electric force varies inversely as
the square of the distance, the forces exerted by these two surface
charges at the point in question are proportional to the solid angle of
the little cone. Hence the forces due to the two areas at opposite ends
of the chord are equal and opposed.
[Illustration: FIG. 1.]
Hence we see that if the whole surface of the sphere is divided into
pairs of elements by cones described through any interior point, the
resultant force at that point must consist of the sum of pairs of equal
and opposite forces, and is therefore zero. For the proof of the
converse proposition we must refer the reader to the _Electrical
Researches of the Hon. Henry Cavendish_, p. 419, or to Maxwell's
_Treatise on Electricity and Magnetism_, 2nd ed., vol. i. p. 76, where
Maxwell gives an elegant proof that if the force in the interior of a
closed conductor is zero, the law of the force must be that of the
inverse square of the distance.[4] From this fact it follows that we can
shield any conductor entirely from external influence by other charged
conductors by enclosing it in a metal case. It is not even necessary
that this envelope should be of solid metal; a cage made of fine metal
wire gauze which permits objects in its interior to be seen will yet be
a perfect electrical screen for them. Electroscopes and electrometers,
therefore, standing in proximity to electrified bodies can be perfectly
shielded from influence by enclosing them in cylinders of metal gauze.
Even if a charged and insulated conductor, such as an open canister or
deep cup, is not perfectly closed, it will be found that a proof-plane
consisting of a small disk of gilt paper carried at the end of a rod of
gum-lac will not bring away any charge if applied to the deep inside
portions. In fact it is curious to note how large an opening may be made
in a vessel which yet remains for all electrical purposes "a closed
conductor." Maxwell (_Elementary Treatise_, &c., p. 15) ingeniously
applied this fact to the insulation of conductors. If we desire to
insulate a metal ball to make it hold a charge of electricity, it is
usual to do so by attaching it to a handle or stem of glass or ebonite.
In this case the electric charge exists at the point where the stem is
attached, and there leakage by creeping takes place. If, however, we
employ a hollow sphere and let the stem pass through a hole in the side
larger than itself, and attach the end to the interior of the sphere,
then leakage cannot take place.
Another corollary of the fact that there is no electric force in the
interior of a charged conductor is that the potential in the interior is
constant and equal to that at the surface. For by the definition of
potential it follows that the electric force in any direction at any
point is measured by the space rate of change of potential in that
direction or E = [+-]dV/dx. Hence if the force is zero the potential V must
be constant.
(iii.) _Association of Positive and Negative Electricities._--The third
leading fact in electrostatics is that positive and negative electricity
are always created in equal quantities, and that for every charge, say,
of positive electricity on one conductor there must exist on some other
bodies an equal total charge of negative electricity. Faraday expressed
this fact by saying that no absolute electric charge could be given to
matter. If we consider the charge of a conductor to be measured by the
number of tubes of electric force which proceed from it, then, since
each tube must end on some other conductor, the above statement is
equivalent to saying that the charges at each end of a tube of electric
force are equal.
The facts may, however, best be understood and demonstrated by
considering an experiment due to Faraday, commonly called the ice pail
experiment, because he employed for it a pewter ice pail (_Exp. Res._
vol. ii. p. 279, or _Phil. Mag._ 1843, 22). On the plate of a gold-leaf
electroscope place a metal canister having a loose lid. Let a metal ball
be suspended by a silk thread, and the canister lid so fixed to the
thread that when the lid is in place the ball hangs in the centre of the
canister. Let the ball and lid be removed by the silk, and let a charge,
say, of positive electricity (+Q) be given to the ball. Let the canister
be touched with the finger to discharge it perfectly. Then let the ball
be lowered into the canister. It will be found that as it does so the
gold-leaves of the electroscope diverge, but collapse again if the ball
is withdrawn. If the ball is lowered until the lid is in place, the
leaves take a steady deflection. Next let the canister be touched with
the finger, the leaves collapse, but diverge again when the ball is
withdrawn. A test will show that in this last case the canister is left
negatively electrified. If before the ball is withdrawn, after touching
the outside of the canister with the finger, the ball is tilted over to
make it touch the inside of the canister, then on withdrawing it the
canister and ball are found to be perfectly discharged. The explanation
is as follows: the charge (+Q) of positive electricity on the ball
creates by induction an equal charge (-Q) on the inside of the canister
when placed in it, and repels to the exterior surface of the canister an
equal charge (+Q). On touching the canister this last charge goes to
earth. Hence when the ball is touched against the inside of the canister
before withdrawing it a second time, the fact that the system is found
subsequently to be completely discharged proves that the charge -Q
induced on the inside of the canister must be exactly equal to the
charge +Q on the ball, and also that the inducing action of the charge
+Q on the ball created equal quantities of electricity of opposite sign,
one drawn to the inside and the other repelled to the outside of the
canister.
_Electrical Capacity._--We must next consider the quality of a conductor
called its electrical capacity. The potential of a conductor has already
been defined as the mechanical work which must be done to bring up a
very small body charged with a unit of positive electricity from the
earth's surface or other boundary taken as the place of zero potential
to the surface of this conductor in question. The mathematical
expression for this potential can in some cases be calculated or
predetermined.
Potential of a sphere.
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 as 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/x^2 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
_
/x
W = | Qx^-2 dx = Q/x (1).
_/[oo]
Capacity of a sphere.
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 [sigma]dS/r, where dS is an element of surface, [sigma] 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 an equal distance R from the centre. Accordingly
the potential _at_ the centre is Q/R. But this must be the potential
_of_ the 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.
Capacity of a thin rod.
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 [sigma] the uniform surface
electric density. Then consider a thin annulus of the wire of width
dx; the charge on it is equal to 2[pi]r[sigma]/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
_l/2
/ 2[pi]r[sigma]
V = 2 | -----------------dx = 4[pi]r[sigma] {log_e (1/2 l + [root][r^2 + 1/4 l^2]) - log_e ^r}.
_/ [root](r^2 + x^2)
0
If, then, r is small compared with l, we have V = 4[pi]r[sigma]log_e
l/r. But the charge is Q = 2[pi]r[sigma], and therefore the capacity
of the thin wire is given by