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direction and magnitude of the resultant electric force.
The direction of the force is best expressed by the device
of lines of force. A line of force must be conceived so
drawn in the electric field that a tangent to it at any point
represents the direction of the electric intensity at the
point. For brevity the expression "force at a point" is
used to signify the intensity of the force sustained by unit
quantity of the active agent at the point, or the electric
intensity at the point.

Lines of electric force always spring from a
positively electrified surface and end in a nega-
tively electrified one. The stress along these
lines is a tension, tending to shorten them. It
is accompanied by a pressure at right angles to
the lines and tending to separate them.

When one electrified body attracts another,
the two are drawn together by these taut lines
of force stretching between them. When two
plates oppositely electrified face each other (Fig.
47), lines of electric force stretch across from
the positive to the negative, and the tension in
the medium tends to draw the plates together.

111. Equal Charges of Opposite Sign. When a body
is electrically excited by friction, the body rubbed and the
rubber are equally electrified, but with charges of opposite
sign. The equality consists in the ability of the one




ELECTUK CHARGES. 157

charge to exactly neutralize the other. If a stick of seal-
ing-wax, provided with a flannel cap with a silk cord
attached (Fig. 48), be excited by turning it around a few
times inside the cap, it will not attract a positively electri-
fied pith-ball if the cap be left on ; but if the cap be with-
drawn by the cord, the sealing-wax will attract the pith
and the cap will repel it.

The electrification of a body consists
in the separation of two equal charges
of opposite sign against their mutual at-
traction. Hence the medium between
them is strained by the operation, and
work is done. A positively charged _
conductor, insulated by supports of glass,
shellac, silk, or other non-conductors, is connected to other
bodies by invisible lines of electric force, springing from the
positive charge and extending to the equal negative one on
surrounding bodies. The slightest charge of positive elec-
tricity at one point always means an equal charge of the
opposite sign as near to it as the conductivity of the
dielectrics permits.

Whatever operations of electrically exciting, discharg-
ing, and the like, may be carried on within an insulated
conducting chamber, no signs of excitation will be ex-
hibited without. The positive and negative excitations
exactly neutralize each other outside the chamber.

112. Electroscopes. An electroscope is an instru-
ment for detecting electric charges. The simplest one,
which was employed by Gilbert, consists of a long straw,
turning freely on a sharp point, which must be insulated
from the earth. A pith-ball suspended by a silk thread is
also a convenient sensitive electroscope.



158



ELECTRICITY AND MAGNETISM.



-The Gold-leaf Electroscope is still more sensitive.
Through the top of a glass jar passes a brass rod, terminat-
ing in a ball above, and bent at right angles below to
receive two strips of gold leaf (Fig. 49). The top of the
jar should be coated with shellac both within and without.
Two strips of tin foil are pasted inside the jar from the

bottom up to the lower level
of the gold leaves to prevent
the latter from sticking to
the glass when they are vio-
lently repelled.

If the knob be touched
with a positively electrified
glass tube the leaves will be
mutually repelled with -f
charges. The approach of
any other charged body will
cause them to diverge more
widely if the charge pre-
Fi g . 49. sented is +, and to approach

each other if it is .

113. Charge External. - When a conductor is electri-
fied by friction or by electricity conveyed to it from some
external source, the charge always resides on the outside.
Biot devised a direct demonstration by fitting to an insu-
lated copper ball two hemispherical copper shells. When
the whole was charged and the shells were then deftly
removed by glass handles, the charge was found to be
entirely removed with them. A simple demonstration
of the law is afforded by a hollow metal sphere with a
hole at the top and insulated on a glass stem (Fig. 50).
It may be tested by means of a proof-plane, which is




ELECTRIC CHARGES.



159



composed of a small metal disk with a shellac or ebonite
handle. If the proof -plane be applied to the outside of the
charged sphere, a small charge
may be removed and tested by an
electroscope. If the proof-plane
l>e passed through the hole in the
sphere and applied to the inner
surface, it will be found on with-
drawal to exhibit no trace of elec-
trification. The proof-plane may
be charged from the outside of
the sphere, and then be made to
touch the interior. It will lose
all its charge and will show none
on withdrawal.

Faraday constructed a cube 12
feet on each side and covered it
with tin foil. He went inside of
it with his electroscopes ; but
while it was charged so that long
flashes were given off from the
outside, he could detect no signs of electrification within.

114. Distribution of Charge. The quantity of elec-
tricity (119) on unit surface of a conductor, or the ratio
of the quantity on any small area to the area itself, is
called the surface density. The distribution of an electric
charge is not such as to give uniform surface density over
an insulated conductor, except in the case of a sphere
remote from other conductors and electrified bodies. The
distribution on conductors of various shapes was investi-
gated by Coulomb by means of the proof-plane and torsion
balance (116). The following is a summary of results:




Fig. 50.



160 ELECTRICITY AND MAGNETISM.

(1) On a cylinder with rounded ends the surface den-
sity is greatest at the ends.

(2) On a flat disk the density is much greater at the
edges than on the flat surfaces, but over the latter the dis_
tribution is fairly uniform except near the edges.

(3) With two spheres in contact the charge is nothing at
the point of contact, increases rapidly between 30 and 60
from that point, and becomes greatest at 180. When the
spheres are of unequal size, the density at corresponding
points is greater on the small sphere than on the large one.

The density is greatest on those parts of a conductor
which project most and have the greatest convexity.
Hence at sharp points, such as that of a needle, the density
is very great, and as a consequence the charge escapes
rapidly from them. It is therefore necessary to round
off all edges of insulated conductors and to make them
smooth.

115. Redistribution of Charge. Coulomb demon-
strated that when a charged conducting sphere is brought
into contact with an identical one in the neutral state,
each will then possess a quantity equal to half of the
original charge. If the second sphere, instead of being
neutral, is itself charged, the final charges are again equal.
Each of them is half the algebraic sum of the initial
charges, so that both spheres will be neutral if those
charges were equal and of opposite sign.

The result will be the same with two like conductors of
any form whatever when one touches the other, provided
they are symmetrical with respect to the point of contact.
If this condition of symmetry is not fulfilled, the charges
will divide unequally, but so that their algebraic sum
always equals that of the initial charge*.



ELECTRIC CHARGE*.



161



Since the charge resides on the outside, if a small charged
sphere be introduced into a larger hollow one, it will give
up its charge entirely to the larger sphere. By this. means
a conductor may be charged by successive additions of
small quantities, or one can increase or decrease the electric
charge on the outside of a closed surface by introducing
within small positive or negative charges.

116. Coulomb's Torsion Balance. The torsion balance
was invented by Coulomb for the pur-
pose of investigating the law of at-
traction and repulsion between two
charges of electricity. The instru-
ment is now obsolete, but it illus-
trates the meaning of the law of
inverse squares which was established
by Coulomb's elaborate experiments.

From a torsion head 7i (Fig. 51)
is suspended a very fine wire, carry-
ing at its lower end a light shellac
rod with a gilt pith-ball b. The shel-
lac rod swings inside a protecting glass
case, around which is a graduated
scale s at the level of the gilt ball. A
shellac rod, carrying another gilt ball
'-. can be introduced through a hole in
the top of the case. The torsion head
is divided into degrees, and is pro-
vided with an index. The rod carrying the torsion wire
can be turned independently of the rest of the head, so that
the index can be held at zero, while the rod and wire are
turned till the movable ball just touches the fixed one
without anv torsion of the wire. Calcium chloride, or




Fig. 51.



162 ELECTRICITY AND MAGNETISM.

some other drying agent, is placed in the case to keep
the air dry.

117. Law of Inverse Squares. When the instrument
has been set as described, the vertical rod is removed, the
attached ball is charged, and is then replaced in the instru-
ment. It touches the ball -b and divides its charge with it.
Repulsion follows, and the ball b moves away till the tor-
sion couple of the suspending wire equals the moment of
the force due to the mutual repulsion. The distance
between the balls is not sensibly different from the arc
of the circle separating them, if the balls are not many
degrees apart. The balls are now made to approach each
other by turning the torsion head and twisting the wire.
The two divided circles then give the whole angle of tor-
sion of the wire. The principle employed in comparing
the forces is that when a wire is twisted, the couple of
torsion is proportional to the angle through which the wire
is twisted. For example, if the moments of the couples
required to twist a wire through 10 and through 20 are
measured, the latter will be found to be twice as great as
the former.

The following data belong to one of Coulomb's experi-
ments : The first deflection of the movable ball was 36.
To reduce it to 18 it was found necessary to turn the head
through 126; and for a further reduction to 8. 5 an addi-
tional rotation of 441 was required. The several relative
distances of the balls were then about as 1 to J to J , and
the torsion of the wire was 36 for the first distance,
126 + 18 = 144 for the second, and 441 + 126 + 8.5 = 575.5
for the third. But 144 is 4 x 36, and 575.5 is nearly 16 x 36 ;
so that as the distance is reduced successively to $ and J ,
the force is increased to 4 and 16 times respectively.



ELECTRIC CHARGES. 163

The law of attraction was also investigated by a similar
method, and was found to hold within the same limits.
Also the dependence of the force on the charge was
examined by touching one of the balls with an insulated
one of the same size. Half of the charge was thus
removed, and the force was found to be reduced to one-
half. If the charge of either ball was reduced, the mutual
force was reduced in the same ratio.




118. Second Law of Electrostatics. The second law
of electrostatic action, established by the experiments of
Coulomb, may be enunciated as follows: The force between
twv ///'/ ///<''-? bodies is directly proportional to the product of
the tiro charges, and inversely proportional to the square of
the distance between them.

The law of distance does not hold unless the dimensions
of the charged conductors are very small in comparison
with the distance between them. The charge on a sphere
acts as if it were collected at its centre (121) only when
the distribution of this charge is not affected by the
influence of neighboring charges. In Coulomb's experi-
ment the actual mean distance of the two charges when
the balls were brought as near together as 8 C .5 was greater
than the distance between the centres of the spheres. The
force between two flat disks near each other does not vary
appreciably with a moderate change in the distance.

If the t\\o quantities q and '/are on infinitesimal spheres,
the distance oi' whose centres is r, then the force between
them may he expressed by the formula



The positive sign corresponds to similar charges, and there-

fore to repulsion, and the negative sign to attraction.



164 ELECTRICITY AND MAGNETISM.



119. The Unit of Quantity. - - The definition of the
electrostatic unit of quantity is derived from the law of
attraction and repulsion. If the force in the foregoing
proportion is to become unity when the distance and the
charges are unity, unit quantity must be defined as fol-
lows: The electrostatic unit of electricity is that quantity
which repels an equal and similar quantity, at a distance of
one centimetre in air, with a force of one dyne.

Since the intensity of an electric force is the force
exerted on unit quantity, it follows that the electric inten-
sity at a point distant r centimetres in air from a charge q
is q /r z . The reason for inserting the expression "in air"
will appear later (165).

120. Indirect Proof of the Law of Inverse Squares.

It has already been pointed out
that no electric force can be detected
inside a hollow conductor. This ex-
perimental fact furnishes the basis
of the most conclusive proof that
the force varies inversely as the
square of the distance.

The following may be considered
Fig 52 as an illustration of the principle

rather than a rigid mathematical dem-
onstration : Let P (Fig. 52) be any point within a charged
conducting sphere, and let a narrow cone of two sheets be
described with P as the apex, and cutting the sphere in
two areas and s / at ab and a'l' respectively. Then, since
the surface density is supposed to be uniform, the quantities
on the two areas are proportional to those areas; but the
areas are proportional to the squares of their respective
distances from P. To prove this latter relation, it must





ELECTKTC CHARGES. 165

be noted, first, that the two areas are sections of the cone
equally inclined to its axis. Let ab and a'b' (Fig. 53) be
oblique sections of a cone making the same angle with the
axis. Their linear dimensions are directly proportional to
the distances PA and PB ; and since the areas of similar
figures are proportional to the squares of their homologous
dimensions, the areas of the two sections are proportional
to the squares of PA and PB.

It follows that the two quantities on s and s' P

are proportional to the squares of Pa and Pa'-
Hence the two forces acting on P are directly
proportional to the squares of Pa and Pa', and
inversely proportional to some function of these
distances. But since there is no force inside a
charged sphere, and since the whole surface may
be divided into a series of such pairs of sections Fj 53
made by a cone, and what is true of the whole
is true of each pair, it follows that the forces due to the
charges on s and s' are equal to each other. But the only
function of the distances which will satisfy this condition
is the inverse square. The forces are proportional to the
acting quantities, which are directly proportional to the
squares of the distances ; the forces are also inversely pro-
portional to the squares of the same distances ; and, being
opposite in direction, the resultant is zero.

121. Force Outside a Charged Sphere. The force or
electric intensity at any point outside a charged sphere,
over which the distribution is uniform, is the same as if
the entire charge were collected at its centre. This propo-
sition admits of simple demonstration.

Let P be the point at a distance D from the centre of
the sphere (Fig. 54). Let a be the surface density, and



160



ELECTRICITY AND MAGNETISM.



let s be the area of a very small element of the surface at
the point B. The quantity on it is so-, and if p is the
distance PB, the force at P due to this element of the
charge is scr/p'. Since the entire surface of the sphere is



Fig. 54.

symmetrical with respect to the line PO, the resultant of
all the forces due to the several elements of the charge
must be along P 0. The component of the. force scr/p' along

this line is g(r

f cos a,

P*
where a is the angle OPB.

Draw BA, making the angle AB equal
to a. Also let o) be the solid angle which
the area s subtends at A. The projection s'
of the area s at right angles to AB subtends
the same angle co at A. Since the angle be-
tween s and s / is a (Fig. 55), we may write



or



s'= (or- = s cos a.



cor




cos a



Substituting this value of s in the expression for/, and



ELECTRIC CHARGE*. 167

The triangles OB A and OBP are similar, and therefore

rR



where D is the distiince J0. Hence by substituting above,

, R 2

f = &-

This is the force due to a single element of the surface.
For the entire surface the force is the sum of the small
forces due to all such elements, or

*=&,*.

The expression 2&> is the entire solid angle subtended by
the surface of the sphere at any point within it, and this is
4?r. Hence






But 7rR 2 (T is the product of surface of the sphere and the
surface density, or the whole charge on the sphere, and D
is the distance between the point P and the centre of the
sphere. Therefore the expression for F is precisely the
same as would be obtained for the force at P if the whole
charge were at the centre of the sphere. It is worth noting
that this demonstration applies equally well to the force of
gravity due to a thin shell of matter, when the shell is of
uniform thickness and density.

122. Force very near a Charged Sphere. If the
point P in Fig. 54 is made to approach the sphere, the point
A also moves toward the surface to meet P ; and when P
is at the surface D equals R and

F= 47TO-,

or the electric intensity is independent of the size of the



168 ELECTRICITY AND MAGNETISM.

sphere, and is numerically equal to 4?r times the surface
density. This result, which is known as Coulomb's Law,
requires modification when the sphere is surrounded by
some other dielectric than air. It applies to any charged
conductor. Since there is no force inside the sphere, the
change of force in passing from a point just outside to the
interior is 4?rcr.

If a plane perpendicular to P be drawn through J., it
will divide the spherical surface into two parts, each of
which subtends at A the same angle 2?r. Hence half the
force is due to the charge to the right of this dividing plane,
and the other half to the charge to the left of it. At the
surface of the sphere one of these charges is contained on
an infinitesimal area, and the other is the charge on all the
rest of the sphere. The force is then the same as that due
to a plane of indefinite extent, tangent at C and charged
on both sides.

123. Force near a Charged Plane Conductor (Th.,
262). Imagine a plane of indefinite extent charged

positively on one side to
a density a. Let P be
the point at which the
force is to be determined
(Fig, 56), and PO the
normal to the plane. Let
8 be any small surface on
the plane, and co the solid
angle which it subtends
Fig . 55 at P. It is the solid

angle at the apex of the

cone made by drawing lines from the boundary of s to
the point P. The force at P due to the charge on this




ELEi'TlUC f'lfMWKS. 169



element is sa/r. and the component of this force along the
normal PO is

/. scr
/=__ cos a,

where a is the angle between the normal and the axis of
the cone.

As in Art. 121, the orthogonal section of the cone
x' = cor and

$' = cor = s cos a.



Therefore s



cos a

Substituting in the equation for/, we have
f= aco.

Since the resultant of all the forces due to the elementary
charges is along the normal, the total intensity of the force
at Pis



But 2o> is the solid angle subtended at P by a plane of
indefinite extent, and this is the angle subtended by a
whole hemisphere, or 2?r. Therefore



In the C.<T.S. system the force is in dynes.

If the plane is limited and the point P indefinitely near
it, the force is again ZTTCT.

Since the force on a + unit above the plane is directed
upward and below the plane downward, in passing through
the plane the force changes by the quantity 4?ro-.

PROBLEMS.

1. Two equal small balls are charged with -|-30 and 6 units of
electricity respectively. Find the mutual force between them when
their centres are C cms. apart, before and after contact with each
other.



170 ELECTRICITY AND MAGNETISM.

2. A charge of 100 units is applied to a sphere of 10 cms. radius.
What is the surface density ?

3. In the last problem, Avhat is the value of the electric intensity
at the surface ?

4. Two small balls, each one gm. in mass, are suspended from
the same point by silk fibres 490 cms. long. If g is 980 dynes, show
that the balls will diverge to a distance of one cm. if each is charged
with one unit of electricity.

5. Two small spheres 10 cms. apart are charged with -}- 5 units
and 5 units respectively. Find the direction and magnitude of the
force acting on a -j- unit at a distance of 10 cms. from both charges.



ELECTRIFICATION BY ISFLUESCE. 171



CHAPTER XII.

ELECTRIFICATION BY INFLUENCE.

124. Fundamental Phenomena. - A charged con-
ductor exerts influence, or acts inductively, on all neigh-
boring bodies. If it be positively charged, lines of electric
force start from it and proceed to an equal negative quan-
tity on adjacent bodies. The influence is exerted along
these lines of force, or lines of tension.

Let an insulated
sphere A (Fig. 57), ^ ^"
charged positively,
be placed near an in-
sulated cylindrical

conductor 5. Light

pith-balls suspended by linen threads at either end of B
will diverge, and the nearer A approaches B the wider the
divergence, unless the charges on A and B unite by a
spark across the air-gap. If A and both ends of B be now
examined by means of a proof-plane and an electroscope,
it will be found that the charge on A has been redistributed,
so that the surface density on the side toward B is greater
than on the remote side ; also the end a of the cylinder will
be found to be negatively charged, the central portion will
be neutral, and the end b will be positively charged. The
density at b will be less than at #, and the neutral line will
be somewhat nearer a than b.




172 ELECTRICITY AND MAGNETISM.

When A is removed or discharged by connecting with
the earth, all signs of electrification on B disappear. The
separation of the positive and negative charges on B
through the influence of the charge on A is called electro-
static induction, or electrification by influence.

125. Charging by Influence. If the conductor B be
connected with the earth while under the inductive influ-
ence of A, the repelled charge will pass off, leaving only
the attracted electricity. This latter charge is said to be
" bound " in distinction from the " free " charge which goes
to the earth. If now A be removed while B remains insu-
lated, the charge on the latter will be distributed over the
whole conductor, and B is said to have been charged by
influence or induction.

The electrification of B represents energy. Work has
been done in removing A against the attraction of the
charge on B. If B uninsulated were to be brought up to
A from a distance, and then removed after insulating it, the
work done by mutual attraction during the approach would
be less than that done against the attraction during the
withdrawal, because the acting charge on B in the latter
movement remains constant, while during the approach of
B to A the charge on B increases from nothing to the
maximum. The working force is then less during the
approach than during the recession.

If when the charge has been insulated on B the posi-
tive on A is discharged to earth, the electrification of B
still represents energy. The energy of the discharge of A
under these conditions is less than that required to charge
it when removed from inductive action on other bodies.
This will be better understood after studying the relation
between energy and potential.



ELECTRIFICATION BY INFLUENCE. 173

126. Electrification with like Charges by Influence.
- Wl leu a body is charged by influence as explained in

the last article, the repelled charge always becomes free,
and the conductor is charged so that the inducing and the
induced charges are of opposite sign. In this case pro-


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Online LibraryHenry S. (Henry Smith) CarhartPhysics for university students (Volume 2) → online text (page 11 of 28)