Physics for university students (Volume 2) online

155. Seat of the Charge. The Leyden jar with re-
movable coatings is due to Franklin. By means of it he
showed that the charge resides on the surface of the glass.




Fig. 82.

A (Fig. 82) is the jar complete, B is the glass vessel,
the outer and D the inner metallic plate. If the jar be
charged in the usual way and be placed on an insulating
support, the inner plate may be removed by lifting it by
the curved rod ; then the outer plate may be removed from
the glass jar. The two plates are then completely dis-
charged. After putting the jar together again, it can be
discharged with a bright flash. The coatings serve as dis-
chargers for the glass. The charge on two small areas of
the glass may be made to unite with a crackling sound
by touching them at the same time with the fingers. " The
two conducting surfaces may therefore be regarded simply
as the boundaries of the intervening dielectric."



CAPACITY AND CONDENSERS. 209

156. Energy expended in charging a Condenser.
The energy expended (138) in transferring Q units of
electricity through a potential difference of V J r is
Q ( V K))- If the charge Q is transferred from the earth,
whose potential is zero, to a conductor whose potential is
FJ the work done is Q V. But in charging a condenser, or
any conductor, the potential is zero at the beginning of the
charging and Fat the end. The mean potential to which
the charge is raised is then F. The work done in charg-
ing the condenser is therefore



But since QCV^

yV v



.

*^p /Q

The energy expended in charging a condenser to a poten-
tial difference Vis one-half the product of the capacity
and the square of the potential difference. Potential
corresponds to height when work is done against gravity.
Thus in building a brick tower of uniform cross-section A,
the mean height to which the bricks are lifted is half the
final height of the tower ; the work done in gravitational
units is one-half the product of the mass M and the height
h of the tower. If with half the cross-section the tower
is carried to twice the height, the Avork done is simply
doubled because the same mass is lifted to double the
mean height. If, however, the tower with the cross-section
A is built to twice the height 2A, the work is quadrupled^
because both the mass lifted and the mean height are
doubled. The constant is the area A, corresponding to
capacity, and the work done varies as the square of the
height.



>v -

157. Energy lost in dividing a Charge. Let O l and
. be the capacities of two condensers, and let the first be



210 ELECTRICITY AND MAGNETISM.

charged with a quantity Q. The energy of the charge is

ici>=i

2 ~2 CT

After the charge has been divided between the two con-
densers by connecting them in parallel, the potential differ-

ence has fallen to Hence the energy is then



It is less than the energy before the division so long as <7 2
has any value in comparison with C\. If the two capacities
are equal, the energy after the division is one-half as great
as before it. The other half is represented by the energy
of the spark at the moment of the division. The lowering
of the potential by the increase of capacity diminishes the
energy represented by a given charge. Energy is always
lost by the division, whatever be the relative capacities of
the condensers, except when the potential differences of the
two are the same before they are joined in parallel ; but in
this case there is no electric flow and no lowering of the
potential difference.

158. Energy of Similar Condensers in Parallel. If
n condensers of the same capacity
are charged in parallel, for ex-
ample, n Leyden jars of the same
size with their outside coatings
connected together, and likewise
all their inside coatings (Fig. 83),
the capacity of the whole is n

times the capacity of a single condenser, because the effect
is simply to increase the size of the coatings.




CAPACITY AXD CONDENSERS. 211



The energy of discharge of a single condenser is k
and for n condensers of the same capacity it is



The en erg}* is thus increased in proportion to the number

of .similar condensers.

159. Energy of Condensers connected in Series.
If several Leyden jars are insulated and the outside of the
Hist is connected to the inside of
the second, etc. (Fig. 84), they are
said to be connected in series or
in " cascade." The inside of the
Hist jar is one side of the compound
condenser, and the outside of the
last one is the other side. If the Fig . 34.

potential difference between the
t\vo sides is F", then the energy of each of the n similar jars

V 2
is J(7_ ; and the energy of the charge in the n jars is




n



W=



Hence the energy of the charge of the n jars in series is 1/w.
of the energy of one of the jars charged to the same poten-
tial difference between its two coatings, ^r



160. Electric Strain. The phenomenon of the residual
charge may be best explained by considering the dielectric

the medium through whose agency the induction takes
place. The charging of a Leyden jar is accompanied by
the straining of the glass. If the potential difference is
raised to a sufficiently high value, the glass maybe strained
beyond the elastic limit and may give way with a disruptive



212 ELECTRICITY AND MAGNETISM.

discharge of the jar. The glass is then shattered at the
point through which the discharge takes place. In the case
of air or other fluid dielectrics, such as insulating oils, the
dielectric may be broken down by a disruptive discharge,
but the damage is immediately repaired by the inflow of
the insulating fluid.

By subjecting plate glass to powerful electrostatic stress
and passing plane polarized light through it at right angles
to the lines of force, Kerr discovered that glass becomes
doubly refracting, and is strained as if it were com-
pressed along lines of force. Quartz behaves in the same
way, but resins and all insulating fluids examined, except
olive oil, act as if extended along lines of force. The
action requires about thirty seconds to reach its maximum
effect, and about the same time is required for complete
recovery. Kohlrausch and others have pointed out the
analogy between Kerr's discovery and the elastic fatigue of
solids after being subjected to a twisting strain. A fibre of
glass if twisted does not immediately regain its initial form
when released from tMe stress, but a slight set remains from
which it slowly recovers.

The glass of a charged jar is then greatly strained, and
it does not at once recover when the jar is discharged. Its
after-recovery from distortion sets free energy which is
represented by the residual charge. Hopkinson has shown
that it is possible to superpose several residual charges of
opposite signs. In the same way a glass fibre may be
twisted first in one direction and then the other, and the
residual twists will appear in reverse order. No residual
charges can be obtained from air condensers, nor from those
with quartz plates. Correspondingly, quartz shows no elas-
tic fatigue after being twisted.

Siemens has shown that the glass of a Leyden jar is



CAPACITY AND CONDENSERS.



213




sensibly warmed by rapid charging and discharging. The
distortion shows a lag behind the electric stress, a phenom-
enon known as hysteresis when applied to magnetic induc-
tion. The result in both cases is the absorption of energy
and the generation of heat. The quantity of heat generated
is proportional to the square of the potential difference to
which the condenser is subjected.

161. Dielectric Polarization (B., 573). Faraday's
theory of induction was that it is an action between con-
tiguous parts of the dielectric, resulting in a certain
polarized state of its particles. In proof of this polariza-
tion he placed bits of dry
silk filaments in turpen-
tine contained in a long n
rectangular glass vessel -
with pointed conductors
entering from opposite
ends. When one of
these was connected with
the earth and the other with a frictional machine, the bits
of silk collected together along lines of induction, forming
long filaments of considerable tenacity. Matteucci de-
monstrated that the dielectric is polarized or charged by
contiguous particles throughout. He formed a condenser
of a large number of thin plates of mica compressed be-
tween two terminal metal plates (Fig. 85). After charg-
ing it and insulating, it was found on removing the mica
plates and examining them that each one was charged
positively on one side and negatively on the other, all the
positive sides being turned toward the positive electrode,
and all the negative ones in the opposite direction.

Maxwell explains the residual charge by assuming that



Fig. 85.



214 ELECTRICITY AND MAGNETISM.

the dielectric is not homogeneous, and that it therefore
becomes electrified at the surfaces separating the non-
homogeneous parts, like the electrification of the mica
plates. The reunion of these charges is gradual after the
first discharge, and their external effect is shown as a
residual charge. "According to this theory all charge is
the residual effect of the polarization of the dielectric "
(Maxwell). In the interior of the dielectric the polariza-
tion is neutralized by adjacent opposite charges ; " it is
only at the surface of the dielectric that the effects of the
charge become apparent."

'

162. Distinction between Conductors and Insulators
(Max., 156). - - The potential difference between the
boundaries of a dielectric is the electromotive force acting
on it. If the dielectric is an imperfect insulator, the state
of constraint is continually giving way or being relaxed.
The medium yields to the electromotive force, and the
potential energy of its distortion is converted into heat.
In good insulators the rate at which this conversion takes
place is very slow.

In conductors the electric polarization gives way as fast
as it is produced by the electromotive force, with a steady
transfer of electricity; this transfer is called an electric
current. The external agency which maintains the cur-
rent is constantly doing work in restoring the polarization,
and the result of expending energy on the conductor is the
generation of heat. Non-conductors are capable of main-
taining the state of electric polarization ; in conductors
this polarization breaks down as fast as it is formed. The
application of an electromotive force to the former causes
a momentary transfer of electricity ; its application to the
latter produces an electric current.



CAPACITY AND CONDENSERS. 215

163. Electric Displacement. Electricity exhibits the
properties of an incompressible fluid. Electric charges
reveal themselves only at the boundaries between conduc-
tors and the dielectric. Lines of induction run from the
positive charge at one boundary through the dielectric to
the negative at the other ; and if we conceive tubes
of induction bounded laterally by lines of induction,
every tube in a dielectric between two conductors has
equal charges on its two ends, or the induced and the
inducing charges are equal to each other. All cases of
electrification are cases of the transfer of electricity. Hence
Maxwell proposed his theory of electric displacement, which
supposes that when an electromotive force acts on a dielec-
tric, as in induction, electricity is transferred or displaced
along tubes of induction. The electromotive force not
only distorts the medium, but transfers electricity by
stretching the dielectric. If a dielectric, polarized by elec-
tric displacement, be left to itself, the elastic reaction pro-
duces a back electromotive force and a reverse electric
transfer to restore the equilibrium. This restoration to
equilibrium constitutes the discharge of the condenser. A
disruptive discharge means the rupture of the dielectric,
usually the air. If the discharge is abrupt, the sudden
release of the dielectric from strain is followed by rapid
electric displacements in opposite directions alternately,
till the energy of the charge is all wasted in heat. This
phenomenon is known as the oscillatory discharge of a
condenser. It was discovered by Joseph Henry in 1842.

164. Electric Transfer always in Closed Circuits.
The electric-displacement theory of Maxwell has led to a
conception of the electric circuit which allows the contrast
between a conducting and a non-conducting circuit to be



216



ELECTRICITY AND MAGNETISM.



expressed in a simple manner. In a circuit made up
entirely of conductors the operation of an electromotive
force causes a continuous flow of electricity ; but if the
circuit is only partly conducting and in part composed of
a dielectric, then the action of an electromotive force in
the circuit produces a transient electric flow through the
conductor and an equivalent displacement through the di-
electric. The amount of the flow depends on the capacity
of the dielectric as a condenser and the magnitude of the
electromotive force, or Q= CV. Through the conductor
electricity is transferred by the process of conduction ;
through the dielectric it is transferred as a displacement,



^vSx

&



that is, it is forced along by straining the medium. Dis-
placement always produces a reactive electromotive force
which counterbalances the direct electromotive force and
effects a discharge when the latter is removed.

Consider a circuit of water-pipes filled with water and
containing a pump P (Fig. 86). If there were no
obstructions in the pipes the motion of the pump would
cause a circulation of water through the system. This
arrangement corresponds to a conducting circuit. But
if we imagine elastic diaphragms stretched across the
enlarged pipe at many points, as , I, e, d, e, the rotation of



CAPAt ITT A.\D CONDENSERS.



217



the pump, so as to produce a flow in the direction of the
arrows, displaces water along the enlarged pipe by stretch-
ing the diaphragms, and causes a transient current through
the remainder of the system. The displacement ceases as
soon as the reaction of the diaphragms equals the force
applied to the pump. The same quantity of water is
transferred across every cross-section of the pipes through-
out the whole system, whether the diaphragms are present
or not. Without the diaphragms the flow would be con-
tinuous ; with them it continues only so long as they yield
to the stress of the water. If the force applied to the
pump be withdrawn, the reaction of the tense diaphragms
produces a counter-flow. The diaphragms represent the
dielectric and the unobstructed pipes the conductor. So
in charging a condenser the same quantity of electricity
is displaced through the dielectric as flows along the con-
ducting 1 part of the circuit.

165. Specific Inductive Capacity. Different dielec-
trics possess different powers
of t ra i is mitting induction
across them. The density of
the charges at the surfaces of
the condensing plates, with a
given difference of potential
between them, depends not
only on the distance between
them, but also on the facility
with which the dielectric per- ^
mits electric displacement.

Let A, B, (Fig. 87), be three insulated conducting
plates. To the back of A and C are suspended pith-balls.
Let B receive a positive charge and let^t and C be charged




Fig. 87.



218 ELECTRICITY AND MAGNETISM.

negatively by induction. If they are touched with the
finger the pith-balls will collapse and remain in contact
with the plates. If now A, for example, be moved nearer
to B, both pith-balls will diverge, the one on A with a
+ charge and the one on C with a one. The diminished
distance between A and B permits increased induction
, which transfers + electricity to the back of A; but the
increased induction on the left of B diminishes it on the
right, and some of the charge on becomes free and
spreads over the back of the plate.

Replace A in the first position, with B charged as before
and the pith-balls not diverging. Interpose between A and
B, without touching them, a thick plate of glass or sulphur.
Both pith-balls will again diverge as if A had been moved
nearer B, showing that the effect is the same as the reduc-
tion of the thickness of the air between the plates. The
capacity of a condenser depends then on the nature of the
insulating medium between the two opposed conductors.
The specific inductive capacity, or dielectric constant, of a
substance is the ratio of the capacity of a condenser with
the substance as the dielectric to its capacity when the
dielectric is air. The dielectric constants of all gases are
nearly the same, but those of solids differ greatly.

166. Faraday's Experiments. The first experiments
on specific inductive capacity were those of Cavendish, but
they were unknown till Faraday had made his discoveries
in the same subject. Faraday's experiments were made
with two exactly similar condensers shown in section in
Fig. 88. The metallic sphere A is supported by the rod M,
and both are insulated from the outer shell B by a plug
of shellac. The shell B is made in two halves which can
be detached from each other, so that the space between



CAPACITY AND CONDENSERS.



219



A and B can be filled either with a solid dielectric or with
a gas.

When both condensers were filled with dry air and one
of them was charged, it divided its
charge equally with the other on
joining them in parallel, its poten-
tial falling to one-half. The ca-
pacities of the two were therefore
the same. The space within one of
the condensers was then filled with
a solid, such as shellac, and the
above experiment was repeated.
The resulting potential was then
han half the initial potential.

Let Fbe the potential of the air
condenser before the division of the
charge, and (/ its capacity. If K
is the specific inductive capacity of
the dielectric in the second con-
denser, the capacity of this condenser will be KC. Let
V be the common potential of the two after the division of
the charge : then




Fig. 88.



Whence



K=



V- V

~V^~



ire -



In this way Faraday obtained for sulphur, as compared
with air, the value 2.2* J, and for shellac, 2.0.

Faraday's discovery of this property of a dielectric led
him to adopt the view that the effects observed in an
electric field are to be ascribed to the action of the dielec-
tric between electrified bodies, and not to the action of an
electrified body on others at a distance.






220 ELECTRICITY AND MAGXETINM.

; '11

167. Recent Results. - - The dielectric constant is
smaller in rapidly oscillating fields than in slowly changing
ones, because of the absorption of the charge which takes
place with the continued application of an electromotive
force in one direction. This fact explains to a certain
extent the great discrepancies which are found among the
results obtained by different observers. The following
table illustrates the difference between the values derived
from rapid and from slow methods :

Rapid. Slow.

Glass 3.013 to 3.243

" dense flint 7.37

" light flint 6.72

Ebonite 2.284 3.15

Gutta-percha 2.462

Paraffin (solid) 1.994 2.29

(liquid) 1.92

Shellac , 2.747

Sulphur 2.579 3.97

Mica 6.64

Turpentine 2.23

Distilled water 76.

Alcohol 26.

Northrup has recently measured the specific inductive
capacity of paraffin and plate glass, both with rapidly
oscillating and slowly changing fields, with the following
results :

Rapid. Slow.

Paraffin 2.25 2.32

Plate glass 5.86 6.25

The following are the specific inductive capacities of
several gases:

Hydrogen 0.999674 Carbon monoxide . 1.001

Carbon dioxide . . . 1.000356 Olefiant gas . . . 1.000722
Sulphur dioxide . . . 1.0037



CAPACITY AND CONDENSERS. 221

168. Effect of the Dielectric on the Electric Intensity
(J. J. T., 116). Consider two parallel-plate condensers
A and B, the plates being at the same distance in the two,
but the dielectric of A being air and that of B a medium
whose specific inductive capacity is K. Let us suppose
the charge per unit area, or the surface density, on the
plates of A and B is the same. Then, since the potentials
are inversely as the capacities when the charges are the
same, and since the capacity of B is K times that of A, it
follows that the potential difference between the plates of
A is K times as great as that between the plates of B.
But in both cases the electric intensity in the dielectric is
the rate of variation of the potential per unit length. Now
as the thickness of the dielectric is the same in A as in
B, while the potential difference of A is K times as great
as in B, it follows that the electric intensity in the air
between the plates of A is K times as great as in the
dielectric of B, or the electric intensity is inversely as the
specific inductive capacity.

We have seen in Art. 152 that the electric intensity F
between two plates in air is 47r<7. Hence in a medium
whose dielectric constant is K,



K'

Thus with given charges the forces in the field are dimin-
ished by introducing a medium with a large specific induc-
tive capacity.

169. Effect of the Dielectric on the Forces between
the Plates. From the equation of the last article,

47TO- = KF= ,

t

where t is the thickness of the dielectric.



222 ELECTRICITY AND MAGNETISM.

KV

Whence T _ - .

kirt

The force 011 unit quantity on one of the plates, due to
the charge on the other, is $F, and 011 unit area it is %Fcr.
Hence the force on either plate per unit area is



With a given charge, or given surface density, the force
between the plates is inversely as the specific inductive
capacity.

Fa V KV KV 1
Again, since T ==-._ = _,

it follows that, with a given potential difference, the force
between the plates is directly proportional to the specific
inductive capacity.

PROBLEMS.

1. Two Leyden jars are charged with quantities as 1 to 4. The
tin-foil surface of the second jar is twice as large as that of the first
and the glass is half as thick. Find the relative energy of the two
charges.

2. An insulated metal ball of 10 cms. radius, and removed from
all other conductors, is charged with 100 units of electricity. What
will be its potential if it be then surrounded by a smooth conducting
shell of 11 cms. radius, and connected to earth?

3. If one of two insulated conducting spheres, 20 cms. in diam-
eter, be charged to a potential of 15 units, and then be. connected with
the other sphere, by means of a long thin wire, find the energy of
the discharge between them.

4. Two Leyden jars of 200 sq. cms. tin-foil surface and glass
1 mm. thick, specific inductive capacity 6.283, are charged to poten-
tials of 100 and 10 units respectively. Find the energy lost in ergs
on connecting them in parallel.

5. Find the capacity of a spherical conductor, the radii of the



CAPACITY A\I) < OM)EXSERS. 223

opposed surfaces being 9 and 10 cms., and the dielectric paraffin,
whose specific inductive capacity is 2.3.

6. Two circular brass plates 30 cms. in diameter are separated
by glass 2 mms. thick and of specific inductive capacity 6. If they
are charged to a potential difference of 1,000 units, find the force of
attraction between them.

7. Tn the last problem, find the surface density on the boundary
between the glass and the plates.



224 ELECTRICITY AND MAGNETISM.



CHAPTER XV.

ATMOSPHERIC ELECTRICITY.

17O. Lightning an Electrical Phenomenon. - While
some of the early philosophers surmised that the lightning
flash was an electrical discharge, yet this view obtained
but little currency till Franklin's suggestion in 1749 to
apply his discovery of the discharging power of points
to the investigation of the problem had actually been car-
ried into effect. In 1752 d'Alibard, acting on Franklin's
suggestion, erected an iron rod 40 feet high, but not con-
nected with the earth, and drew sparks from passing
clouds. About the same time (1752) Franklin sent up
his famous kite by means of a linen thread, during a pass-
ing storm, and held it by means of a silk ribbon between
his hand and a key attached to the thread. When the
thread had been wetted by the rain, sparks were drawn
from the key and a Leyden jar was charged. The next
year Richmann, of St. Petersburg, was killed by light-
ning while experimenting with a rod similar to that of
d'Alibard.

171. The High Potential of Thunder Clouds. - - The
source of the electrification of the atmosphere and of clouds
remains as yet unsettled. But given ever so slight an
electrification of aqueous vapor, it is not difficult to ac-
count for the high potential exhibited \yy thunder clouds. 1