LECTURE II. 

THE ELECTRIC FIELD. 

7. Let, in Fig. 7, a generator G transmit electric power over 
line A into a receiving circuit L. 

While power flows through 
the conductors A, power is con- 
sumed in these conductors by 
conversion into heat, repre- 

sented by i?r. This, however, 

Fig. 7. is not all, but in the space 

surrounding the conductor cer- 
tain phenomena occur: magnetic and electrostatic forces appear. 


Fig. 8. — Electric Field of Conductor. 

The conductor is surrounded by a magnetic field, or a magnetic 
flux, which is measured by the number of lines of magnetic force <J>. 
With a single conductor, the lines of magnetic force are concentric 
circles, as shown in Fig. 8. By the return conductor, the circles 

10 


THE ELECTRIC FIELD. 


11 


are crowded together between the conductors, and the magnetic 
field consists of eccentric circles surrounding the conductors, as 
shown by the drawn lines in Fig. 9. 

An electrostatic, or, as more properly called, dielectric field, issues 
from the conductors, that is, a dielectric flux passes between the 
conductors, which is measured by the number of lines of dielectric 
force ty. With a single conductor, the lines of dielectric force are 
radial straight lines, as shown dotted in Fig. 8. By the return 
conductor, they are crowded together between the conductors, 
and form arcs of circles, passing from conductor to return conduc- 
tor, as shown dotted in Fig. 9. 


Fig. 9. — Electric Field of Circuit. 

The magnetic and the dielectric field of the conductors both are 
included in the term electric field, and are the two components of 
the electric field of the conductor. 

8. The magnetic field or magnetic flux of the circuit, <£, is pro- 
portional to the current, i, with a proportionality factor, L, which 
is called the inductance of the circuit. 


= Li. 


(1) 


The magnetic field represents stored energy w. To produce it, 
power, p, must therefore be supplied by the circuit. Since power 
is current times voltage, 

p = e'i. (2) 


12 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

To produce the magnetic field $ of the current i, a voltage ef 
must be consumed in the circuit, which with the current i gives 
the power p, which supplies the stored energy w of the magnetic 
field <i>. This voltage er is called the inductance voltage, or voltage 
consumed by self-induction. 

Since no power is required to maintain the field, but power is 
required to produce it, the inductance voltage must be propor- 
tional to the increase of the magnetic field: 

:' ; (3) 

or by (1), 

(4) 
If i and therefore $ decrease, -r and therefore e' are negative; 

that is, p becomes negative, and power is returned into the circuit. 
The energy supplied by the power p is 

w = I p dt, 
or by (2) and (4), 

w = I Li di; 

hence 

L* (^ 

w = T (5) 

is the energy of the magnetic field 

$ = Li 

of the circuit. 

9. Exactly analogous relations exist in the dielectric field. 

The dielectric field, or dielectric flux, ty} is proportional to the 
voltage 6, with a proportionality factor, C, which is called the 
capacity of the circuit: 

f = Ce. (6) 

The dielectric field represents stored energy, w. To produce it, 
power, p, must, therefore, be supplied by the circuit. Since power 
is current times voltage, 

p = i'e. (7) 

To produce the dielectric field ty of the voltage e, a current ir 
must be consumed in the circuit, which with the voltage e gives 


THE ELECTRIC FIELD. 13 

the power p, which supplies the stored energy w of the dielectric 
field ^. This current i' is called the capacity current, or, wrongly, 
charging current or condenser current. 

Since no power is required to maintain the field, but power is 
required to produce it, the capacity current must be proportional 
to the increase of the dielectric field: 


or by (6), 

i' = C^. (9) 

de 
If e and therefore ^ decrease, -j- and therefore if are negative; 

that is, p becomes negative, and power is returned into the circuit. 
The energy supplied by the power p is 

w=j*pdt, (10) 

or by (7) and (9), 

w = I Cede; 
hence 

rw 

« = £ (ID 

is the energy of the dielectric field 

t = Ce 
of the circuit. 

As seen, the capacity current is the exact analogy, with regard 
to the dielectric field, of the inductance voltage with regard to the 
magnetic field; the representations in the electric circuit, of the 
energy storage in the field. 

The dielectric field of the circuit thus is treated and represented 
in the same manner, and with the same simplicity and perspicuity, 
as the magnetic field, by using the same conception of lines of 
force. 

Unfortunately, to a large extent in dealing with the dielectric 
fields the prehistoric conception of the electrostatic charge on the 
conductor still exists, and by its use destroys the analogy between 
the two components of the electric field, the magnetic and the 


14 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

dielectric, and makes the consideration of dielectric fields un- 
necessarily complicated. 

There obviously is no more sense in thinking of the capacity 
current as current which charges the conductor with a quantity 
of electricity, than there is of speaking of the inductance voltage 
as charging the conductor with a quantity of magnetism. But 
while the latter conception, together with the notion of a quantity 
of magnetism, etc., has vanished since Faraday's representation 
of the magnetic field by the lines of magnetic force, the termi- 
nology of electrostatics of many textbooks still speaks of electric 
charges on the conductor, and the energy stored by them, without 
considering that the dielectric energy is not on the surface of the 
conductor, but in the space outside of the conductor, just as the 
magnetic energy. 

10. All the lines of magnetic force are closed upon themselves, 
all the lines of dielectric force terminate at conductors, as seen in 
Fig. 8, and the magnetic field and the dielectric field thus can be 
considered as a magnetic circuit and a dielectric circuit. 

To produce a magnetic flux <£, a magnetomotive force F is required. 
Since the magnetic field is due to the current, and is proportional 
to the current, or, in a coiled circuit, to the current times the num- 
ber of turns, magnetomotive force is expressed in current turns or 
ampere turns. 

F = ni. (12) 

If F is the m.m.f., I the length of the magnetic circuit, energized 
by F, ,£ 

/ = 7 (13) 

is called the magnetizing force, and is expressed in ampere turns per 
cm. (or industrially sometimes in ampere turns per inch). 

In empty space, and therefore also, with very close approxi- 
mation, in all nonmagnetic material, / ampere turns per cm. length 
of magnetic circuit produce 3C = 4 TT/ 10"1 lines of magnetic force 
per square cm. section of the magnetic circuit. (Here the factor 
10"1 results from the ampere being 10"1 of the absolute or cgs. 
unit of current.) 

(14) 


* The factor 4 *• is a survival of the original definition of the magnetic field 
intensity from the conception of the magnetic mass, since unit magnetic mass 
was defined as that quantity of magnetism which acts on an equal quantity at 


THE ELECTRIC FIELD. 15 

is called the magnetic-field intensity. It is the magnetic density, 
that is, the number of lines of magnetic force per cm2, produced 
by the magnetizing force of / ampere turns per cm. in empty space. 
The magnetic density, in lines of magnetic force per cm2, pro- 
duced by the field intensity 3C in any material is 

& = /z3C, (15) 

where ju is a constant of the material, a " magnetic conductivity," 
and is called the permeability. ^ = 1 or very nearly so for most 
materials, with the exception of very few, the so-called magnetic 
materials: iron, cobalt, nickel, oxygen, and some alloys and oxides 
of iron, manganese, and chromium. 

If then A is the section of the magnetic circuit, the total magnetic 
flux is 

$ = A®. (16) 

Obviously, if the magnetic field is not uniform, equations (13) 
and (16) would be correspondingly modified; / in (13) would be 
the average magnetizing force, while the actual magnetizing force 
would vary, being higher at the denser, and lower at the less dense, 
parts of the magnetic circuit: 

'-"• 

In (16), the magnetic flux $ would be derived by integrating the 
densities (B over the total section of the magnetic circuit. 

ii. Entirely analogous relations exist in the- dielectric circuit. 

To produce a dielectric flux ^, an electromotive force e is required, 
which is measured in volts. The e.m.f. per unit length of the 
dielectric circuit then is called the electrifying force or the voltage 
gradient, and is 

G = f- (18)- 

unit distance with unit force. The unit field intensity, then, was defined as 
the field intensity at unit distance from unit magnetic mass, and represented 
by one line (or rather "tube") of magnetic force. The magnetic flux of unit 
magnetic mass (or "unit magnet pole") hereby became 4w lines of force, and 
this introduced the factor 4 TT into many magnetic quantities. An attempt 
to drop this factor 4 TT has failed, as the magnetic units were already too well 
established. 

The factor 1Q-1 also appears undesirable, but when the electrical units 
were introduced the absolute unit appeared as too large a value of current as 
practical unit, and one-tenth of it was chosen as unit, and called "ampere." 


16 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

This gives the average voltage gradient, while the actual gradient 
in an ummiform field, as that between two conductors, varies, 
being higher at the denser, and lower at the less dense, portion of 
the field, and is 


then is the dielectric-field intensity, and 

D = KK (20) 

would be the dielectric density, where K is a constant of the material, 
the electrostatic or dielectric conductivity, and is called the spe- 
cific capacity or permittivity. 

For empty space, and thus with close approximation for air and 
other gases, 

1 

K — ~9» 

VL 

where 

v = 3 X 1010 

is the velocity of light. 

It is customary, however, and convenient, to use the permit- 
tivity of empty space as unity: K = 1. This changes the unit of 

dielectric-field intensity by the factor — , and gives: dielectric-field 
intensity, 

dielectric density, 


= T^-oJ (21) 

4 Try2 


D = KK, (22) 

where K = 1 for empty space, and between 2 and 6 for most solids 
and liquids, rarely increasing beyond 6. 
The dielectric flux then is 

^ = AD. (23) 

12. As seen, the dielectric and the magnetic fields are entirely 
analogous, and the corresponding values are tabulated in the 
following Table I. 

* The factor 4 TT appears here in the denominator as the result of the factor 
4*- in the magnetic-field intensity 5C, due to the relations between these 
quantities. 


THE ELECTRIC FIELD. 
TABLE I. 


17 


Magnetic Field. 

Dielectric Field. 

Magnetic flux: 
4> = Li 108 lines of magnetic force. 

Dielectric flux: 
^ = Ce lines of dielectric force. 

Inductance voltage: 

e'^n-^. 1Q-8 = L -jj volts. 
at at 

Capacity current: 
., _ d^ _ „ di 
dt dt 

Magnetic energy: 

Li2. . 
w = -n- joules. 

Dielectric energy: 

Ce2 
w = -=- joules. 

Magnetomotive force: 
F = ni ampere turns. 

Electromotive force: 
e = volts. 

Magnetizing force: 

F 

f = -r ampere turns per cm. 

Electrifying force or voltage gra- 
dient: 

a 

G = j volts per cm. 

Magnetic-field intensity: 
3C = 47r/10-1 lines of magnetic 
force per cm2. 

Dielectric-field intensity: 

K = - — - lines of dielectric force 
4 Try2 

per cm2. 

Magnetic density: 
(B = M5C lines of magnetic force 
per cm2. 

Dielectric density: 
D = nK lines of dielectric force 
per cm2. 

Permeability: /* 

Permittivity or specific capacity: K 

Magnetic flux: 
$ = A($> lines of magnetic force. 

Dielectric flux: 
^ = AD lines of dielectric force. 

v = 3 X 10 10 = velocity of light. 

The powers of 10, which appear in some expressions, are reduc- 
tion factors between the absolute or cgs. units which are used for 
$, 3C, CB, and the practical electrical units, and used for other 
constants. 

As the magnetic field and the dielectric field also can be con- 
sidered as the magnetic circuit and the dielectric circuit, some 
analogy exists between them and the electric circuit, and in Table 
II the corresponding terms of the magnetic circuit, the dielectric 
circuit, and the electric circuit are given. 


18 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

TABLE II. 


Magnetic Circuit. 

Dielectric Circuit. 

Electric Circuit. 

Magnetic flux (magnetic 

Dielectric flux (dielectric 

Electric current: 

current): 

current) : 

<£ = lines of magnetic 

^ = lines of dielectric 

i = electric cur- 

force. 

force. 

rent. 

Magnetomotive force: 

Electromotive force: 

Voltage: 

F = ni ampere turns. 

e = volts. 

e = volts. 

Permeance: 

M = 4?F 

Permittance or capacity: 

Conductance: 

Inductance: 

4irV2f , 

i 

Q — - mnos. 

~~F~ ' ~T 

henry. 

Reluctance: 

(Elastance ?): 

Resistance: 

F 

1 e 

e 

& 

C 4*v*l>- 

T ~~" T OIIIXIS. 

Magnetic energy: 

Dielectric energy: 

Electric power: 

w=— = — !Q-* joules. 

Ce2 e^ . , 
w = -JT- = -jr- joules. 

p = ri2 = ge2 — ei 
watts. 

Magnetic density: 

Dielectric density: 

Electric-current 

density: 

(B = -j =/z JClinespercm2. 

A. 

D = ^ = K/ninespercm2. 

7 = -j = yG am- 

A. 

perespercm2. 

Magnetizing force: 

Dielectric gradient: 

Electric gradient: 

F 
/ = j ampere turns per 

G = j volts per cm. 

G =j volts per cm. 

cm. 

Magnetic-field intensity: 

Dielectric-field inten- 

sity: 

OC = AT/. 

j^ "" . 

Permeability: 

Permittivity or specific 

Conductivity: 

capacity: 

*-|- 

K_D 

y = ~ mho — cm. 

Cr 

Reluctivity: 

(Elastivity ?): 

Resistivity: 

p = & 

1 . K* 

1 G , 
p = - = -?ohm — cm. 
y I 

Specific magnetic energy: 

Specific dielectric energy: 

Specific power: 

Awf2 /(B1A_8 

KGZ GD . , 

Po = p/2 = G2 = GI 

W^O ~~* ^ — c\ ^-^ 

WQ — -. ; — ~~^: — JOUieS 

joules per cm3. 

4irV2 2 ' 

per cm3. 

watts per cm3.