e. m. f. is called a self-induced e. m. f. This self-induced e. m. f.
* Supposed to have zero resistance for the sake of simplicity of statement.
ELEMENTS OF PHYSICS.
When a current is increasing in a coil the reaction (self-induced
e. m. f.) is in a direction opposite to the current. When the cur-
rent is decreasing the self-induced e. m. f. is in the same direction
as the current.
Remark: The identity of equations (45)a and (45)0 is a proof
of equation (44).
63. Proposition. The inductance of a coil wound on a given
spool is proportional to the square of the number of turns, Z, of wire.
For example, a given spool wound with No. 16 wire has 500
turns and an inductance of, say, 0.025 henry; the same spool
wound with No. 28 wire would have about ten times as many
turns and its inductance would be about 100 times as great or
Proof : To double the number of turns on a given spool would everywhere double
the field intensity for the same current, and therefore the energy of the field would
everywhere be quadrupled for given current so that the inductance would be quad
rupled according to equation (39).
64. Proposition. The inductance of a coil of given shape is
proportional to its linear dimensions, the number of turns of wire
being unchanged. For example, a given coil has an inductance
of 0.022 henry ; and a coil three times as large in length, diam-
eter, etc., has an inductance of 0.066 henry.
Proof : Consider a coil a and a similar
coil A twice as large. See Fig. 33. Con-
sider corresponding volume elements Az/
^j i/ and A F of the magnetic fields due to a and
to A respectively. Then AF
Further the magnetic field intensity at Az/ is
twice as great for given current as at A V
according to equation (201) (text-book) so
that the kinetic energy per unit volume is
four times as great at Az> as at A V. There-
fore the amount of energy in the element
A V is twice as great as in Az/, or the total
energy of the coil A is twice as great as the energy of the coil a for given current.
Consequently the inductance of A is twice as great as the inductance of a accord-
ing to equation (39).
65. Calculation of inductance in terms of magnetic flux per
unit current. According to equation (44) the inductance of a
coil is equal to the quotient where A 7 is the magnetic flux
through the coil* due to current i in the coil. There are impor-
tant cases in which the flux through a coil due to given current
may be easily calculated and therefore the inductance of such a
coil is easily determined.
Long solenoid. Consider a long cylindrical coil of wire of
mean radius r and having n turns of wire per unit length. The
field intensity in this coil is/"= Apni according to equation (276),
text -book, and the area oi the opening of the coil is zrr 2 so that the
flux through the opening is ^7i 2 r 2 ni( = N f \ If we consider length
/ of the coil having In turns, the flux through this portion of the
coil is 4x 2 r 2 n 2 /i( = ln.N f ) which divided by i gives the inductance
of the coil, so that
L = 4 7T VV/. (46)
This equation is strictly true for very long coils only. For
short coils the value of L given by this equation is too great ;
however, this equation is very useful in
enabling one to calculate easily the ap-
proximate inductance of even a short
Coil wound on an iron core. Consider
a coil of Z turns of wire wound on an
iron ring of peripheral length / and of
sectional area q. (See Fig. 34.) The
magnetomotive force of this coil, with
current i, is 47? Zi according to equation
(275), the flux through the rod is
i l (
N f )
* That is the flux through a mean turn multiplied by the number of turns of wire.
54 ELEMENTS OF PHYSICS.
according to equation (286), and the flux through the coil is
Therefore, the inductance of the coil is
Remark : The permeability p. of iron decreases with increasing
magnetizing force. Therefore, the inductance of a coil wound
on an iron core is not a definite constant as in case of a coil
without an iron core.
66. Growth and decay of current in an inductive circuit. Con-
sider a circuit of inductance L and resistance R to which a con-
stant e. m. f, E, is connected. A perceptible interval of time
elapses before the full value, -, of the current is established in
the circuit. During this time the current is growing, and the
following equation is satisfied at each instant :
E=Ri + L* ( (48)
in which i is the instantaneous value of the growing current ; Ri
is the part of E which overcomes the resistance of the circuit,
and L is the part of E which causes the current to increase, ac-
cording to equation (41).
If a circuit of inductance L and resistance R with a given cur-
rent is left to itself without any e. m. f. to maintain the current
the current dies away, or decays, and the e. m. f, Ri, which, at
each instant, overcomes the resistance is the self-induced e. m. f.
L - , so that at each instant Ri = L or
Example, An e. m. f. of 1 10 volts acts on a coil of which the
inductance is 0.04 henry and the resistance is 3 ohms. At the
instant that the e. m. f. begins to act the actual current i in the
coil is zero and the whole of the e. m. f. acts to increase the cur-
rent so that 1 10 volts
0.04 henry x , or
per second. When the growing current has reached a value of
30 amperes, Rt is equal to 90 volts and the remainder of the 1 10
volts acts to cause the current to increase, that is 30 volts =
0.04 henry x , or = 750 amperes per second.
If a current is established in this coil and the coil left to itself,
short-circuited, without any e. m. f. to maintain the current ; then,
as the decaying current reaches a value of, say, 30 amperes, the
e. m. f. Ri is 90 volts and this e. m. f. is equal to L -r so that
is 2250 amperes per second.
The algebraic expression for the
growing current in an inductive current
= *"*"' (50)
in which e is the Naperian base, and i
is the value of the current / seconds
after the e. m. f. E is connected to the
circuit. This equation is true, for in the
first place i=o when / = o ; and in the
second place it satisfies equation (50) as
may be shown by differentiation. The ordinates of the curve, Fig. 35, show the suc-
cessive values of the growing current in an inductive circuit.
The algebraic expression for the decaying cur-
rent in a short-circuited inductive circuit is
\ J 3. 4 > J
HVHDRSWHS Or * 3KCOND
HUNDKEDTWS OF A SECOND
in which / is the value of the decaying at the
instant from which time is reckoned (/ = o) and
i is the value of the current / seconds
afterwards. The ordinates of the
curve, Fig. 36, show the successive
values of a decaying current in a
short-circuited inductive circuit.
56 ELEMENTS OF PHYSICS.
67. Circulation of charge during growth or decay of a current in an
inductive Circuit. During the growth of a currrent in an inductive circuit which
is connected to a constant e. m. f. , the current i is less than its final value / ( = J
so that the quantity of electric charge which has passed through the circuit by the
time the current has reached its final value is less than it would have been had the
current started with its full final value. This deficiency of charge is an important
consideration in some of the methods for measuring inductance ; it is
and it is equal to the charge which passes through the circuit while a current of initial
value / is decaying to zero. In this equation L is the inductance and R the resis-
tance of the circuit.
Proof of equation (52). The amount by which the actual current falls short of
/ * E\ - *
/ ( = J at time / is, by equation (50), equal to 2e L and it is this defi-
ciency in current which produces the deficiency in the charge so that
Xt=ao a, T *
g z'* .<#?= Q. E. D.
68. Mutual Inductance. The subject of mutual inductance is
discussed in articles 544-549, text-book. Of these articles 548
and 549 may be omitted. Equation (298) should read
W= iLtf + U.A,' + Mi^. (298)
Remark : In practical work the conception of Mutual Induc-
tance is not of great importance inasmuch as the inductive rela-
tion of two coils is merely a matter of the sending of magnetic
flux through the one by a current in the other.
tlbe flDacmtllan Company.
THE ELEMENTS OF PHYSICS.
EDWARD L. NICHOLS, B.S., Ph.D.,
WILLIAM S. FRANKLIN, M.S.,
Projessor of Physics and Electrical Engineering at the , ><tural
College, Ames, Iowa.
The following '
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THE ELEMENTS OF PHYSICS.
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