Suppose we have one coil of wire (Coil 1) connected to an AC source.
(See Figure 1.) Let N
1
be the number of turns of wire in Coil 1 and be the time-varying current through Coil 1. Let voltage supplied by the AC source. (Suppose
( )
be the time-varying
( )
is sinusoidal –
( ) shaped like sine or cosine – for example.) field
G
B t
1
( )
in Coil 1. Thus, there will be a time-varying magnetic flux through Coil 1.
Now suppose we bring another coil – Coil 2, of N
2
turns – to a position near Coil 1, as shown in Figure 1. Some of the time-varying magnetic flux produced by Coil 1 will now pass through Coil 2, inducing an emf in
Coil 2, by Faraday’s law. Thus, a time-varying current
( )
will be induced in Coil 2 due to the changing magnetic flux produced by Coil 1.
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Ch. 30: Inductance
Let
Φ
12
be the magnetic flux that Coil 1 causes through each turn of
Coil 2 . The total flux that Coil 1 causes – through all turns of Coil 2 – is then N
2
Φ
12
.
We find experimentally that N
2
Φ
12
is just proportional to i
1
, the current in
Coil 1. We express this mathematically as follows:
N
2
Φ =
The constant of proportionality, M
12
is called the mutual inductance .
(1)
2

Ch. 30: Inductance
Note:
•
M
12
depends on geometry : o cross-sectional areas of the coils o # of turns in each coil o positions of the coils (how far apart they are, e.g.)
•
M
12
also depends on whether iron or any other materials (especially easily magnetizable ones) are present.
•
M does not depend on time, as long as the coils are at fixed
12 positions.
3

Ch. 30: Inductance
By Faraday’s law, the emf induced in Coil 2 is:
ε
2
= −
N
2 d
Φ dt
12
= −
N
2 d dt ⎝
⎛
⎜
N
2
ε
2
= −
M
12 di
1 dt
⎟
⎞
⎠
Similarly, if there is a changing current induced in Coil 2,
(2)
( )
, it causes some changing magnetic flux to pass through Coil 1. If the flux that
Coil 2 causes through each turn of Coil 1 is called
Φ
21
, then the flux through all turns of Coil 1 is N
1
Φ
21
.
It turns out (experimentally) that the total flux that Coil 2 causes through
Coil 1 is proportional to i
2
:
N
1
Φ =
(3)
And Faraday’s law says this changing flux induces an emf in Coil 1:
ε
1
= −
N
1 d
Φ dt
21
= −
N
1 d dt
⎛
⎜
⎝ N
1
⎟
⎞
⎠
ε = −
1
M
21 di
2 dt
(4)
4

Ch. 30: Inductance
It turns out (won’t prove this here) that the two mutual inductances above are, in fact, equal :
M
21
=
M
12
, so I will just call them both M . With this fact, (2) and (4) become:
ε
ε
2
1
= −
= −
M
M di
1 dt di
2 dt
(5)
(6)
Note the unit for M :
A s "Henry", H
5

Ch. 30: Inductance
Even just a single coil – or even any single circuit – has some inductance.
Consider, for example, a single coil driven by an AC source that causes a time-varying current
( )
to flow through the coil. This changing current causes a changing magnetic flux through the coil. Let
Φ ( )
be the flux through one turn. If the coil has coil is N
Φ ( )
.
N turns, then the total flux through the
The total flux through the coil is proportional to the current
( )
. We express this fact mathematically as:
N
Φ ( ) = ( )
From Faraday’s law, this changing flux induces an emf in the coil:
ε = −
N d
Φ dt
( )
= −
N d dt ⎝
⎜
⎛
Li
N ⎠
⎟
⎞
ε = −
L di dt
(7)
(8)
6

Ch. 30: Inductance
Note the minus sign (Lenz’s law). This means that if
( ) increases in the sense shown in Figure 2, the emf induced is in the polarity shown in
Fig. 2. This emf is of such a polarity as to make
( )
increase not so much as it otherwise would.
Thus the coil “fights against” any attempt to change the current through it.
Again, the self-inductance , L , is a constant having to do with geometry, # of turns, presence of iron, etc.
The unit for L is, once again, the henry.
We call any coil an inductor .
7