INDUCTORS AND INDUCTANCE

PHY110W MAGNETISM INDUCTANCE
INDUCTORS AND INDUCTANCE
A capacitor is a device used to set up a known electric
field (in which energy can be stored).
Capacitance is defined as
C
=
q
V
An inductor is a device used to set up a
known magnetic field (in which energy
can be stored).
For an inductor (ie a solenoid) of N turns, carrying a
current i, and producing a magnetic flux Φ, its
inductance L is defined as
L
NΦ
i
= (31-30)
where the product NΦ is known as the flux linkage.
SI unit:
the henry, H [1 H = 1 T.m 2 /A]
Calculating inductance:
• Determine the magnetic field caused by the current i
in the conductor
• Find the magnitude of the flux through the loop
( Φ= B⋅dA)
∫
• Calculate the flux linkage, NΦ
• Determine the inductance,
L =
NΦ
i
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PHY110W MAGNETISM INDUCTANCE
INDUCTANCE OF A SOLENOID
l
B
For a long solenoid of cross-sectional area A, with n
turns per metre, the magnetic field near its centre is
given by
B
=
µ ni
0
and the flux linkage for this section of the solenoid is
NΦ = ( nl )( BA)
Hence the inductance per unit length of a solenoid is
L
l
2
=
NΦ
= µ 0nA
(31-33)
il
Note that, just as for capacitance, inductance depends
only on the geometry of the device.
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PHY110W MAGNETISM INDUCTANCE
SELF-INDUCTION
Changing current in a coil produces a changing
magnetic flux (which induces an emf) …
… not only in another nearby coil, but also in itself!
From Faraday's law, this self-induced emf, E L , is
( Φ)
d N
dt
E L =−
(31-36)
where NΦ = Li (31-35)
di
∴ E L =−L (31-37)
dt
Note that the magnitude of the induced emf does not
depend on the current strength through the coil – but
the rate at which the current is changing.
As usual, the direction of the induced emf is given by
Lenz's law.
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PHY110W MAGNETISM INDUCTANCE
RL CIRCUITS
Initially, an inductor acts to oppose changes in the
current through it. A long time later, it acts like
ordinary conducting wire.
Applying Kirchhoff's loop rule to
the adjacent circuit as the
constant emf E is switched on …
E
increasing i
R
L
E L
E
− iR − L di = dt
0
solving which for i we get
i
⎛
=
E
1 −
R
⎜
⎝
e
−Rt L
⎞
⎟
⎠
(31-42)
which can be written as
i
⎛
=
E
−
R ⎜
⎝
−t
τ
L
1 e
⎞
⎟
⎠
(31-43)
where τ L is the inductive time constant (τ L = L / R )
At time t = 0, i = 0. But as t → ∞, i → E / R .
−1
At time t = τ L i =
E
( 1− e ) = 0,63
E .
R
R
Graphically:
i
E
/ R
0.63 E / R
τ L
t
33

PHY110W MAGNETISM INDUCTANCE
RL CIRCUITS contd
When the battery is switched out of
the circuit, the inductor opposes
the subsequent reduction in
current by developing an emf in the
direction of the original current (i 0 ).
Current decays according to
decreasing i
R
L
E L
E
R
−t
τ
L
i = e = i e
0
−t
τ
L
(31-47)
At time t = 0, i 0 = E / R . But as t → ∞, i → 0.
−1
At time t = τ L i =
E
e = 0,37
E .
R R
Graphically:
i
E
/ R
0.37 E / R
τ L
t
When the current in the circuit is decreasing, the
inductive time constant is the time it takes the current
in the circuit to fall to 37% of its original value.
When the current in the circuit is increasing, the
inductive time constant is the time it takes the current
in the circuit to reach 63% of its peak value.
34

PHY110W MAGNETISM INDUCTANCE
ENERGY STORED IN
A MAGNETIC FIELD
Just as a capacitor stores energy in its internal electric
field, according to:
U
E
2
1 q
= or
2 C
UE
=
1 CV 2
2
… so an inductor stores energy in its magnetic field.
If the inductor has zero resistance, the potential
difference across it is V = –E.
At any time, the power supplied to the device is
P = Vi = − E i = Li di
dt
and the increase in energy supplied during a time
interval dt is
dU B = P dt = Li di
(31-50)
The total energy required to increase the current from
zero to i is thus
B
i
1 2
∫ (31-51)
2
0
U = L idi = Li
35

PHY110W MAGNETISM INDUCTANCE
ENERGY DENSITY OF
A MAGNETIC FIELD
In the same way as for the electric field, we can
represent stored magnetic potential energy in terms of
the energy density (ie the energy per unit volume) of
the field.
If, for example the inductor is a long solenoid, of length
d, cross-sectional area A, and n turns per unit length,
the flux is
Φ B = BA = µ 0 niA
The inductance of the solenoid is then
L
=
NΦ
=
i
0
2
µ n lA
from which the stored energy is
2 2
1 2 µ 0ni
UB
= Li = lA
2 2
Recognising that lA is the volume of the magnetic field,
and substituting for the field strength, (B = µ 0 ni) the
energy density per unit volume, u, is
u
B
2 2
µ
2
0ni
= = 1 B
(31-56)
2 2 µ 0
(cf
uE
2
= 1 ε 2
0E
)
36

PHY110W MAGNETISM INDUCTANCE
MUTUAL INDUCTION
To distinguish it from self-induction, the induction of an
emf in another nearby coil (caused by changing current
– and hence changing magnetic flux) in a coil is better
referred to as mutual induction.
The mutual inductance M 21 of coil 2 with respect to
coil 1 is given by
M
21
N
Φ
2 21
= (31-62)
i1
and hence the emf induced in coil 2 due to the
changing current in coil 1 is
di
=−M dt
1
E 2 21
(31-63)
Similarly,
di
=−M dt
2
E 1 12
(31-64)
where
M
12
=
N Φ
i
1 12
2
But actually M 12 = M 21 = M (31-65)
SI unit:
(as for self-induction)
the henry, H [1 H = 1 T.m 2 /A]
37