32n 30 inductance
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30.1 Mutual Inductance Derivation
The flux F2 in coil 2 is proportional to
the current in coil 1 or vice versa
(WHY?) and the constant of
proportionality is the mutual
inductance
=
=
The induced emf in the coils are
=
=
Arlyn D. Macasero-Roque 30 Inductance 1
The SI unit of inductance is Tm2/A,
defined as henry:
-
Example 1:
In one form of Tesla coil (a high-voltage regulator), a long solenoid
with length l and cross-sectional area A is closely wound with N1turns of wire. A coil with N turns surrounds it at its center. Find the
mutual inductance.
Solution:
Remember that a long solenoid carrying a
current i1 produces a magnetic field B1 that
points along the axis of the solenoid. The
field magnitude B is proportional to I and to
n1 , the number of turns per unit length:
0 1 11 0 1 1
N iB n i
l
Arlyn D. Macasero-Roque 30 Inductance 2
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The flux through a cross section of the
solenoid equals B1A. Since a very long
solenoid produces no magnetic field
outside of its coil, this is also equal to the
flux trough each turn of the outer,surrounding coil, no matter what the
cross-sectional area of the outer coil. The
mutual inductance is then given as:
22 2 0 1 1 0 1 22 1
1 1 1
BN N N i A AN NN B AM
i i i l l
F
Arlyn D. Macasero-Roque 30 Inductance 3
Example 1:
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Example 2:
Two coils have mutual inductance M = 0.275 H. The
current in the first coil increases at a uniform rate of
0.0500 A/s.
(a) What is the induced emf in the second coil?
(b) Suppose the current described is in the second coil
rather than in the first. What is the induced emf in the
first coil?
12 (0.275 H)( 0.0500 A/s) 0.0138 V
diM
dt
21 (0.275 H)( 0.0500 A/s) 0.0138 V
diM
dt
Arlyn D. Macasero-Roque 30 Inductance 4
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Example 3:
Two toroidal solenoids are wound on the same form so
that the magnetic field of one passes through the
windings of the other. Solenoid 1 has 800 turns and
solenoid 2 has 300 turns. When the current in solenoid
1 is 3.55 A, the average flux through each turn of
solenoid 2 is 0.0280 Wb.
(a) What is the mutual inductance of the pair of
solenoids?
(b) When the current in solenoid 2 is 1.60 A, what is the
average flux through each turn of solenoid 1?
2 2
1
(300)(0.0280 Wb)2.37 H
3.55 A
NM
i
F
321
1
(1.60 A)(2.37 H) 4.73 10 Wb
800
iM
N
F
Arlyn D. Macasero-Roque 30 Inductance 5
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In mutual inductance, if two coils are near each other, a current in the first coil produces a magnetic flux through the second coil. If we
change this flux by changing current, an induced emf appears in the second coil. But this change of current ALSO changes the flux within
the first coil, so an induced emf appears in the first coil as well.
This process is called self-induction, and the emf that appears is
called self-induced emf.
30.2 Self Inductance
Arlyn D. Macasero-Roque 30 Inductance 6
The inductance
is then defined
as
BNLi
F
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Recall that a capacitor can be used to produce electric field between
conductors and that the parallel-plate arrangement was used as a
basic type of capacitor.
Similarly, an inductor can be used to produce a magnetic field and
that a solenoid can be used as a basic type of inductor.
30.2 Inductors
Arlyn D. Macasero-Roque 30 Inductance 7
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The inductance of a LONG SOLENOID of cross-section is
=
The inductance per unit length near its center is
=
The inductance of a TOROIDAL SOLENOID is
=
.
Inductance, like capacitance, depends only on the geometry.
Arlyn D. Macasero-Roque 30 Inductance 8
30.2 Inductance of Derivation
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Energy Stored in an Inductor
Let the current at some instant be i and let its rate of change be di/dt; the current is increasing, so di/dt > 0. The voltage between the
terminals a and b of the inductor at this instant is Vab = Ldi/dt, and the rate P at which energy is being delivered to the inductor (equal to
the instantaneous power supplied by the external source) is
ab
diP V i Li
dt
30.3 Magnetic Field Energy
Arlyn D. Macasero-Roque 30 Inductance 9
After the current has reached its final steady value I, di/dt = 0 and no more energy is input to the inductor. When there is no current, the
stored energy U is zero; when the current is I, the energy is LI2. Inductor with current I, energy is stored. Resistor with current I, energy is
dissipitated.
The total energy U supplied while
the current increases from zero to a
final value I is
= 0
=
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Self-Inductance of an Ideal Toroid within its coil
=
2
2
The energy stored in the toroidal solenoid with current
=1
2
2
22
The energy density stored in the toroid with volume = 2A
=
=1
222
2 2
=
(in vacuum) =
(in material)
30.3 Magnetic Energy Density of a Toroid
Arlyn D. Macasero-Roque 30 Inductance 10
The energy in an inductor is actually stored in the magnetic field within the coil, just as the energy of a capacitor is stored in the electric field
between plates.
-
From Faradays Law:
The direction of L is found fromLenzs Law: the self-induced emf acts
to oppose the change in the current.
The above circuit diagrams
illustrate that self induction
opposes the change in the
current.
We can define a self-induced potential
difference VL across the inductor. If the
inductor is ideal (zero resistance),
VL = L.
L
diL
dt
+
- +
-
Arlyn D. Macasero-Roque 30 Inductance 11
30.2 Self-Induced Potential Difference
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Arlyn D. Macasero-Roque 30 Inductance 12
30.2 Inductors as Circuit Elements
The potential difference across a resistor depends on the current
while the potential difference across an inductor depends on the
rate of change of the current.
-
Example 4:
An inductor has an inductance of 0.540 H and carries a
current in the direction as shown and is decreasing at a
steady rate of 0.0300 A/s.
(a) What is the self-induced emf?
(b) Which end of the inductor (A or B) is at a higher
potential?
From Lenzs law we find A to be the higher potential
i
A B
(0.540 H)( 0.0300 A/s) 0.0162 VLdi
Ldt
Arlyn D. Macasero-
Roque
30 Inductance 13
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Example 5:
The inductor has inductance 0.260 H and carries a
current in the direction as shown in the figure below.
The current is changing at a constant rate. (a) The
potential between points and is Vab = 1.04 V, withpoint at higher potential. Is the current increasing ordecreasing? (b) If the current at = . is 12.0 A,what is the current at = . ?
a) From Lenz Law, i should be decreasing in order to
induced an emf in that direction.
b) (di/dt) is negative and hence i2 = 4 A
Arlyn D. Macasero-
Roque
30 Inductance 14
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When the switch S1 is closed on a,
the current in the resistor starts to
rise. If the inductor were not in the
circuit, the current would rapidly rise
from zero to a steady value of /R butbecause the inductor is present, the
current would gradually rise from
zero.
In this circuit, the inductor acts to
oppose changes in the current
through it. Some time later, it acts like
an ordinary connecting wire when the
current approaches /R.
Arlyn D. Macasero-Roque 30 Inductance 15
30.4 R-L Circuits
-
Loop rule gives us
+ = 0
The initial rate of change of current is
=
The final rate of change of current is
= 0
And the steady-state current is
=
Arlyn D. Macasero-Roque 30 Inductance 16
30.4 Current Growth in R-L Circuits
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This differential equation has a solution given by
Defining the time constant as
We can rewrite the solution as
63%
Arlyn D. Macasero-Roque 30 Inductance 17
30.4 Current Growth in R-L Circuits
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Reversing the switch to point b, removes the
battery from the circuit and the current through
the resistor begins to decrease. However, it
cannot drop immediately to zero but must decay
to zero over time.
Without the battery, the loop rule equation now becomes
The solution to this equation is
37%
Arlyn D. Macasero-
Roque
30 Inductance 18
30.4 Current Decay in R-L Circuits
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Example 6:
A sensitive electronic device of 175-W resistance is to be
connected to an emf source by a switch. The device is
designed to operate at a 36-mA current, but to avoid damage
the current must not rise above 4.9 mA in the first 58 s after
the switch is closed. So to protect the device, it is connected
in series with an inductor.
(a) Assuming negligible internal resistance, what emf must be
used?
(b) What inductance is required?
(c) What is the time constant?
(0.036 A)(175 ) 6.3 VIR W
( / ) (175 / )58 s6.3 V( ) (1 ) 4.9 mA (1 ) 69 mH175
R L t Li t e e LR
69 mH390 s
175 L
L
R
W
Arlyn D. Macasero-Roque 30 Inductance 19
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(a) What is the initial current in the resistor just after closing the switch at b?
(b) What is the current in the resistor after 0.200 ms?
(c) What is the potential difference across the inductor after 0.200 ms?
(d) How long does it take the current to decrease to half its initial value?
Example 7:
( / ) 0120 V( ) (0) 0.300 A400
R L ti t e i eR
W
( / ) (400 /0.200 H)0.200 ms
0( ) (0.3 A) 0.201 AR L ti t I e e
(0.201 A)(400 ) 80.4 VLV iR W
Suppose = 120V, R = 400 W, and L =0.200 H. The switch is closed at a until
a constant current is established. The
switch is then closed at b.
( / ) 00
0 0
0.5( ) 0.200 H( ) ln ln 0.347 ms
400
R L t IL i ti t I e tR I I
W
Arlyn D. Macasero-Roque 30 Inductance 20
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Example 8:
(a) Just after closing the switch at a, what are the voltages across R and
L?
(b) After several time constants with the switch still at a, what are the
voltages across R and L?
(c) What are the voltages across R and L during the time when i = 0.0500
A?
Suppose = 80.0V, R = 400 W, and
L = 0.200 H. Initially, there is no
current in the circuit. The switch is
closed at a.
Arlyn D. Macasero-Roque 30 Inductance 21
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Arlyn D. Macasero-Roque 30 Inductance 22
31.6 Transformers
Composed of 2 coils or windings, electrically insulated from each other
but wound on the same core.
All magnetic field lines are confined to the iron core (possible with
cores of large permeability)
TRANSFORMERS can easily step-up/step-down AC voltages (more
difficult with DC voltages).
-
Neglecting resistance in the windings,
the induced emfs in the primary and
secondary coils are:
=
and =
The magnetic flux is the same for
each coil, hence
=
and since 1 and 2 both oscillate at the same frequency as the acsource, then
1 11 1 2 2
2 2
V N
V I V IV N
(from energy considerations)
Step-up: N2 > N1 Step-down: N2 < N1
31.6 Transformers
Arlyn D. Macasero-Roque 30 Inductance 23
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When the switch is closed, the potential difference across R
is
1) V(N2/N1)
2) V(N1/N2)
3) V
4) 0
Check Your Understanding!
Arlyn D. Macasero-Roque 30 Inductance 24
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The primary coil of a transformer is connected to a battery, a
resistor and a switch. The secondary coil is connected to an
ammeter. When the switch s closed, the ammeter reads
1) Zero current
2) A nonzero current for a short instant
3) A steady current
Check Your Understanding!
Arlyn D. Macasero-Roque 30 Inductance 25
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30.1 Mutual Inductance Derivation
In the figure, the current i1 in coil 1 sets
up a magnetic field B, some of which
pass through coil 2. The flux F2 in coil
2 is proportional to the current in coil 1
(why?)
22 = 211
=
where M21 is a proportionality constant.
Arlyn D. Macasero-Roque 30 Inductance 26
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30.1 Mutual Inductance Derivation
When F2 and i1 change with time, we
can take time derivatives on both
sides:
22 =
211
2
2 = 21
1
=
Similarly,
=
Arlyn D. Macasero-Roque 30 Inductance 27
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Consider a long solenoid of cross-section A. What is the
inductance per unit length near its center?
First we calculate the flux linkage NFB set up by the current in the
solenoid. The flux linkage can be expressed as
=
where n is the number of turns per unit length and B is the
magnetic field within the solenoid:
=
Arlyn D. Macasero-Roque 30 Inductance 28
30.2 Inductance of a Long Solenoid Derivation
-
The inductance is then
==
=
Thus, the inductance per unit length for a long solenoid near the
center is
=
Arlyn D. Macasero-Roque 30 Inductance 29
30.2 Inductance of a Long Solenoid
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Figure (a) shows a toroidal solenoid, which we may describe as a
solenoid bent into a torus (doughnut) with rectangular cross section.
From the symmetry we see the field forms concentric circles inside the
toroid as shown in figure (b).
path of integration
Arlyn D. Macasero-Roque 30 Inductance 30
30.2 Inductance of a Toroid
-
According to Amperes law, the
magnetic field within the toroid is:
=
The flux can be obtained by
integration:
= =
=
2
=
So the inductance of a toroidal
solenoid is expressed as:
=
Arlyn D. Macasero-Roque 30 Inductance 31
30.2 Inductance of a Toroid Derivation
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