Lecture 22. Inductance. Magnetic Field Energy.
Outline:
Self-induction and self-inductance.
Inductance of a solenoid.
The energy of a magnetic field.
Alternative definition of inductance.
Mutual Inductance.
1
- CH2 exam average ~54 %.
- To the right are VERY tentative correlations
between percentage scores and letter grades in
Physics 227. The instructors reserve the right
to adjust these boundaries (both up and down)
before final grades are assigned at the end of
the semester.
A ≥ 85
80 ≤ B+ < 85
70 ≤ B < 80
65 ≤ C+ < 70
53 ≤ C < 65
40 ≤ D < 53
40 < F
2
Maxwell’s Equations
 Gauss’s Law: charges are sources of electric field (and a non-zero net electric
field flux through a closed surface); field lines begin and end on charges.
𝑄𝑒𝑒𝑒𝑒
𝜀0
�
𝐸 ∙ 𝑑𝐴⃗ =
�
𝐵 ∙ 𝑑𝐴⃗ = 0
�
𝐸 ∙ 𝑑𝑙⃗ = −
�
𝐵 ∙ 𝑑𝑙⃗ = 𝜇0 �
𝑠𝑠𝑠𝑠
 No magnetic monopoles; magnetic field lines form closed loops.
𝑠𝑠𝑠𝑠
 Faraday’s Law of electromagnetic induction; a time-dependent ΦB generates E.
𝑙𝑜𝑜𝑜
𝑑
� 𝐵 ∙ 𝑑𝐴⃗
𝑑𝑑 𝑠𝑢𝑢𝑢
𝑑Φ𝐵
ℰ=−
𝑑𝑑
 Generalized Ampere’s Law; B is produced by both currents and time-dependent ΦE.
𝑙𝑜𝑜𝑜
𝑠𝑢𝑢𝑢
𝐽⃗ + 𝜖0
 The force on a (moving) charge.
𝐹⃗ = 𝑞 𝐸 + 𝑣⃗ × 𝐵
𝜕𝐸
∙ 𝑑𝐴⃗
𝜕𝜕
3
Induced E.M.F. and its consequences
𝑑Φ𝐵
𝐹𝐹𝐹𝐹𝐹𝐹𝐹: ℰ = −
𝑑𝑑
1.
2.
𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝑩 𝒕
3.
𝐼 𝑡
𝐼 𝑡
𝑩 𝒕
8
(Self) Inductance
Variable
battery
𝐼 𝑡
𝑩 𝒕
The back
e.m.f. :
𝓔 =−
Inductance:
(or self-inductance)
𝑑Φ𝐵
𝐹𝐹𝐹𝐹𝐹𝐹𝐹: ℰ = −
𝑑𝑑
The flux depends on the current, so a wire loop with a
changing current induces an “additional” e.m.f. in itself
that opposes the changes in the current and, thus, in the
magnetic flux (sometimes called the back e.m.f.).
𝒅𝜱𝑩
𝒅𝒅
= −𝑳
𝒅𝒅
𝒅𝒅
𝜱
𝑳≡
𝑰
𝐿 – the coefficient of proportionality between
Φ and 𝐼, purely geometrical quantity.
Units: T⋅m2/A = Henry (H)
The self-inductance L is
always positive.
This definition applies when there is a single current path
(see below for an inductance of a system of distributed currents 𝐽 𝑟 ).
Inductance as a
circuit element:
9
Inductance of a Long Solenoid
Long solenoid of radius r and length l with the total number of turns 𝑁:
𝐿≡
Φ𝐵
𝐼
l
Note that
The magnetic flux generated by current 𝐼:
𝑁
𝜇0 𝑁 2 𝐼 2
Φ𝐵 = 𝐵𝜋𝑟 𝑁 = 𝐵 = 𝜇0 𝐼 =
𝜋𝑟 = 𝜇0 𝑛2 𝐼 l 𝜋𝑟 2
l
l
2
Φ𝐵 𝜇0 𝑁 2 2
𝐿=
=
𝜋𝑟 = 𝜇0 𝑛2 l 𝜋𝑟 2
𝐼
l
(n – the number of
turns per unit length)
- the inductance scales with the solenoid volume l 𝜋𝑟 2 ;
𝑁
- the inductance scales as N2: B is proportional to l and the net flux ∝ N.
Let’s plug some numbers: l =0.1m, N=100, r=0.01m
100
𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 = 4𝜋 ∙ 10−7 ∙
0.1
How significant is the induced e.m.f.? For
𝑑𝑑
~104 𝐴/𝑠 (let’s say, ∆𝐼~ 1 mA in 0.1 µs)
𝑑𝑑
2
∙ 0.1 ∙ 𝜋 0.01
2
≈ 40𝜇𝜇
𝑑𝑑 4 ∙ 10−5 𝐻 × 1 ∙ 104 𝐴
ℰ =𝐿 =
= 0.4𝑉
𝑑𝑑
𝑠
10
Inductance vs. Resistance
Inductor affects the current flow
only if there is a change in current
(di/dt ≠ 0). The sign of di/dt
determines the sign of the
induced e.m.f. The higher the
frequency of current variations,
the larger the induced e.m.f.
13
Energy of Magnetic Field
To increase a current in a solenoid, we have to do some work: we work against the back e.m.f.
Let’s ramp up the current from 0 to the final value I (𝑖 𝑡 is
the instantaneous current value):
V
𝑖 𝑡
𝑖 𝑡
𝑉 𝑡
= 𝐿
𝑑𝑑 = 𝑃 𝑡 𝑑𝑑
The net work to ramp up
the current from 0 to 𝐼 :
𝑑𝑑 𝑡
𝑑𝑑
𝑃 𝑡 =𝑉 𝑡 𝑖 𝑡 =𝐿
𝑑𝑑 𝑡
𝑖 𝑡
𝑑𝑑
P(power) - the rate at which energy is being
delivered to the inductor from external sources.
𝑓
𝐼
1 2
𝑈 = � 𝑃 𝑡 𝑑𝑑 = � 𝐿𝐿 ∙ 𝑑𝑑 = 𝐿𝐼
2
𝑖
0
𝟏 𝟐
𝑼 = 𝑳𝑰
𝟐
This energy is stored in the magnetic field created by the current.
𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 = 𝜇0 𝑛2 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣
1
𝐵2
2
2
𝑈𝐵 = 𝜇0 𝑛 𝑣𝑣𝑣𝑣𝑣𝑣 ∙ 𝐼 = 𝐵 = 𝜇0 𝑛𝑛 =
∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐵 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣
2𝜇0
2
The energy density in
a magnetic field:
𝑩𝟐
𝒖𝑩 =
𝟐𝝁𝟎
(compare with 𝑢𝐸 =
This is a general result (not solenoid-specific).
𝜀0 𝐸 2
2
)
14
Quench of a Superconducting Solenoid
MRI scanner: the magnetic field up to 3T
within a volume ~ 1 m3.
The magnetic field energy:
𝑈 = 𝑣𝑣𝑣𝑣𝑣𝑣 ∙ 𝑢𝐵 = 1𝑚
The capacity of the LHe
dewar: ~2,000 liters.
3
3𝑇
2
𝑊𝑊
2 ∙ 4𝜋 ∙ 10−7
𝐴∙𝑚
≈ 4𝑀𝑀
The heat of vaporization of liquid helium: 3 kJ/liter.
Thus, ~ 1000 liters will be evaporated during the quench.
Magnet quench: http://www.youtube.com/watch?v=tKj39eWFs10&feature=related
15
Another (More General) Definition of Inductance
Alternative definition of inductance:
1 2
𝐿𝐼 =
2
�
𝑎𝑙𝑙 𝑠𝑠𝑠𝑠𝑠
𝐵2
𝑑𝜏
2𝜇0
𝑳≡
𝟐𝑼𝑩
𝑰𝟐
Advantage: it works for a distributed current flow (when it’s unclear which
loop we need to consider for the flux calculation).
16
Example: inductance per unit length for a coaxial cable
𝐼
𝑏
𝑎
𝐼
Uniform current density in the central conductor of radius a:
𝑏
1
𝑈𝐵 =
� 𝐵 𝑟
2𝜇0
0
2
𝑑𝜏
𝐵 𝑟 =
Straight Wires: Do They Have Inductance?
Using 𝑳 = 𝟐𝑼𝑩 /𝑰𝟐 :
𝑏
𝜇0 𝐼
1
𝑈𝐵 =
�
2𝜋𝜋
2𝜇0
𝑎
2
𝑏
𝜇0 𝐼
1
𝑑𝜏 =
�
2𝜋𝜋
2𝜇0
𝑎
Using 𝑳 = 𝜱/𝑰:
𝜇 𝐼
~l 𝐵 ≈ 0
2𝜋𝜋
l
l
Φ ≈ l�
𝑎
2
𝜇0 𝐼
,𝑎 < 𝑟 < 𝑏
2𝜋𝜋
𝜇0 𝐼𝑟
,
2𝜋𝑎2
𝑟<𝑎
𝜇0 𝐼 2
l
∙ l ∙ 2𝜋𝜋𝜋𝜋 ≈ (𝑏~l ) ≈
∙ l ∙ 𝑙𝑙
4𝜋
𝑎
𝜇0 𝐼
𝝁𝟎 𝑰
l
𝑑𝑑 =
∙ l ∙ 𝒍𝒍
2𝜋𝜋
𝟐𝝅
𝒂
at home
𝐿=
2𝑈𝐵
𝐼2
l
𝝁𝟎
𝑳𝑯 ≈
l ∙ 𝒍𝒍
𝟐𝝅
𝒂
𝜇0
l
l
−7
𝐿𝐻 ≈
l ∙ 𝑙𝑙 ≈ 2 ∙ 10 l 𝑚 ∙ 𝑙𝑙
2𝜋
𝑎
𝑎
l = 1𝑐𝑐 𝐿 ≈ 10−9 𝐻
17
Inductance vs. Capacitance
𝐿=
Φ𝐵 2𝑈𝐵
= 2
𝐼
𝐼
1 2 1 Φ𝐵 2
𝑈𝐵 = 𝐿𝐼 =
2
2 𝐿
𝐵2
𝑈𝐵 =
∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐵 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣
2𝜇0
𝐶=
𝑄 2𝑈𝐸
= 2
𝑉
𝑉
1 2 1 𝑄2
𝑈𝐸 = 𝐶𝑉 =
2
2 𝐶
𝜖0 𝐸 2
𝑈𝐸 =
∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐸 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣
2
18
Mutual Inductance
Let’s consider two wire loops at rest. A time-dependent current in loop 1 produces a
time-dependent magnetic field B1. The magnetic flux is linked to loop 1 as well as
loop 2. Faraday’s law: the time dependent flux of B1 induces e.m.f. in both loops.
loop 2
𝑩𝟏
The e.m.f. in loop 2 due to the time-dependent I1 in loop 1:
ℰ2 = −
𝐼1
loop 1
Primitive “transformer”
𝑑Φ1→2
𝑑𝑑
Φ1→2 = 𝑀1→2 𝐼1
The flux of B1 in loop 2 is proportional to the current I1 in
loop 1. The coefficient of proportionality 𝑀1→2 (the so-called
mutual inductance) is, similar to L, a purely geometrical
quantity; its calculation requires, in general, complicated
integration. Also, it’s possible to show that
𝑀1→2 = 𝑀2→1 = 𝑀
so there is just one mutual inductance 𝑀.
𝑀 can be either positive or negative, depending on the choices made for the senses of
transversal about loops 1 and 2. Units of the (mutual) inductance – Henry (H).
ℰ2 = −𝑀
𝜕𝐼1
𝜕𝜕
𝑀=
Φ1→2
𝐼1
19
Calculation of Mutual Inductance
l
𝑀1→2 = 𝑀2→1 = 𝑀
Calculate M for a pair of coaxial solenoids
(𝑟1 = 𝑟2 = 𝑟, l 1 = l 2 = l, 𝑛1 and 𝑛2 - the
densities of turns).
𝐵1 = 𝜇0 𝑛1 𝐼1
the flux per one turn
2
𝑁2 Φ
𝜇
𝑁
𝑁
𝜋𝑟
0
1
2
𝑀=
= 𝑁2 𝜇0 𝑛1 𝜋𝑟 2 =
l
𝐼1
2 𝜋𝑟 2
𝜇
𝑁
0
𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 =
l
Experiment with two coaxial coils.
Φ = 𝐵1 𝜋𝑟 2 = 𝜇0 𝑛1 𝐼1 𝜋𝑟 2
𝑀=
𝐿1 𝐿2
The induced e.m.f. in the secondary coil is much greater if we use a ferromagnetic core:
for a given current in the first coil the magnetic flux (and, thus the mutual inductance)
will be ~𝐾𝑚 times greater.
𝑑𝐼1
𝑑𝐼1
ℰ2 = 𝑀∗
= 𝐾𝑚 𝑀
𝑑𝑑
𝑑𝑑
no-core mutual inductance
20
Next time: Lecture 23. RL and LC circuits.
§§ 30.4 - 6
21
Appendix I: Inductance of a Circular Wire Loop
2𝑎
Example: estimate the inductance of a circular wire loop
with the radius of the loop, b, being much greater than
the wire radius a.
𝑏
The field dies off rapidly (~1/r) with distance from the wire, so it won’t be a big mistake
to approximate the loop as a straight wire of length 2πb, and integrate the flux through
a strip of the width b:
b
𝜇0 𝐼
𝐵≈
2𝜋𝜋
𝑏
Φ ≈ 2𝜋𝜋 �
2πb
𝑎
𝐿 𝑏 = 1𝑚, 𝑎 =
𝜇0 𝐼
𝑏
𝑑𝑑 = 𝜇0 𝑏𝐼 ∙ 𝑙𝑙
2𝜋𝜋
𝑎
10−3 𝑚
≈ 4𝜋 ∙
Note that we cannot assume that the wire radius is zero:
𝑏
𝜇0 𝐼
𝑏
Φ~ �
𝑑𝑑 ~𝑙𝑙
2𝜋𝜋
"0"
- diverges!
10−7
𝐿=
Φ
𝑏
≈ 𝜇0 𝑏 ∙ 𝑙𝑙
𝐼
𝑎
𝑊𝑊
1𝑚 ∙ 𝑙𝑙103 ≈ 10𝜇𝜇
𝐴∙𝑚
What to do: introduce some
reasonably small non-zero radius.
"0"
22