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.

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Energy Transformations in EM Waves and Circuits

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The energy of EM waves traveling in vacuum is stored in both 𝐸 and 𝐵 fields (in equal amounts).

Circuit elements that store the 𝐸 field energy - capacitors.

Circuit elements that store the 𝐵 field energy – inductors.

By combining capacitors and inductors, we can build the EM oscillators (e.g., generators of EM waves).

Maxwell’s Equations

� 𝐸 ∙ 𝑑𝑙𝑙𝑜𝑜𝑜

= −𝑑𝑑𝑑� 𝐵 ∙ 𝑑𝐴

𝑠𝑢𝑢𝑢

Faraday’s Law of electromagnetic induction; a time-dependent ΦB generates E.

� 𝐵 ∙ 𝑑𝑙𝑙𝑜𝑜𝑜

= 𝜇0 � 𝐽 + 𝜖0𝜕𝐸𝜕𝑑 ∙ 𝑑𝐴

𝑠𝑢𝑢𝑢

Generalized Ampere’s Law; B is produced by both currents and time-dependent ΦE.

�⃗� = 𝑞 𝐸 + �⃗� × 𝐵

The force on a (moving) charge.

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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.

No magnetic monopoles; magnetic field lines form closed loops.

� 𝐸 ∙ 𝑑𝐴𝑠𝑢𝑢𝑢

=𝑄𝑒𝑒𝑒𝑙𝜀0

� 𝐵 ∙ 𝑑𝐴𝑠𝑢𝑢𝑢

= 0

ℰ = −𝑑Φ𝐵

𝑑𝑑

Induced E.M.F. and its consequences

8

𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝑩 𝒆

𝐹𝐹𝐹𝐹𝑑𝐹𝐹: ℰ = −𝑑Φ𝐵

𝑑𝑑

1. 2.

3. 𝑩 𝒆

𝐼 𝑑

𝐼 𝑑

𝑩 𝒆

(Self) Inductance

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.).

Inductance: (or self-inductance)

This definition applies when there is a single current path (see below for an inductance of a system of distributed currents 𝐽 𝐹 ).

𝑳 ≡𝜱𝑰

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𝑩 𝒆

𝐼 𝑑

Variable battery

𝐿 – the coefficient of proportionality between Φ and 𝐼, purely geometrical quantity. 𝓔 = −

𝒅𝜱𝑩

𝒅𝒆= −𝑳

𝒅𝑰𝒅𝒆

Units: T⋅m2/A = Henry (H)

The back e.m.f. :

Inductance as a circuit element:

𝐹𝐹𝐹𝐹𝑑𝐹𝐹: ℰ = −𝑑Φ𝐵

𝑑𝑑

Inductance of a Long Solenoid

The magnetic flux generated by current 𝐼:

𝐿 =Φ𝐵

𝐼=𝜇0𝑁2

l 𝜋𝐹2 = 𝜇0𝑛2l 𝜋𝐹2

l

(n – the number of turns per unit length)

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Long solenoid of radius r and length l with the total number of turns 𝑁:

𝐿 ≡Φ𝐵

𝐼

Φ𝐵 = 𝐵𝜋𝐹2𝑁 = 𝐵 = 𝜇0𝑁l𝐼 =

𝜇0𝑁2𝐼l

𝜋𝐹2 = 𝜇0𝑛2𝐼 l 𝜋𝐹2

- the inductance scales with the solenoid volume l 𝜋𝐹2; - the inductance scales as N2: B is proportional to 𝑁l and the net flux ∝ N.

Let’s plug some numbers: l =0.1m, N=100, r=0.01m

𝐿 = 𝜇0𝑛2l 𝜋𝐹2 = 4𝜋 ∙ 10−7 ∙1000.1

2

∙ 0.1 ∙ 𝜋 0.01 2 ≈ 40𝜇𝜇

How significant is the induced e.m.f.? For 𝑑𝑑𝑑𝑑

~104𝐴/𝑠 (let’s say, ∆𝐼~ 10 mA in 1 µs) ℰ = 𝐿

𝑑𝐼𝑑𝑑 =

4 ∙ 10−5 𝜇 × 1 ∙ 104𝐴𝑠 = 0.4𝑉

Note that

Inductance vs. Resistance

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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.

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.

𝑈 = �𝑃 𝑑 𝑑𝑑𝑢

𝑖

= �𝐿𝐿 ∙ 𝑑𝐿𝑑

0

=12 𝐿𝐼

2

V Let’s ramp up the current from 0 to the final value 𝐼

(𝐿 𝑑 is the instantaneous current value):

𝑉 𝑑 = 𝐿𝑑𝐿 𝑑𝑑𝑑 𝑃 𝑑 = 𝑉 𝑑 𝐿 𝑑 = 𝐿

𝑑𝐿 𝑑𝑑𝑑 𝐿 𝑑

P(power) - the rate at which energy is being delivered to the inductor from external sources.

The net work to ramp up the current from 0 to 𝐼 :

𝐿 𝑑

𝐿 𝑑 𝑑𝑈 = 𝑃 𝑑 𝑑𝑑

This energy is stored in the magnetic field created by the current.

𝐿 = 𝜇0𝑛2l 𝜋𝐹2 = 𝜇0𝑛2 ∙ 𝑣𝑣𝑙𝑣𝑣𝑣

𝑈𝐵 =12 𝜇0𝑛

2 𝑣𝑣𝑙𝑣𝑣𝑣 ∙ 𝐼2= 𝐵 = 𝜇0𝑛𝐼 =𝐵2

2𝜇0∙ 𝑣𝑣𝑙𝑣𝑣𝑣 = 𝑣𝐵 ∙ 𝑣𝑣𝑙𝑣𝑣𝑣

The energy density in a magnetic field:

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𝒖𝑩 =𝑩𝟐

𝟐𝝁𝟎 (compare with 𝑣𝐸 = 𝜀0𝐸2

2 )

This is a general result (not solenoid-specific).

𝑼 =𝟏𝟐𝑳𝑰𝟐

Quench of a Superconducting Solenoid

MRI scanner: the magnetic field up to 3T within a volume ~ 1 m3. The magnetic field energy:

Magnet quench: http://www.youtube.com/watch?v=tKj39eWFs10&feature=related

𝑈 = 𝑣𝑣𝑙𝑣𝑣𝑣 ∙ 𝑣𝐵 = 1𝑣3 3𝑇 2

2 ∙ 4𝜋 ∙ 10−7 𝑊𝑊𝐴 ∙ 𝑣

≈ 4𝑀𝐽

The heat of vaporization of liquid helium: 3 kJ/liter. Thus, ~ 1000 liters will be evaporated during the quench.

The capacity of the LHe dewar: ~2,000 liters.

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Another (More General) Definition of Inductance

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𝑈𝐵 =12𝐿𝐼2 = �

𝐵2

2𝜇0𝑑𝜏

𝑎𝑙𝑙 𝑠𝑜𝑎𝑒𝑒

Advantage: it works for a distributed current flow (when it’s unclear which loop we need to consider for the flux calculation).

𝑳 ≡𝟐𝑼𝑩𝑰𝟐

Alternative definition of inductance:

Example: inductance per unit length for a coaxial cable

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Uniform current density in the central conductor of radius a:

𝑈𝐵 =1

2𝜇0� 𝐵 𝐹 2𝑑𝜏𝑏

0

𝐵 𝐹 =

𝜇0𝐼2𝜋𝐹

, 𝐹 < 𝐹 < 𝑊

𝜇0𝐼𝐹2𝜋𝐹2

, 𝐹 < 𝐹

Straight Wires: Do They Have Inductance?

𝐼

𝐼

𝐹 𝑊

𝐿 =2𝑈𝐵𝐼2

Using 𝑳 = 𝜱/𝑰: 𝐿 𝜇 ≈

𝜇02𝜋 l ∙ 𝑙𝑛

l𝐹 ≈ 2 ∙ 10−7l 𝑣 ∙ 𝑙𝑛

l𝐹

l = 1𝑐𝑣 𝐿 ≈ 10−9𝜇

𝑈𝐵 =1

2𝜇0�

𝜇0𝐼2𝜋𝐹

2𝑑𝜏

𝑏

𝑎

=1

2𝜇0�

𝜇0𝐼2𝜋𝐹

2∙ l ∙ 2𝜋𝐹𝑑𝐹

𝑏

𝑎

≈ (𝑊~l ) ≈𝜇0𝐼2

4𝜋∙ l ∙ 𝑙𝑛

l𝐹

l

~l 𝐵 ≈𝜇0𝐼2𝜋𝐹 Φ ≈ l�

𝜇0𝐼2𝜋𝐹 𝑑𝐹

l

𝑎

=𝝁𝟎𝑰𝟐𝝅 ∙ l ∙ 𝒆𝒆

l𝒆

𝑳 𝑯 ≈𝝁𝟎𝟐𝝅

l ∙ 𝒆𝒆l𝒆

at home

Using 𝑳 = 𝟐𝑼𝑩/𝑰𝟐:

Inductance vs. Capacitance

𝑈𝐵 =𝐵2

2𝜇0∙ 𝑣𝑣𝑙𝑣𝑣𝑣 = 𝑣𝐵 ∙ 𝑣𝑣𝑙𝑣𝑣𝑣

𝐿 =Φ𝐵𝐼 =

2𝑈𝐵𝐼2

𝑈𝐵 =12𝐿𝐼2 =

12Φ𝐵

2

𝐿

𝐶 =𝑄𝑉

=2𝑈𝐸𝑉2

𝑈𝐸 =12𝐶𝑉2 =

12𝑄2

𝐶

𝑈𝐸 =𝜀0𝐸2

2 ∙ 𝑣𝑣𝑙𝑣𝑣𝑣 = 𝑣𝐸 ∙ 𝑣𝑣𝑙𝑣𝑣𝑣

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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.

ℰ2 = −𝑑Φ1→2𝑑𝑑 Φ1→2 = 𝑀1→2𝐼1

𝑩𝟏

loop 1 𝐼1

loop 2

Primitive “transformer”

The e.m.f. in loop 2 due to the time-dependent I1 in loop 1:

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.

ℰ2 = −𝑀𝜕𝐼1𝜕𝑑 𝑀 =

Φ1→2𝐼1

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).

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Calculation of Mutual Inductance

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Calculate M for a pair of coaxial solenoids (𝐹1 = 𝐹2 = 𝐹, l 1 = l 2 = l, 𝑛1 and 𝑛2 - the densities of turns).

𝐵1 = 𝜇0𝑛1𝐼1 Φ = 𝐵1𝜋𝐹2 = 𝜇0𝑛1𝐼1𝜋𝐹2

𝑀 =𝑁2Φ𝐼1

= 𝑁2𝜇0𝑛1𝜋𝐹2 =𝜇0𝑁1𝑁2𝜋𝐹2

l

the flux per one turn

l

𝑀1→2 = 𝑀2→1 = 𝑀

𝐿 = 𝜇0𝑛2l 𝜋𝐹2 =𝜇0𝑁2 𝜋𝐹2

l 𝑀 = 𝐿1𝐿2

Experiment with two coaxial coils.

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.

ℰ2 = 𝑀∗ 𝑑𝐼1𝑑𝑑 = 𝐾𝑚𝑀

𝑑𝐼1𝑑𝑑

no-core mutual inductance

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Next time: Lecture 23. RL and LC circuits. §§ 30.4 - 6

Appendix I: Inductance of a Circular Wire Loop

Note that we cannot assume that the wire radius is zero:

- diverges! Φ~ �𝜇0𝐼2𝜋𝐹 𝑑𝐹

𝑏

"0"

~𝑙𝑛𝑊

"0" What to do: introduce some

reasonably small non-zero radius.

Example: estimate the inductance of a circular wire loop with the radius of the loop, b, being much greater than the wire radius a. 𝑊

2𝐹

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:

2πb

b 𝐵 ≈𝜇0𝐼2𝜋𝐹

Φ ≈ 2𝜋𝑊 �𝜇0𝐼2𝜋𝐹

𝑑𝐹𝑏

𝑎

= 𝜇0𝑊𝐼 ∙ 𝑙𝑛𝑊𝐹

𝐿 =Φ𝐼≈ 𝜇0𝑊 ∙ 𝑙𝑛

𝑊𝐹

𝐿 𝑊 = 1𝑣, 𝐹 = 10−3𝑣 ≈ 4𝜋 ∙ 10−7𝑊𝑊𝐴 ∙ 𝑣 1𝑣 ∙ 𝑙𝑛103 ≈ 10𝜇𝜇

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