23. Self and Mutual inductance

Announcements:

Assignment 3 due Monday (instead of today)

Lab 3 (AC circuits) will start Monday after the break

The practice quiz website will be up this w-e

Last class before the break!!!!

Where will you go over the break?

1. Go skiing

2. Work on the Faraday machine

3. Go to sun and sea

4. Go home

5. Stay in Montreal

6. I don’t know yet

7. Other

Joseph Henry

1797 – 1878

American physicist

First director of the Smithsonian

First president of the Academy of Natural Science

Improved design of electromagnet

Constructed one of the first motors

Discovered self-inductance

▪ Didn’t publish his results

Unit of inductance is named in his honor

Section 32.1

Inductance Units

The SI unit of inductance is the henry (H)

Named for Joseph Henry

A

sV1H1

=

Section 32.1

Self-Inductance, consequence of Faraday’s law

An induced emf is always proportional to the time rate of change of the current.

▪ The emf is proportional to the flux, which is proportional to the field and the field is

proportional to the current.

L is a constant of proportionality called the self-inductance of the coil.

This creates a voltage = 𝜀𝐿 across the coil.

▪ It depends on the geometry of the coil and other physical characteristics.

𝜀𝐿 = −𝐿𝑑𝐼

𝑑𝑡

Section 32.1

L

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

𝑑

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝜇0𝐼𝑁

𝑑

𝑑

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝜇0𝐼𝑁

𝑑and Φ𝐵 = 𝑆 ∙ 𝐵

𝑑

S

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝜇0𝐼𝑁

𝑑and Φ𝐵 = 𝑆 ∙ 𝐵

⇒ 𝐿 =𝜇0𝑁

2𝑆

𝑑𝑑

S

Inductance of a Coil

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝜇0𝐼𝑁

𝑑and Φ𝐵 = 𝑆 ∙ 𝐵

⇒ 𝐿 =𝜇0𝑁

2𝑆

𝑑= 𝜇0𝑛

2𝑉𝑑

S

𝑛 =𝑁

𝑑

Volume of inductor

Inductance of a Coil adding an iron bar inside

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝝁𝐼𝑁

𝑑and Φ𝐵 = 𝑆 ∙ 𝐵

⇒ 𝐿 =𝝁𝑁2𝑆

𝑑= 𝝁𝑛2𝑉

𝑑𝑛 =

𝑁

𝑑

Volume of inductor

Permeability of the material

Iron

Inductance of a Coil adding an iron bar inside

A closely spaced coil of N turns carrying current I has an inductance of

The inductance is a measure of the opposition to a change in current.

𝐿 =𝑁Φ𝐵

𝐼

Section 32.1

𝐵 = 𝝁𝐼𝑁

𝑑and Φ𝐵 = 𝑆 ∙ 𝐵

⇒ 𝐿 =𝝁𝑁2𝑆

𝑑= 𝝁𝑛2𝑉

𝑑𝑛 =

𝑁

𝑑

Volume of inductor

Permeability of the material

Iron

𝜇 = 5000𝜇0Huge enhancement of the magnetic field!

Magnetic energy

Power in inductor:

Power in inductor:

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡

Energy in inductor:

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡

Energy in inductor:

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖

Energy in inductor:

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

𝐵

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑𝐵

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑𝐵

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑=

𝑁2𝜇0𝑉𝐼2

2𝑑2𝐵

Volume inside

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑=

𝑁2𝜇0𝑉𝐼2

2𝑑2𝐵

Volume inside

𝐵 = 𝜇0𝐼𝑁

𝑑

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑=

𝑁2𝜇0𝑉𝐼2

2𝑑2=

𝑉𝐵2

2𝜇0𝐵

Volume inside

𝐵 = 𝜇0𝐼𝑁

𝑑

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑=

𝑁2𝜇0𝑉𝐼2

2𝑑2=

𝑉𝐵2

2𝜇0𝐵

Volume inside

𝑢 =𝑈

𝑉Magnetic field energy

density in vacuum

Power in inductor:

𝑈 = න𝑃 ∙ 𝑑𝑡 = න𝐿𝑖𝑑𝑖

𝑑𝑡∙ 𝑑𝑡 = න

0

𝐼

𝐿𝑖 ∙ 𝑑𝑖 =1

2𝐿𝐼2

Energy in inductor:

L=𝑁2𝜇0𝑆

𝑑→ 𝑈 =

𝑁2𝜇0𝑆𝐼2

2𝑑=

𝑁2𝜇0𝑉𝐼2

2𝑑2=

𝑉𝐵2

2𝜇0𝐵

Volume inside

𝑢 =𝑈

𝑉=

𝐵2

2𝜇0

Magnetic field energy

density in vacuum

Energy density of electromagnetic field in vacuum:

𝑢 =𝐵2

2𝜇0+𝜀0𝐸

2

2

Magnetic field contribution

Electric field contribution

Energy density of electromagnetic field in vacuum:

𝑢 =𝐵2

2𝜇0+𝜀0𝐸

2

2

Magnetic field contribution

Electric field contribution

Energy density of electromagnetic field in materials:

𝑢 =𝐵2

2𝜇+𝜀𝐸2

2

permeability

permittivity

ϕ1 = 𝑳𝐼1 Self-inductance (a current creates its own magnetic flux)

I1

11 LI=

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

ϕ1 = 𝐿𝐼1 Self-inductance (a current creates its own magnetic flux)

11 LI=

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

ϕ2 = 𝑴𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

12 MI=

I1

ϕ2 = 𝐿𝐼2 Self-inductance (a current creates its own magnetic flux)

ϕ2 = 𝑀𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

21 MI=

222 IL=I2

Reciprocity: The mutual inductance

M of 1 onto 2 is the same as 2 onto 1

ϕ2 = 𝐿𝐼2 Self-inductance (a current creates its own magnetic flux)

ϕ2 = 𝑀𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

I2

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

= −𝐿𝑑𝐼1𝑑𝑡

𝑒𝑚𝑓2 = −𝑑ϕ2

𝑑𝑡= −𝑀

𝑑𝐼1𝑑𝑡

I1

11 LI=

12 MI=

ϕ2 = 𝐿𝐼2 Self-inductance (a current creates its own magnetic flux)

ϕ2 = 𝑀𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

I2

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

= −𝐿𝑑𝐼1𝑑𝑡

𝑒𝑚𝑓2 = −𝑑ϕ2

𝑑𝑡= −𝑀

𝑑𝐼1𝑑𝑡

Unit of inductance: Henry [H]

I1

11 LI=

12 MI=

ϕ2 = 𝐿𝐼2 Self-inductance (a current creates its own magnetic flux)

ϕ2 = 𝑀𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

I2

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

= −𝐿𝑑𝐼1𝑑𝑡

𝑒𝑚𝑓2 = −𝑑ϕ2

𝑑𝑡= −𝑀

𝑑𝐼1𝑑𝑡

→ 𝐼2 =𝑒𝑚𝑓2𝑅2

A change in current 𝑰𝟏 will induce 𝑰𝟐

Resistance

(impedance) of 𝐼2loop

I1

11 LI=

12 MI=

ϕ2 = 𝐿𝐼2 Self-inductance (a current creates its own magnetic flux)

ϕ2 = 𝑀𝐼1 Mutual-inductance (a current creates a magnetic flux elsewhere)

I2

𝑒𝑚𝑓1 = −𝑑ϕ1𝑑𝑡

= −𝐿𝑑𝐼1𝑑𝑡

𝑒𝑚𝑓2 = −𝑑ϕ2

𝑑𝑡= −𝑀

𝑑𝐼1𝑑𝑡

→ 𝐼2 =𝑒𝑚𝑓2𝑅2

A change in current 𝑰𝟏 will induce 𝑰𝟐

Resistance

(impedance) of 𝐼2loop

Primary coil

11 LI=

12 MI=

I1

powerelectronictips.com

60Hz

100kHz

𝑁2 loops, multiplies the magnetic flux!

𝑒𝑚𝑓2 = −𝑁2𝑑ϕ2

𝑑𝑡= −𝑀

𝑑𝐼1𝑑𝑡

→ 𝑀 =𝑁2ϕ2

𝐼1

Example:

Mutual inductance

(between coil 1 and 2)

Example:

L=𝑁1ϕ𝐵1

𝑖1=

𝑁12𝜇0𝐴𝑖1

𝑙𝑖1=

𝑁12𝜇0𝐴

𝑙

Mutual inductance

(between coil 1 and 2)

Self inductance (coil 1)

LC circuit

Courtesy of M. Devoret

2000 qubits !

LC circuit

LC circuit

LC circuit

𝑞

𝐶

LC circuit Kirchhoff loop:

−𝐿𝑑𝐼

𝑑𝑡−𝑞

𝐶= 0

LC circuit Kirchhoff loop:

−𝐿𝑑𝐼

𝑑𝑡−𝑞

𝐶= 0

𝐼 =𝑑𝑞

𝑑𝑡

LC circuit Kirchhoff loop:

−𝐿𝑑𝐼

𝑑𝑡−𝑞

𝐶= 0

𝐼 =𝑑𝑞

𝑑𝑡

𝐿𝑑2𝑞

𝑑𝑡2+𝑞

𝐶= 0

LC circuit Kirchhoff loop:

−𝐿𝑑𝐼

𝑑𝑡−𝑞

𝐶= 0

𝐼 =𝑑𝑞

𝑑𝑡

𝐿𝑑2𝑞

𝑑𝑡2+𝑞

𝐶= 0

Second order differential equation (like the harmonic oscillator)

Digital Lab