Lecture 9Physics 2018/2019

Magnetism

Magnetic fields are invisible fields that exert a vector force,characterized by both strength and direction, and are produced bymagnetic objects or changing electric fields.

Bar magnets Earth

S

S S

N

NN

The simplest magnetic structure that can exist in nature is the magnetic dipole.

There exists no separate north or south pole of magnets.

Magnetic fields are produced by electric currents, which can bemacroscopic currents in wires, or microscopic currents associatedwith electrons in atomic orbits.

𝜇 𝑟 < 1Diamagnetic materials

𝐵 < 𝐵0

Paramagnetic materials𝐵 > 𝐵0

Ferromagneticmaterials

𝐵 ≫ 𝐵0

𝜇 𝑟 > 1

𝜇 𝑟 ≫ 1

𝐵 = 𝜇0𝜇𝑟𝐻𝐵 = 𝜇𝑟𝐵0

𝑇𝑐 − 𝐶𝑢𝑟𝑖𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒

1

Magnetic flux density (magnetic induction) B, Magnetic field strength H

𝐵0 = 𝜇0𝐻 𝐵 = 1 𝑇(𝑡𝑒𝑠𝑙𝑎) = 1𝑘𝑔𝑠−2𝐴−1

𝐵 = 𝜇 𝑟𝜇 0𝐻 𝐻 =𝐴

𝑚𝐵 = 𝜇0 1 + 𝜒 𝐻 𝜒 − 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑠𝑢𝑠𝑐𝑒𝑝𝑡𝑖𝑏𝑖𝑙𝑖𝑡𝑦μ0 = 4π×10−7 N A-2 or TmA-1

μd <1 diamagnetic materials

μp >1 paramagnetic materials

μf >>1 ferromagnetic materials

south pole

north pole

Magnetic field is represented by field lines:

1. direction of the tangent to a magnetic field line at any point

gives the direction of B at that point

2. the spacing of the lines is a measure of the magnitude of B.

N

S

Vector field, field lines never cross, do not start and stop anywhere –closed loops, field lines direction – from north pole to south pole

Lorentz force – charged particle in magnetic field

𝐹𝑚 = 𝑞𝑣𝐵𝑠𝑖𝑛𝛼 𝛼 − ∡ 𝑣 𝑎𝑛𝑑 𝐵

𝐹𝑚 = 𝑞𝐿

𝑡𝐵sin 𝛼 = 𝐼𝐿𝐵 sin 𝛼

Hendrik Antoon Lorentz(1853 –1928)

Mass spectroscopy

𝐹 = 𝑚𝑎 = 𝑚𝑣2

𝑟= 𝑞𝑣𝐵𝑠𝑖𝑛𝛼

𝑣⊾𝐵

𝑟 =𝑚

𝑞

𝑣

𝐵𝑠𝑖𝑛𝛼 = 11

2𝑚𝑣2 = 𝑞𝑈

𝑣 =2𝑞𝑈

𝑚

𝑟 =𝑚

𝑞

2𝑈

𝐵2

parameters of the equipment

Magnetic field along a straight wire

Ampere’s law The magnetic field on the perimeter of a region is proportional to the current, that passes through the region.

𝑐𝑙𝑜𝑠𝑒𝑑𝑝𝑎𝑡ℎ

𝐵∥∆𝐿 = 𝜇0𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ර

𝐶

𝐵𝑑𝑙 = 𝜇0𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑

𝜇0 = 4𝜋. 10−7𝑇𝑚/𝐴

𝐵∥ = 𝑐𝑜𝑛𝑠𝑡

𝐵2𝜋𝑟 = 𝜇0𝐼𝑒𝑛𝑐𝑙.

𝐵 =𝜇0𝐼𝑒𝑛𝑐𝑙.

2𝜋𝑟

André-Marie Ampère1775-1836

Orientation of magnetic field linesRight hand rule

Magnetic field of two straight wires𝐹1 = 𝑞1𝑣1𝑥𝐵2

𝐹2 = 𝑞2𝑣2𝑥𝐵1

𝐹𝑖 = 𝐼𝑖 𝑡 𝑣𝑖 𝐵𝑗 𝑠𝑖𝑛90° = 𝐼𝑖 𝑙𝑖 𝐵 𝑗

𝐵𝑗 =𝜇0𝐼𝑗

2𝜋𝑎

𝐹 =𝜇0𝐼1𝐼2𝑙

2𝜋𝑎=

4𝜋. 10−7. 1.1.1

2𝜋. 1

𝐹 = 2. 10−7𝑁

Ampere’s law – magnetic field inside a current loop and solenoid

𝑐𝑙𝑜𝑠𝑒𝑑 𝑝𝑎𝑡ℎ

𝐵∥Δ𝐿 = 𝜇0𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑

𝐵∥ ≠ 𝑐𝑜𝑛𝑠𝑡

𝐵 =𝜇0𝐼

2𝑅

𝑁

𝑐𝑙𝑜𝑠𝑒𝑑 𝑝𝑎𝑡ℎ

𝐵∥Δ𝐿 = 𝑁𝜇0𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑

𝑁𝐵∥

𝑐𝑙𝑜𝑠𝑒𝑑 𝑝𝑎𝑡ℎ

Δ𝐿 = 𝑁𝜇0𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑

𝐵𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑𝐿𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 = 𝑁𝜇 0𝐼

𝐵𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 =𝑁𝜇 0𝐼

𝐿𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑

Magnetic flux – electromagnetic induction, Faraday’s law

Faraday’s law of electromagnetic induction: the magnitude of the inducedelectromotive force (emf) equals to the rate of the change of the magnetic flux

Φ𝐵 = 𝐵𝑆 Φ𝐵 = 𝐵𝑆𝑐𝑜𝑠𝜃 Φ𝐵 = 0

magnetic flux 𝜙 = 1 𝑊𝑏 𝑤𝑒𝑏𝑒𝑟 = 1𝑇𝑚2

Φ𝐵 = 𝐵. Ԧ𝑆Φ𝐵 = 𝐵𝑆𝑐𝑜𝑠𝜃

Induced voltage

𝜀𝑖𝑛𝑑(𝑈) = −𝑑Φ𝐵

𝑑𝑡

Lenz’s law

The magnetic field produced by an induced current always opposes any

changes in the magnetic flux. 𝑈𝑖𝑛𝑑 = −𝑑Φ𝐵

𝑑𝑡

Induced voltage, solenoid’s inductance

𝑈𝑖𝑛𝑑 = −𝑑Φ𝐵

𝑑𝑡= −

𝑑𝑁𝐵𝑆

𝑑𝑡

𝐵 =𝑁𝜇0𝐼

𝑙

𝑈𝑖𝑛𝑑 = −𝑁2𝜇0𝑆

𝑙

𝑑𝐼

𝑑𝑡= −𝐿

𝑑𝐼

𝑑𝑡

L depends on the geometry of the conductor, high L – solenoids

Electric circuit with direct current (DC), I=const. → magnetic flux Φ=const.

emf Uind is induced only at swithching on and switching off the current

𝑆 𝑎𝑟𝑒𝑎 𝑜𝑓𝑠𝑖𝑛𝑔𝑙𝑒 𝑙𝑜𝑜𝑝

Total fluxΦ𝑡𝑜𝑡𝑎𝑙 = 𝑁𝐵𝑆

Φ = 𝐵𝑆𝑐𝑜𝑠𝜃 𝑈 = −𝑑Φ

𝑑𝑡𝜃 = 𝑓(𝑡)

𝑈𝑚𝑎𝑥

𝐼𝑚𝑎𝑥𝑈𝑈𝑚𝑎𝑥

𝑈 𝐼𝑚𝑎𝑥

∼

Alternating current - AC (harmonic motion)

𝑈 = −𝑑Φ𝐵

𝑑𝑡= −𝐵𝑆

𝑑𝑐𝑜𝑠𝜔𝑡

𝑑𝑡𝜔 = 2𝜋𝑓𝑈 = 𝐵𝑆𝜔 𝑠𝑖𝑛𝜔𝑡 = 𝑈𝑚𝑎𝑥𝑠𝑖𝑛𝜔𝑡

𝐼 =𝑈

𝑅=

𝑈𝑚𝑎𝑥𝑠𝑖𝑛𝜔𝑡

𝑅= 𝐼𝑚𝑎𝑥𝑠𝑖𝑛𝜔𝑡

The voltage and the current are in phase (𝜑 = 0).

U

𝑃𝑚𝑎𝑥 = 𝑈𝑚𝑎𝑥𝐼𝑚𝑎𝑥

Power in AC resistor circuit

𝑃 = 𝑈𝐼𝑃 = 𝑈𝑚𝑎𝑥 sin 𝜔𝑡 𝐼𝑚𝑎𝑥 sin 𝜔𝑡𝑃 = 𝑈𝑚𝑎𝑥𝐼𝑚𝑎𝑥𝑠𝑖𝑛2(𝜔𝑡)

Time average 𝑃𝑎𝑣 = 0

(𝑈𝑎𝑣)2 =1

2(𝑈𝑚𝑎𝑥)2

𝑈𝑟𝑚𝑠 =𝑈𝑚𝑎𝑥

2= 0,71𝑈𝑚𝑎𝑥

(𝐼𝑎𝑣)2 =1

2(𝐼𝑚𝑎𝑥)2

𝐼𝑟𝑚𝑠 =𝐼𝑚𝑎𝑥

2= 0,71𝐼𝑚𝑎𝑥

𝑃𝑟𝑚𝑠 =𝑈𝑚𝑎𝑥

2

𝐼𝑚𝑎𝑥

2=

1

2𝑈𝑚𝑎𝑥𝐼𝑚𝑎𝑥

AC circuit with capacitor, capacitive reactance

∼ =

𝑄 = 𝐶𝑈 𝐼 =𝑑𝑄

𝑑𝑡

𝐼 = 𝐶𝑑𝑈

𝑑𝑡= 𝐶

𝑑𝑈𝑚𝑎𝑥𝑠𝑖𝑛𝜔𝑡

𝑑𝑡

𝐼 = 𝐶𝑈𝑚𝑎𝑥𝜔 𝑐𝑜𝑠𝜔𝑡 = 𝐼𝑚𝑎𝑥𝑐𝑜𝑠𝜔𝑡

𝐼𝑚𝑎𝑥 = 𝐶 𝑈𝑚𝑎𝑥𝜔

𝑋𝐶 =𝑈𝑚𝑎𝑥

𝐼𝑚𝑎𝑥=

1

𝜔𝐶=

1

2𝜋𝑓𝐶

𝑈 = 𝑈𝑚𝑎𝑥𝑠𝑖𝑛𝜔𝑡

𝐼 = 𝐼𝑚𝑎𝑥cos(𝜔𝑡) = 𝐼𝑚𝑎𝑥 sin 𝜔𝑡 + ൗ𝜋2

𝜑 = ൗ𝜋2 = 90° − 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓𝑡

𝑈𝑚𝑎𝑥

U

AC circuit with inductor

∼

𝑈 = −𝐿𝑑𝐼

𝑑𝑡𝐼 = 𝐼𝑚𝑎𝑥 sin 𝜔𝑡

𝑈 = −𝐼𝑚𝑎𝑥𝐿ω cos 𝜔𝑡 𝑈 = −𝑈𝑚𝑎𝑥𝑐𝑜𝑠𝜔𝑡

𝜑 = −𝜋

2= −90° − 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓𝑡

𝑈𝑚𝑎𝑥 = 𝐼𝑚𝑎𝑥𝜔𝐿

𝑋𝐿 =𝑈𝑚𝑎𝑥

𝐼𝑚𝑎𝑥= 𝜔𝐿

𝑋𝐿 = 2𝜋𝑓𝐿

U

RLC circuit, Z – impedance of the circuit

𝑍 =𝑈

𝐼= 𝑅2 + 𝜔𝐿 −

1

𝜔𝐶

2

𝑈𝑅 = 𝐼𝑅𝑈𝑅𝑖𝑛 − 𝑝ℎ𝑎𝑠𝑒

𝑈𝐿 = 𝐼𝜔𝐿𝑈𝐿 𝑙𝑒𝑎𝑑𝑠 𝐼

𝑈𝐶 = 𝐼1

𝜔𝐶𝑈𝐶 𝑙𝑎𝑔𝑠 𝐼

𝑈𝑅 𝑈𝐿 𝑈𝐶

U

𝑈𝐿

𝑈𝑅

𝑈𝐶

𝑈𝑅

𝑈𝐶

𝑈𝐿

𝑋𝑅

𝑋𝐿 − 𝑋𝐶

𝑍𝑈𝑆

𝑈𝑆 = 𝑈𝑅2 + 𝑈𝐿 − 𝑈𝐶

2 𝑡𝑔𝜑 =𝑈𝐿 − 𝑈𝐶

𝑈𝑅

𝑍 = 𝑅2 + 𝑋𝐿 − 𝑋𝐶2 𝑡𝑔𝜑 =

𝑋𝐿−𝑋𝐶

𝑅

𝑈𝐿 = 𝐼𝜔𝐿

𝑈𝑅 = 𝐼𝑅

𝑈𝐶 = 𝐼1

𝜔𝐶

RLC circuit - resonance

𝑍 =𝑈

𝐼= 𝑅2 + 𝜔𝐿 −

1

𝜔𝐶

2

𝑍 = 𝑍𝑚𝑖𝑛 = 𝑅 ⇒ 𝜔𝐿 −1

𝜔𝐶= 0

Resonance condition –Thompson’s law

𝜔 =1

𝐿𝐶𝑓 =

1

2𝜋

1

𝐿𝐶

Power of AC circuit𝑃 = 𝑈𝐼 = 𝑈𝑚𝑎𝑥 sin 𝜔𝑡 + 𝜑 . 𝐼𝑚𝑎𝑥 sin 𝜔𝑡

2 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 = cos 𝛼 − 𝛽 − cos 𝛼 + 𝛽

𝑃 =1

2𝑈𝑚𝑎𝑥. 𝐼𝑚𝑎𝑥 c𝑜𝑠 𝜔𝑡 + 𝜑 − 𝜔𝑡 − cos 𝜔𝑡 + 𝜑 + 𝜔𝑡

𝑃 =1

2𝑈𝑚𝑎𝑥. 𝐼𝑚𝑎𝑥 c𝑜𝑠 𝜑 −

1

2𝑈𝑚𝑎𝑥. 𝐼𝑚𝑎𝑥cos 2𝜔𝑡 + 𝜑

ത𝑃 =𝑈𝑚𝑎𝑥

2

𝐼𝑚𝑎𝑥

2𝑐𝑜𝑠𝜑 = 𝑈𝑟𝑚𝑠𝐼𝑟𝑚𝑠𝑐𝑜𝑠𝜑

-1000

-500

0

500

1000

1500

0.00 0.01 0.02 0.03 0.04 0.05 0.06

P (

W)

t (s)

Magnetic fileds of microscopic currents

a. Orbital magnetic moment of the electrons

b. Spin of the electrons (EPR)

c. Magnetic moment of protons and neutrons – ~1000 times weaker than the magnetic moment of electrons (NMR)

Orbital moment 𝐿 = Ԧ𝑟𝑥 Ԧ𝑝 = 𝑚𝑒 Ԧ𝑟𝑥 Ԧ𝑣

Magnetic moment

Ԧ𝜇 = −𝑔𝐿𝑒

2𝑚𝑒𝐿 𝜇𝐿𝑧 = −𝑔𝐿

𝑒ℏ

2𝑚𝑒𝑚𝑙 = −𝑚𝑙𝜇𝐵

g-factor 𝑔𝐿 = 1

Bohr magneton 𝜇 𝐵 =𝑒ℏ

2𝑚𝑒= 9,274015𝑥10−24J/T

Spin of the electrons

𝜇𝑠 = −𝑔𝑠𝑒

𝑚𝑒

Ԧ𝑆

𝜇𝑆𝑧 = −𝑔𝑠

𝑒ℏ

2𝑚𝑒𝑚𝑠 = −2𝑚𝑠𝜇𝐵

𝑔𝑆- g factor𝑔𝑠 = −2,0023

Nuclear magnetic moment

𝜇 = 𝑔𝑒

2𝑚𝑝𝐼 𝜇𝑧 = 𝑔

𝑒ℏ

2𝑚𝑝𝑚𝑙 = 𝑔𝜇𝑁𝑚𝐼 𝜇𝑁 = 5,05084𝑥10−27𝐽/𝑇

𝑝𝑟𝑜𝑡𝑜𝑛 𝑔 = 5,5856947

Summary𝐵 = 𝜇0𝜇𝑟𝐻

𝐹 = 𝑞𝐸 + 𝑞𝑣𝑥𝐵

𝑟 =𝑚

𝑞

2𝑈

𝐵2

straight conductor

𝐵 =𝜇0𝐼𝑒𝑛𝑐𝑙.

2𝜋𝑟𝐹𝑖 = 𝐼𝑖 𝑙𝑖 𝐵 𝑗

𝐹 =𝜇0𝐼1𝐼2𝑙

2𝜋𝑟Loop

𝐵 =𝜇0𝐼

2𝑅Solenoid

𝐵𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 =𝑁𝜇 0𝐼

𝐿𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑

Φ𝐵 = 𝐵𝑆𝑐𝑜𝑠𝜃

𝜀𝑖𝑛𝑑(𝑈) = −𝑑Φ𝐵

𝑑𝑡

𝑈𝑖𝑛𝑑 = −𝐿𝑑𝐼

𝑑𝑡

𝑋𝐶 =1

𝜔𝐶=

1

2𝜋𝑓𝐶

𝑋𝐿 = 𝜔𝐿 = 2𝜋𝑓𝐿

𝑍 = 𝑅2 + 𝜔𝐿 −1

𝜔𝐶

2

𝑡𝑔𝜑 =𝑋𝐿 − 𝑋𝐶

𝑅

𝜔 =1

𝐿𝐶𝑓 =

1

2𝜋

1

𝐿𝐶

ത𝑃 = 𝑈𝑟𝑚𝑠𝐼𝑟𝑚𝑠𝑐𝑜𝑠𝜑