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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.
π π < 1Diamagnetic materials
π΅ < π΅0
Paramagnetic materialsπ΅ > π΅0
Ferromagneticmaterials
π΅ β« π΅0
π π > 1
π π β« 1
π΅ = π0πππ»π΅ = πππ΅0
ππ β πΆπ’πππ π‘πππππππ‘π’ππ
1
Magnetic flux density (magnetic induction) B, Magnetic field strength H
π΅0 = π0π» π΅ = 1 π(π‘ππ ππ) = 1πππ β2π΄β1
π΅ = π ππ 0π» π» =π΄
ππ΅ = π0 1 + π π» π β ππππππ‘ππ π π’π ππππ‘ππππππ‘π¦ΞΌ0 = 4ΟΓ10β7 N A-2 or TmA-1
ΞΌd <1 diamagnetic materials
ΞΌp >1 paramagnetic materials
ΞΌf >>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 βclosed loops, field lines direction β from north pole to south pole
Lorentz force β charged particle in magnetic field
πΉπ = ππ£π΅π πππΌ πΌ β β‘ π£ πππ π΅
πΉπ = ππΏ
π‘π΅sin πΌ = πΌπΏπ΅ sin πΌ
Hendrik Antoon Lorentz(1853 β1928)
Mass spectroscopy
πΉ = ππ = ππ£2
π= ππ£π΅π πππΌ
π£βΎπ΅
π =π
π
π£
π΅π πππΌ = 11
2ππ£2 = ππ
π£ =2ππ
π
π =π
π
2π
π΅2
parameters of the equipment
Magnetic field along a straight wire
Ampereβs law The magnetic field on the perimeter of a region is proportional to the current, that passes through the region.
ππππ πππππ‘β
π΅β₯βπΏ = π0πΌππππππ ππ ΰΆ»
πΆ
π΅ππ = π0πΌππππππ ππ
π0 = 4π. 10β7ππ/π΄
π΅β₯ = ππππ π‘
π΅2ππ = π0πΌππππ.
π΅ =π0πΌππππ.
2ππ
André-Marie Ampère1775-1836
Orientation of magnetic field linesRight hand rule
Magnetic field of two straight wiresπΉ1 = π1π£1π₯π΅2
πΉ2 = π2π£2π₯π΅1
πΉπ = πΌπ π‘ π£π π΅π π ππ90Β° = πΌπ ππ π΅ π
π΅π =π0πΌπ
2ππ
πΉ =π0πΌ1πΌ2π
2ππ=
4π. 10β7. 1.1.1
2π. 1
πΉ = 2. 10β7π
Ampereβs law β magnetic field inside a current loop and solenoid
ππππ ππ πππ‘β
π΅β₯ΞπΏ = π0πΌππππππ ππ
π΅β₯ β ππππ π‘
π΅ =π0πΌ
2π
π
ππππ ππ πππ‘β
π΅β₯ΞπΏ = ππ0πΌππππππ ππ
ππ΅β₯
ππππ ππ πππ‘β
ΞπΏ = ππ0πΌππππππ ππ
π΅π ππππππππΏπ πππππππ = ππ 0πΌ
π΅π πππππππ =ππ 0πΌ
πΏπ πππππππ
Magnetic flux β electromagnetic induction, Faradayβs law
Faradayβs law of electromagnetic induction: the magnitude of the inducedelectromotive force (emf) equals to the rate of the change of the magnetic flux
Ξ¦π΅ = π΅π Ξ¦π΅ = π΅ππππ π Ξ¦π΅ = 0
magnetic flux π = 1 ππ π€ππππ = 1ππ2
Ξ¦π΅ = π΅. Τ¦πΞ¦π΅ = π΅ππππ π
Induced voltage
ππππ(π) = βπΞ¦π΅
ππ‘
Lenzβs law
The magnetic field produced by an induced current always opposes any
changes in the magnetic flux. ππππ = βπΞ¦π΅
ππ‘
Induced voltage, solenoidβs inductance
ππππ = βπΞ¦π΅
ππ‘= β
πππ΅π
ππ‘
π΅ =ππ0πΌ
π
ππππ = βπ2π0π
π
ππΌ
ππ‘= βπΏ
ππΌ
ππ‘
L depends on the geometry of the conductor, high L β solenoids
Electric circuit with direct current (DC), I=const. β magnetic flux Ξ¦=const.
emf Uind is induced only at swithching on and switching off the current
π ππππ πππ πππππ ππππ
Total fluxΞ¦π‘ππ‘ππ = ππ΅π
Ξ¦ = π΅ππππ π π = βπΞ¦
ππ‘π = π(π‘)
ππππ₯
πΌπππ₯πππππ₯
π πΌπππ₯
βΌ
Alternating current - AC (harmonic motion)
π = βπΞ¦π΅
ππ‘= βπ΅π
ππππ ππ‘
ππ‘π = 2πππ = π΅ππ π ππππ‘ = ππππ₯π ππππ‘
πΌ =π
π =
ππππ₯π ππππ‘
π = πΌπππ₯π ππππ‘
The voltage and the current are in phase (π = 0).
U
ππππ₯ = ππππ₯πΌπππ₯
Power in AC resistor circuit
π = ππΌπ = ππππ₯ sin ππ‘ πΌπππ₯ sin ππ‘π = ππππ₯πΌπππ₯π ππ2(ππ‘)
Time average πππ£ = 0
(πππ£)2 =1
2(ππππ₯)2
ππππ =ππππ₯
2= 0,71ππππ₯
(πΌππ£)2 =1
2(πΌπππ₯)2
πΌπππ =πΌπππ₯
2= 0,71πΌπππ₯
ππππ =ππππ₯
2
πΌπππ₯
2=
1
2ππππ₯πΌπππ₯
AC circuit with capacitor, capacitive reactance
βΌ =
π = πΆπ πΌ =ππ
ππ‘
πΌ = πΆππ
ππ‘= πΆ
πππππ₯π ππππ‘
ππ‘
πΌ = πΆππππ₯π πππ ππ‘ = πΌπππ₯πππ ππ‘
πΌπππ₯ = πΆ ππππ₯π
ππΆ =ππππ₯
πΌπππ₯=
1
ππΆ=
1
2πππΆ
π = ππππ₯π ππππ‘
πΌ = πΌπππ₯cos(ππ‘) = πΌπππ₯ sin ππ‘ + ΰ΅π2
π = ΰ΅π2 = 90Β° β πβππ π π βπππ‘
ππππ₯
U
AC circuit with inductor
βΌ
π = βπΏππΌ
ππ‘πΌ = πΌπππ₯ sin ππ‘
π = βπΌπππ₯πΏΟ cos ππ‘ π = βππππ₯πππ ππ‘
π = βπ
2= β90Β° β πβππ π π βπππ‘
ππππ₯ = πΌπππ₯ππΏ
ππΏ =ππππ₯
πΌπππ₯= ππΏ
ππΏ = 2πππΏ
U
RLC circuit, Z β impedance of the circuit
π =π
πΌ= π 2 + ππΏ β
1
ππΆ
2
ππ = πΌπ ππ ππ β πβππ π
ππΏ = πΌππΏππΏ πππππ πΌ
ππΆ = πΌ1
ππΆππΆ ππππ πΌ
ππ ππΏ ππΆ
U
ππΏ
ππ
ππΆ
ππ
ππΆ
ππΏ
ππ
ππΏ β ππΆ
πππ
ππ = ππ 2 + ππΏ β ππΆ
2 π‘ππ =ππΏ β ππΆ
ππ
π = π 2 + ππΏ β ππΆ2 π‘ππ =
ππΏβππΆ
π
ππΏ = πΌππΏ
ππ = πΌπ
ππΆ = πΌ1
ππΆ
RLC circuit - resonance
π =π
πΌ= π 2 + ππΏ β
1
ππΆ
2
π = ππππ = π β ππΏ β1
ππΆ= 0
Resonance condition βThompsonβs law
π =1
πΏπΆπ =
1
2π
1
πΏπΆ
Power of AC circuitπ = ππΌ = ππππ₯ sin ππ‘ + π . πΌπππ₯ sin ππ‘
2 π πππΌ π πππ½ = cos πΌ β π½ β cos πΌ + π½
π =1
2ππππ₯. πΌπππ₯ cππ ππ‘ + π β ππ‘ β cos ππ‘ + π + ππ‘
π =1
2ππππ₯. πΌπππ₯ cππ π β
1
2ππππ₯. πΌπππ₯cos 2ππ‘ + π
ΰ΄€π =ππππ₯
2
πΌπππ₯
2πππ π = ππππ πΌπππ πππ π
-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 β ~1000 times weaker than the magnetic moment of electrons (NMR)
Orbital moment πΏ = Τ¦ππ₯ Τ¦π = ππ Τ¦ππ₯ Τ¦π£
Magnetic moment
Τ¦π = βππΏπ
2πππΏ ππΏπ§ = βππΏ
πβ
2ππππ = βππππ΅
g-factor ππΏ = 1
Bohr magneton π π΅ =πβ
2ππ= 9,274015π₯10β24J/T
Spin of the electrons
ππ = βππ π
ππ
Τ¦π
πππ§ = βππ
πβ
2ππππ = β2ππ ππ΅
ππ- g factorππ = β2,0023
Nuclear magnetic moment
π = ππ
2πππΌ ππ§ = π
πβ
2ππππ = πππππΌ ππ = 5,05084π₯10β27π½/π
ππππ‘ππ π = 5,5856947
Summaryπ΅ = π0πππ»
πΉ = ππΈ + ππ£π₯π΅
π =π
π
2π
π΅2
straight conductor
π΅ =π0πΌππππ.
2πππΉπ = πΌπ ππ π΅ π
πΉ =π0πΌ1πΌ2π
2ππLoop
π΅ =π0πΌ
2π Solenoid
π΅π πππππππ =ππ 0πΌ
πΏπ πππππππ
Ξ¦π΅ = π΅ππππ π
ππππ(π) = βπΞ¦π΅
ππ‘
ππππ = βπΏππΌ
ππ‘
ππΆ =1
ππΆ=
1
2πππΆ
ππΏ = ππΏ = 2πππΏ
π = π 2 + ππΏ β1
ππΆ
2
π‘ππ =ππΏ β ππΆ
π
π =1
πΏπΆπ =
1
2π
1
πΏπΆ
ΰ΄€π = ππππ πΌπππ πππ π