Lecture 9 - uniba.sk...Magnetic fields are produced by electric currents, which can be macroscopic...
Transcript of Lecture 9 - uniba.sk...Magnetic fields are produced by electric currents, which can be macroscopic...
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Lecture 9Physics 2018/2019
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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.
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Magnetic fields are produced by electric currents, which can bemacroscopic currents in wires, or microscopic currents associatedwith electrons in atomic orbits.
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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
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1
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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
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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
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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)
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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
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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
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Orientation of magnetic field linesRight hand rule
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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
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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
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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‘
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Lenzβs law
The magnetic field produced by an induced current always opposes any
changes in the magnetic flux. ππππ = βπΞ¦π΅
ππ‘
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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
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Ξ¦ = π΅ππππ π π = βπΞ¦
ππ‘π = π(π‘)
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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).
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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₯
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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
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π = ππππ₯π ππππ‘
πΌ = πΌπππ₯cos(ππ‘) = πΌπππ₯ sin ππ‘ + ΰ΅π2
π = ΰ΅π2 = 90Β° β πβππ π π βπππ‘
ππππ₯
U
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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
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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
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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
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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
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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)
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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)
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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
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Spin of the electrons
ππ = βππ π
ππ
Τ¦π
πππ§ = βππ
πβ
2ππππ = β2ππ ππ΅
ππ- g factorππ = β2,0023
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Nuclear magnetic moment
π = ππ
2πππΌ ππ§ = π
πβ
2ππππ = πππππΌ ππ = 5,05084π₯10β27π½/π
ππππ‘ππ π = 5,5856947
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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