Current in an LC circuit Period: Frequency: Current in an LC Circuit.
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Transcript of Current in an LC circuit Period: Frequency: Current in an LC Circuit.
LC
tQQ cos0
dt
dQI
LC
t
LC
QI sin0
Current in an LC circuit
Period: LCT 2
Frequency: f 1 / 2 LC
Current in an LC Circuit
Question (Chap. 23)
A. There will be no current in the circuit at any time because of the opposing emf in the inductor.
B. The current in the circuit will maximize at time t when the capacitor will have charge Q(t)=0.
C. The current in the circuit will maximize at time t when capacitor will have full charge Q(t)=Q0.
D. The current will decay exponentially.
Q0
A capacitor C was charged and contains charge +Q0 and –Q0 on each of its plates, respectively. It is then connected to an inductor (coil) L. Assuming the ideal case (wires have no resistance) which is true?
QuestionTwo metal rings lie side-by-side on a table. Current in the left ring runs clockwise and is increasing with time. This induces a current in the right ring. This current runs
A) ClockwiseB) Counterclockwise
when viewed from above
Single home current: 100 A serviceVwires=IRwires
Transformer: emfHV IHV = emfhomeIhome
Single home current in HV: <0.1 APower loss in wires ~ I2
Faraday’s Law: Applications
Maxwell’s Equations
0
ˆ
insideqdAnE
pathinsideIldB _0
Gauss’s law for electricity
Gauss’s law for magnetism
Complete Faraday’s law
Ampere’s law(Incomplete Ampere-Maxwell law)
0ˆ AnB
∮𝐸 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ ��𝑑 𝐴 ]
No current inside
0 ldB
Current pierces surface
IldB 0
r
IB
2
40
Irr
IldB 0
0 22
4
pathinsideIldB _0Ampere’s Law
Time varying magnetic field leads to curly electric field.
Time varying electric field leads to curly magnetic field?
dAnEelec ˆ
00
0cosQ
AA
Qelec
dt
dQ
dt
d elec
0
1
I
0
1
I
dt
dI elec
0 ‘equivalent’ current
pathinsideIldB _0 combine with current in Ampere’s law
Maxwell’s Approach
Four equations (integral form) :
Gauss’s law
Gauss’s law for magnetism
Faraday’s law
Ampere-Maxwell law
0
ˆ
insideqdAnE
dAnBdt
dldE ˆ
dt
dIldB elec
pathinside 0_0
+ Lorentz force BvqEqF
Maxwell’s Equations
0ˆ AnB
Time varying magnetic field makes electric field
Time varying electric field makes magnetic field
Do we need any charges around to sustain the fields?
Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation?
Solution plan: • Propose particular configuration• Check if it is consistent with Maxwell’s eqs• Show the way to produce such field• Identify the effects such field will have on matter• Analyze phenomena involving such fields
Fields Without Charges
Key idea: Fields travel in space at certain speedDisturbance moving in space – a wave?
1. Simplest case: a pulse (moving slab)
A Simple Configuration of Traveling Fields
0
ˆ
insideqdAnE
0ˆdAnE
Pulse is consistent with Gauss’s law
0ˆ AnB
Pulse is consistent with Gauss’s law for magnetism
A Pulse and Gauss’s Laws
dt
demf mag
Since pulse is ‘moving’, B depends on time and thus causes E
Area doesnot move
tBhvmag
Bhvdt
d
tmagmag
emf
EhldEemf
E=Bv
Is direction right?
A Pulse and Faraday’s Law
dt
dIldB elec
pathinside 0_0
=0
tEhvelec
Ehvdt
d
telecelec
BhldB
EvhBh 00
vEB 00
A Pulse and Ampere-Maxwell Law
vEB 00 E=Bv
vBvB 00
2001 v
m/s 8
00
1031
v
Based on Maxwell’s equations, pulse must propagate at speed of light
E=cB
A Pulse: Speed of Propagation
Question
At this instant, the magnetic flux Fmag through the entire rectangle is:
A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh
Question
What is around the full rectangular path?
A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2EvDt; E)2EvDt
Exercise
If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field?
E cB
Force on charge q moving with velocity v perpendicular to B:
E 3x108 m / s 1x10 5 T 3000V / m
𝐹𝑚𝑎𝑔
𝐹𝑒𝑙
=𝑣𝐵𝐸
¿𝑣𝐵𝑐𝐵
=𝑣𝑐
Electromagnetic pulse can propagate in spaceHow can we initiate such a pulse?
Short pulse of transverseelectric field
Accelerated Charges
1. Transverse pulse propagates at speed of light
2. Since E(t) there must be B
3. Direction of v is given by: BE
E
Bv
Accelerated Charges
We can qualitatively predict the direction.What is the magnitude?
Magnitude can be derived from Gauss’s law
Field ~ -qa
rc
aqEradiative 2
04
1
1. The direction of the field is opposite to qa
2. The electric field falls off at a rate 1/r
Magnitude of the Transverse Electric Field
An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field.
1. The direction of the field is opposite to qa
a
E
BE
2. The direction of propagation is given by
B
Exercise
Circular motion: Is there radiation emitted? v
aClassical physics says “YES”Þ orbiting particle must lose energy!Þ speed decreasesÞ particle comes closer to center
Classical model of atom:
Electrons should fall on nucleus!
To explain the facts - introduction ofquantum mechanics:Electrons can move around certain orbits only and emit E/M radiation only when jumping from one orbit to another
Stability of Atoms
fT
f
/12
Acceleration:
tydt
yda sin2
max2
2
rc
aqEradiative 2
04
1
jsin4
12
2max
0
trc
qyEradiative
Sinusoidal E/M field
Sinusoidal Electromagnetic Radiation
Sinusoidal E/M Radiation: Wavelength
fT
f
/12
Freeze picture in time:
Instead of period canuse wavelength:
cTf
c
Example of sinusoidal E/M radiation:
atomsradio stationsE/M noise from AC wires
Need to create oscillating motion of electrons
Radio frequencyLC circuit: can produce oscillating motion of chargesTo increase effect: connect to antenna
Visible lightHeat up atoms, atomic vibration can reach visible frequency rangeTransitions of electrons between different quantized levels
E/M Radiation Transmitters
How can we produce electromagnetic radiation of a desired frequency?
AC voltage(~300 MHz)
nolight
E/M radiation can be polarized along one axis…
…and it can be unpolarized:
Polarized E/M Radiation