MIDTERM 3 UTC 4.132 Thu-Nov 15, 7:00PM - 9:00PM
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Transcript of MIDTERM 3 UTC 4.132 Thu-Nov 15, 7:00PM - 9:00PM
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MIDTERM 3
UTC 4.132 Thu-Nov 15, 7:00PM - 9:00PM
Course Summaries Unit 1, 2, 3 ProvidedTA session Monday Homework Review
(attendance optional)
Bring pencils, calculators (memory cleared)
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Chapter 24
Classical Theory of Electromagnetic Radiation
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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 ]
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No current inside
0 ldB
Current pierces surface
IldB 0
rIB 2
40
IrrIldB 0
0 224
pathinsideIldB _0Ampere’s Law
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Time varying magnetic field leads to curly electric field.
Time varying electric field leads to curly magnetic field?
dAnEelec ˆ
00
0cosQA
AQ
elec
dtdQ
dtd elec
0
1
I
0
1
I
dtdI elec
0 ‘equivalent’ current
pathinsideIldB _0 combine with current in Ampere’s law
Maxwell’s Approach
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dtdIldB elec
pathinside 0_0
Works!
The Ampere-Maxwell Law
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Four equations (integral form) :
Gauss’s law
Gauss’s law for magnetism
Faraday’s law
Ampere-Maxwell law
0
ˆ
insideqdAnE
dAnBdtdldE ˆ
dtdIldB elec
pathinside 0_0
+ Lorentz force BvqEqF
Maxwell’s Equations
0ˆ AnB
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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
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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
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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
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dtd
emf mag
Since pulse is ‘moving’, B depends on time and thus causes E
Area doesnot move
tBhvmag
Bhvdt
dt
magmag
emf
EhldEemf
E=Bv
Is direction right?
A Pulse and Faraday’s Law
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dtdIldB elec
pathinside 0_0
=0
tEhvelec
Ehvdt
dt
elecelec
BhldB
EvhBh 00
vEB 00
A Pulse and Ampere-Maxwell Law
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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
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Question
At this instant, the magnetic flux mag through the entire rectangle is:
A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh
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Question
In a time t, what is mag?
A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h
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Question
emf = mag/t = ?
A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h
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Question
What is around the full rectangular path?
A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2Evt; E)2Evt
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Question
emf dmag
dtBvh
rEgd
rl Eh—
What is E?
A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x
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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=𝑣𝐵
𝐸¿𝑣𝐵𝑐𝐵=
𝑣𝑐
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Direction of speed is given by vector product BE
Direction of Propagation
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Electromagnetic pulse can propagate in spaceHow can we initiate such a pulse?
Short pulse of transverseelectric field
Accelerated Charges
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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
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We can qualitatively predict the direction.What is the magnitude?
Magnitude can be derived from Gauss’s law
Field ~ -qa
rcaqEradiative 2
041
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
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Field of an accelerated charge
1
2
3
4
vT
𝑎A B
Φ𝑆
𝛼
Φ𝐵
Φ𝐴
Accelerates for t, then coasts for T at v=at to reach B.
cT
ct
𝜃𝑣𝑇sin𝜃
r>cT ; outer shell
inner shell of acceleration zone
> since B is closer, but = since areas compensate
Φ𝐵+Φ𝐴=0 No chargeΦ𝑆=0
𝐸𝑆
𝐸𝑟𝑎𝑑
𝐸𝑡𝑎𝑛
tan (𝛼)=𝐸 𝑡𝑎𝑛
𝐸𝑟𝑎𝑑¿𝑣𝑇𝑠𝑖𝑛(𝜃)
𝑐𝑡
𝐸𝑡𝑎𝑛=𝐸𝑟𝑎𝑑𝑣𝑇𝑠𝑖𝑛 (𝜃)
𝑐𝑡
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Field of an accelerated charge
1
2
3
4
vT
𝑎A B
Φ𝑆
𝛼
Φ𝐵
Φ𝐴
cT
ct
𝜃𝑣𝑇sin𝜃
𝐸𝑆
𝐸𝑟𝑎𝑑
𝐸𝑡𝑎𝑛
𝐸𝑡𝑎𝑛=𝐸𝑟𝑎𝑑𝑣𝑇𝑠𝑖𝑛 (𝜃)
𝑐𝑡
𝐸𝑟𝑎𝑑=1
4𝜋 𝜀0
𝑞𝑟2
𝐸𝑡𝑎𝑛= 14𝜋 𝜀0
𝑞𝑟2𝑣𝑇𝑠𝑖𝑛 (𝜃)
𝑐𝑡
𝑎=𝑣 /𝑡𝐸𝑡𝑎𝑛= 𝑞
4𝜋 𝜀0
𝑎𝑠𝑖𝑛(𝜃)𝑐2𝑟
c
𝑎𝑠𝑖𝑛 (𝜃 )=𝑎⊥𝐸𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑣𝑒=
𝑞4𝜋 𝜀0
−𝑎⊥
𝑐2𝑟