Download 24.1-4, 24.11

Current in an LC Circuit
 t 
Q  Q0 cos

 LC 
I 
dQ
dt
Current in an LC circuit
Q0
 t 
I
sin 

LC  LC 
Period:
T  2 LC

Frequency: f  1 / 2 LC

Question (Chap. 23)
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?
Q0
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.
Question
Two 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) Clockwise
B) Counterclockwise
when viewed from above
Faraday’s Law: Applications
Single home current: 100 A service
Vwires=IRwires
Transformer: emfHV IHV = emfhomeIhome
Single home current in HV: <0.1 A
Power loss in wires ~ I2
Faraday’s Law: Applications
Faraday’s Law: Applications
Induction
microphone
Chapter 24
Classical Theory of
Electromagnetic Radiation
Maxwell’s Equations

 E  nˆdA 
q

 B  nˆA  0
inside
0
Gauss’s law for electricity
Gauss’s law for magnetism
𝑑
𝐸 ∙ 𝑑𝑙 = −
𝑑𝑡
𝐵 ∙ 𝑛𝑑𝐴
Complete Faraday’s law
 
Ampere’s law
B

d
l


I
0  inside_ path

(Incomplete Ampere-Maxwell law)
Ampere’s Law
 
 B  dl  0  Iinside_ path
Current pierces surface
No current inside
 
 B  dl  0
0 2 I
B
4 r
 
 B  dl  0 I
  0 2 I
 B  dl  4 r 2r  0 I
Maxwell’s Approach
Time varying magnetic field leads to curly electric field.
Time varying electric field leads to curly magnetic field?
 elec

  E  nˆdA
 Q 
Q
 A cos 0 
 elec  
0
 A 0 
d elec 1 dQ 1

 I
dt
 0 dt  0
I
d elec
‘equivalent’ current
I  0
dt
combine with current in Ampere’s law
 
 B  dl  0  Iinside_ path
The Ampere-Maxwell Law
 
d elec 

 B  dl  0  I inside_ path   0 dt 
Works!
Maxwell’s Equations
Four equations (integral form) :
q
Gauss’s law

 E  nˆdA 
Gauss’s law for magnetism

 B  nˆA  0
Faraday’s law
Ampere-Maxwell law
+ Lorentz force
inside
0


 
d 
 E  dl   dt  B  nˆdA
 
d elec 

 B  dl  0  I inside_ path   0 dt 


 
F  qE  qv  B
Fields Without Charges
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
A Simple Configuration of Traveling Fields
Key idea: Fields travel in space at certain speed
Disturbance moving in space – a wave?
1. Simplest case: a pulse (moving slab)
A Pulse and Gauss’s Laws

 E  nˆdA 
q
inside
0

 E  nˆdA  0
Pulse is consistent with Gauss’s law

 B  nˆA  0
Pulse is consistent with Gauss’s law
for magnetism
A Pulse and Faraday’s Law
emf  
d mag
dt
Since pulse is ‘moving’, B depends
on time and thus causes E
 mag  Bhv t
mag d mag

 Bhv
t
dt
emf
 
emf   E  dl  Eh
Is direction right?
Area does
not move
E=Bv
A Pulse and Ampere-Maxwell Law
=0
 
d elec 

 B  dl  0  I inside_ path   0 dt 
elec  Ehvt
 elec d elec

 Ehv
t
dt
 
 B  dl  Bh
Bh  0 0 Evh
B  0 0vE
A Pulse: Speed of Propagation
B  0 0vE
E=Bv
B  0 0vBv
1  0 0v 2
v
1
0 0
 3  108 m/s
E=cB
Based on Maxwell’s equations, pulse must propagate at speed of light
Question
At this instant, the magnetic flux mag through the entire
rectangle is:
A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh
Question
In a time t, what is mag?
A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h
Question
emf = mag/t = ?
A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h
Question
What is
𝐸 ∙ 𝑑 𝑙 around the full rectangular path?
A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2Evt; E)2Evt
Question
d mag
emf 
 Bvh
dt
r r
𝐸 ∙dl𝑑 𝑙 =
Eh𝐸ℎ
—
 Eg
What is E?
A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x
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
E  3x108 m / s  1x105 T  3000V / m
Force on charge q moving with velocity v perpendicular to B:
𝐹𝑚𝑎𝑔 𝑣𝐵
𝑣𝐵 𝑣
=
=
=
𝐹𝑒𝑙
𝐸
𝑐𝐵 𝑐
Direction of Propagation
Direction of speed is given by vector product
 
EB
Accelerated Charges
Electromagnetic pulse can propagate in space
How can we initiate such a pulse?
Short pulse of transverse
electric 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: E  B
E
v
B
Magnitude of the Transverse Electric Field
We can qualitatively predict the direction.
What is the magnitude?
Magnitude can be derived
from Gauss’s law
Field ~ -qa

Eradiative 

1  qa
40 c 2 r
1. The direction of the field is opposite to qa
2. The electric field falls off at a rate 1/r
Exercise
a
An electron is briefly accelerated in the
direction shown. Draw the electric and
magnetic vectors of radiative field.
E
B
1. The direction of the field is opposite to qa
 
2. The direction of propagation is given by E  B
Stability of Atoms
v
Circular motion: Is there radiation emitted?
Classical physics says “YES”
 orbiting particle must lose energy!
 speed decreases
 particle comes closer to center
Classical model of atom:
Electrons should fall on nucleus!
To explain the facts - introduction of
quantum mechanics:
Electrons can move around certain orbits
only and emit E/M radiation only when
jumping from one orbit to another
a
Sinusoidal Electromagnetic Radiation
Acceleration:
d2y
a  2   ymax  2 sin t 
dt


1  qa
Eradiative 
40 c 2 r

f 
2
T  1/ f

1 qymax  2
Eradiative 
sin t  ĵ
2
40 c r
Sinusoidal E/M field
Sinusoidal E/M Radiation: Wavelength
Instead of period can
use wavelength:
  cT  c
f

f 
2
T  1/ f
Example of sinusoidal E/M
radiation:
atoms
radio stations
E/M noise from AC wires
Freeze picture in time:
Electromagnetic Spectrum
E/M Radiation Transmitters
How can we produce electromagnetic radiation of a desired frequency?
Need to create oscillating motion of electrons
Radio frequency
LC circuit: can produce oscillating motion of charges
To increase effect: connect to antenna
Visible light
Heat up atoms, atomic vibration can reach visible frequency range
Transitions of electrons between different quantized levels
Polarized E/M Radiation
AC voltage
(~300 MHz)
E/M radiation can be polarized
along one axis…
no
light
…and it can be unpolarized:
Polarized Light
Making polarized light
Turning polarization
Polaroid sunglasses and camera filters:
reflected light is highly polarized: can block it
Considered: using polarized car lights and polarizers-windshields