Electricity and Magnetism

1
Electricity and Magnetism

• Announcements
• Review
• Self and mutual inductance (30-2,30-1)
• Energy in B-Field (30-3)
• LR circuit (30-4)
• LRC circuits and Oscillations (30-4,30-5)
• AC circuits (31-1 to 31-6)
• Displacement current (32-1)
• Maxwells equations (32-3)
• EM waves (32-4 to 32-7)

2
Announcements

• Quiz 4
• Fri, 5/10 Walker Gym
• Evening Review today
• 730 930 PM Room 1-190
• By popular demand Formula sheet by tomorrow
  morning
• There shall be no test after the Friday
  preceeding the reading period
• no makeup?

3
Review for Quiz 4
4
Mutual Inductance

• Transformer action

xs /xp Ns/Np
B
xs - Ns dFB/dt
Secondary
Ns
same
Primary
Np

xp - Np dFB/dt
IAC I0 sin(wt)
Flux through single turn
5
Mutual Inductance
xs /xp Ns/Np

• Transformer action
• Transformers allow change of amplitude for AC
  voltage
• ratio of secondary to primary windings
• Constructed such that FB identical for primary
  and secondary
• What about general case of two coils?

6
Mutual Inductance
FB B B I1
B
Def.M12 N2 FB/I1
I1
N2
N1

7
Mutal Inductance

• Coupling is symmetric M12 M21 M
• M depends only on Geometry and Material
• Mutual inductance gives strength of coupling
  between two coils (conductors)
• M relates x2 and I1 (or x1 and I2)
• Units M V/(A/s) V s /A H (Henry)

x2 - N2 dFB/dt - M dI1/dt
8
Example Two Solenoids
Length l
Q How big is M N2 FB/I1 ? A M m0 N1N2 A/l

N2
N1
Area A
9
Demo Two Coils
Radio
Speaker

• Signal transmitted by varying B-Field
• Coupling depends on Geometry (angle, distance)

10
Demo Marconi Coils
Secondary
Spring
NP
Primary
NP ltlt NS
Iron
NS
Switch
Battery
11
Self Inductance
Circuit sees flux generated by it self
B
Def. L N FB/I Self-Inductance
I

12
Example Solenoid
B
Q How big is L ? A L m0 N2 A/L
I

13
Self Inductance

• L is also measured in H
• L connects induced EMF and variation in current
• x - L dI/dt
• Remember Lenz Rule
• Induced EMF will act against change in
  current -gt effective inertia
• Delay between current and voltage

14
Demo Levitating coil
IAC I0 sin(wt)
Iind -dFB/dt -cos(wtf)
IAC
t
Without delay (f 0) No net force
Iind
15
Demo Levitating coil
IAC I0 sin(wt)
Iind -dFB/dt -cos(wtf)
IAC
t
With delay (f gt0) Net repulsion (currents
are opposite most of the time)
Iind
16
RL Circuits
L
L dI/dt
V
R
2
R I
1
V0
Kirchoffs Rule V0 xind R I -gt V0 L dI/dt
R I Q What is I(t)?
17
RL Circuits
I(t)
63
I(t)V0/R 1-exp(-t/t)
t L/R
t
x(t)
x(t)V0 exp(-t/t)
37
t L/R
t
18
RL Circuits

• Inductance leads to delay in reaction of
  current to change of voltage V0
• All practical circuits have some L and R
• change in I never instantaneous

19
Back EMF
L
R
2
1

• What happens if we move switch to position 2?

V0
20
I(t)
I(t)V0/R exp(-t/t)
63
t L/R
t
x(t)
t L/R
37
t L/R
t
1
2
21
RL circuit

• L counteracts change in current both ways
• Resists increase in I when connecting voltage
  source
• Resists decrease in I when disconnecting voltage
  source
• Back EMF
• Thats what causes spark when switching off e.g.
  appliance, light

22
Energy Storage in Inductor

• Energy in Inductor
• Start with Power P x I L dI/dt I dU/dt
• -gt dU L dI I
• -gt U ½ L I2
• Where is the Energy stored?
• Example Solenoid
• U/Volume ½ B2/m0

23
RLC circuits

• Combine everything we know...
• Resonance Phenomena in RLC circuits
• Resonance Phenomena known from mechanics (and
  engineering)
• Great practical importance
• video...

24
Summary of Circuit Components

V
V(t)
R
VR IR
L
VL L dI/dt
C
VC 1/C Idt
25
R,L,C in AC circuit

• AC circuit
• I(t) I0 sin(wt)
• V(t) V0 sin(wt f)
• Relationship between V and I can be characterized
  by two quantities
• Impedance Z V0/I0
• Phase-shift f

same w!
26
AC circuit
f/w
2p/w
I(t)I0 sin(wt)
I0
V0
V(t)V0 sin(wt f)
Impedance Z V0/I0 Phase-shift f
27
First Look at the components

I(t)
C


I(t)
I(t)
R
L
V I R
V Q/C 1/C Idt
V L dI/dt

• Z R
• 0
• V and I in phase

• Z 1/(wC)
• - p/2
• V lags I by 90o

• Z w L
• p/2
• I lags V by 90o

28
RLC circuit
L

V(t)
R
C
29
RLC circuit
L
V L dI/dt - IR - Q/C 0 L d2Q/dt2 -1/C Q
R dQ/dt V

V(t)
R
C
2nd order differential equation
30
RLC circuit
L
V L dI/dt - IR - Q/C 0 L d2Q/dt2 -1/C Q
R dQ/dt V

V(t)
R
C
Spring
Friction
Inertia
Water
Fext
m d2x/dt2 -k x f dx/dt Fext
Spring
Mass m
31
Resonance
I0
Imax V0/R
w
p/2
f
w
Like L
Like C
-p/2
w (LC)1/2
High Frequency
Low Frequency
32
RLC circuit

• V0 sin(wt) I0wL -1/(wC) cos(wt f) R
  sin(wt f)
• Solution (requires two tricks)
• I0 V0/(wL -1/(wC)2 R2)1/2 V0/Z
• tan(f) wL -1/(wC)/R
• -gt For wL 1/(wC), Z is minimal and f 0
• i.e. w0 1/(LC)1/2 Resonance Frequency

33
Resonance

• Practical importance
• Tuning a radio or TV means adjusting the
  resonance frequency of a circuit to match the
  frequency of the carrier signal

34
LC-Circuit

• What happens if we open switch?

L dI/dt - Q/C 0 L d2Q/dt2 Q/C 0
L
C
d2x/dt2 w02 x 0
Harmonic Oscillator!
V0
35
LC-Circuit
1/2 L I2
L
1/2 k x2
1/2 m v2
C
Spring k
Mass m
1/2 Q2/C
Potential Energy Kinetic Energy
Energy in E-Field Energy in B-Field
Oscillation
Oscillation
36
LC-Circuit
L
C
Spring k
Mass m
d2Q/dt2 1/(LC) Q 0 w02 1/(LC)
d2x/dt2 k/m x 0 w02 k/m
37
LC-Circuit
1/2 L I2
L

• Total energy U(t) is conserved
• Q(t) cos(wt)
• dQ/dt sin(wt)
• UL (dQ/dt)2 sin2
• UC Q(t)2 cos2
• cos2(wt) sin2(wt) 1

C
1/2 Q2/C
Energy in E-Field Energy in B-Field
Oscillation
38
Electromagnetic Oscillations

• In an LC circuit, we see oscillations
• Q Can we get oscillations without circuit?
• A Yes!
• Electromagnetic Waves

Energy in E-Field Energy in B-Field
39
Displacement Current

• Amperes Law broken How can we fix it?

I
I
Q C V
Displacement Current ID e0 dFE/dt
40
Displacement Current

• Extension of Amperes Law

I
I
Q C V
Displacement Current ID e0 dFE/dt Changing
field inside C also produces B-Field!
41
Displacement Current

• Example calculation B(r) for r gt R

D
I
I
R
Q C V
-gt B(r) R2/(2rc2) dV/dt
42
Maxwells Equations

• Symmetry between E and B
• although there are no magnetic monopoles
• Basis for radio, TV, electric motors, generators,
  electric power transmission, electric circuits etc

43
Maxwells Equations
1/c2

• M.E.s predict electromagnetic waves, moving with
  speed of light
• Major triumph of science

44
Electromagnetic Waves

• Until end of semester
• What are electromagnetic waves?
• How does their existence follow from Maxwells
  equations?
• What are the properties of E.M. waves?
• Prediction was far from obvious
• No hint that E.M. waves exist
• Involves quite a bit of math

45
Reminder on waves

• Types of waves
• Transverse
• Longitudinal
• compression/decompression

46
Reminder on waves

• For a travelling wave (sound, water)
• Q What is actually moving?
• -gt Energy!
• Speed of propagation v l f
• Wave equation

Couples variation in time and space
47
Reminder on waves
At a moment in time
Wavelength l
Amplitude
D(X)
Position x
At a point in space
Period T 1/f
Amplitude
D(t)
Time t
48
Wave Equation

• Wave equation
• Speed of propagation v l f
• How can we derive a wave equation from Maxwells
  equations?

Couples variation in time and space
49
Wave properties

• What do we want to know about waves
• Speed of propagation?
• Transverse or longitudinal oscillation?
• What is oscillating?
• What are typical frequencies/wavelengths?

50
Differential Form of M.E.
Gauss, Stokes
51
Differential Form of M.E.
Flux/Unit Volume
Charge Density
Loop Integral/Unit Area
Current Density
52
Maxwells Equations in Vacuum

• Look at Maxwells Equations without charges,
  currents

Now completely symmetric!
53
Maxwells Equations in Vacuum
Solve for a simple geometry
I.
x
II.
III.
z
VI.
y
Allow variations only in z-direction
54
Electromagnetic Waves

• We found wave equations

same for Ex, Bx
v c
E and B are oscillating!
55
Electromagnetic Waves

• Note (Ex, By) and (Ey, Bx) independent

Ey, Bx
Ex, By
E B
56
Plane waves

• Example solution Plane waves
• We can express other functions as linear
  combinations of sin,cos
• White light is combination of waves of
  different frequency
• Demo...

57
Plane waves

• Example solution Plane waves

-
58
E.M. Wave Summary

• E B and perpendicular to direction of
  propagation
• Transverse waves
• Speed of propagation v c l f
• E/B c
• E.M. waves travel without medium

59
Typical E.M. wavelength

• FM Radio
• f 100 MHz
• l c/f 3m

60
Typical E.M. wavelength

• FM Radio
• f 100 MHz
• l c/f 3m
• Antenna O(m)

61
Typical E.M. wavelength

• FM Radio
• f 100 MHz
• l c/f 3m
• Antenna O(m)
• Cell phone?

62
Typical E.M. wavelength

• FM Radio
• f 100 MHz
• l c/f 3m
• Antenna O(m)
• Cell phone
• Antenna O(0.1m)
• f c/ l 3 GHz

63
Energy in E.M. Waves

• Remember
• Energy/Volume given by 1/2 e0 E2 and
• 1/2 B2/m0
• Energy density for E.M. wave
• u e0 E2
• What about power?