Lecture 21-1 Resonance For given peak, R, L, and C, the current amplitude I peak will be at the maximum when the impedance Z is at the minimum. Resonance

Lecture 21-Lecture 21-11Resonance

For given peak, R, L, and C, the current amplitude Ipeak will be at the maximum when the impedance Z is at the minimum.

1res

LC Resonance angular

frequency:

This is called resonance.

i.e., load purely resistive ε and I in phase

22peak L CpeakI

Z

R X X

, ,1 peak

res peakres

aL Z nRC

d IR

L CX X

Lecture 21-Lecture 21-22Transformer

• AC voltage can be stepped up or down by using a transformer.

• AC current in the primary coil creates a time-varying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit.

Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load)

1 2

1 2

Bturn

d V V

dt N N

step-up

step-down

1 2 1 2N N V V

1 2 1 2N N V V

With resistive load R in secondary, current I2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must

somehow be exactly cancelled because is a defined voltage source. This occurs by another current I1 which is induced on the primary side due to I2.

01 LV

Lecture 21-Lecture 21-33Maxwell’s Equations (so far)

Gauss’s law0

inside

S

QE d A

��

Gauss’ law for magnetism 0S

B d A ��

Faraday’s law B

C

dE dl

dt

��

Ampere’s law*0C

B dl I��

Lecture 21-Lecture 21-44Parallel-Plate Capacitor Revisited

0 SSB dl I

��

0

0 0E

dQ dV A dVI C

dt dt d dtdE d

Adt dt

0E

d

dI

dt

will work.

Q

-Q

E��

0E ��

Lecture 21-Lecture 21-55Displacement Current

James Clerk Maxwell proposed that a changing electric field induces a magnetic field, in analogy to Faraday’s law: A changing magnetic field induces an electric field.

Ampere’s law is revised to become Ampere-Maxwell law

0 0 0 0( ) EdC

dB dl I I I

dt

��

is the displacement current.0

Ed

dI

dt

where

Lecture 21-Lecture 21-66Maxwell’s Equations

Basis for electromagnetic waves!

0

inside

S

QE d A

�� 0

SB d A ��

B

C

dE dl

dt

�� 0 0 0C

EB l Id

dtd

��

Lecture 21-Lecture 21-77Electromagnetic Waves From Faraday’s Law

m

m

E

B k

c

E B k��

sin( )mE E kx t y ��

B

C

dE dl

dt

�� y z

E B

x t

sin( )mB B kx t z ��

Lecture 21-Lecture 21-88Electromagnetic Waves From Ampère’s Law

E B k��

sin( )mB B kx t z ��

0 0E

C

dB dl

dt

�� 0 0

yzEB

x t


0 0

/m

m

E kc

B

Lecture 21-Lecture 21-99Electromagnetic Wave Propagation in Free Space

So, again we have a traveling electromagnetic wave

0 0

1

m

m

m

m

Ec

B k

E

B c

0 0

1c

speed of light

in vacuum

70

12 2 20

4 10 ( / )

8.85 10 ( / )

T m A

C N m

83.00 10 ( / )c m s

Speed of light in vacuum is currently defined rather than measured (thus defining meter and also the vacuum permittivity).

2

1B E

x c tE B

x t

Ampere’s Law

Faraday’s Law

2 2

2 2 2

1B B

x c t

Wave Equation

Lecture 21-Lecture 21-1010Plane Electromagnetic Waves

where sin( )mB B kx t z ��


2 2

2 2 2

1B B

x c t

2 2

2 2 2

1E E

x c t

x

E B k��

• Transverse wave

• Plane wave (points of given phase form a plane)

• Linearly polarized (fixed plane contains E)

Lecture 21-Lecture 21-1111 Non-scored Test Quiz

Electromagnetic wave travel in space where E is electric field, B is magnetic field. Which of the following diagram is true?

z

x

y

travel directionB

E

z

x

ytravel direction

B

E

z

x

y

travel direction

B

E

z

x

ytravel direction

B

E

(a).

(d).

(b).

(c).

Lecture 21-Lecture 21-1212Energy Density of Electromagnetic Waves

• Electromagnetic waves contain energy. We know already expressions for the energy density stored in E and B fields:

20

1,

2Eu E2

0

1

2B

Bu

EM wave

/

/m mB E c

B E c

22

020

1

2 2B EEu uE

c

• So Total energy density is

0

0

22

00 0

E B

B EBu u Eu B

cE

22

2

00

2

0000

rms rms rmsrms

EB E Bu E

c

B

cE

B

Lecture 21-Lecture 21-1313Energy Propagation in Electromagnetic Waves

• Energy flux density

= Energy transmitted through unit time per unit area

0 0

1 rms rmsE BEB

Pu c

A

• Intensity I = Average energy flux density (W/m2)

Define Poynting vector

0

1S E B

��

Direction is that of wave propagation

average magnitude is the intensity

20

22

00 0

1 1

2 2 2m m m

m

S I c E

B E Bc E c

Lecture 21-Lecture 21-1414Radiation Pressure

Electromagnetic waves carry momentum as well as energy. In terms of total energy of a wave U, the momentum is U/c.

During a time interval t , the energy flux through area A is U =IA t .

/ /p U c IA c t momentum imparted

p IAF

t c

2

0/ // (2 )r mp I c BF A radiation pressure EXERTED

If radiation is totally absorbed:

If radiation is totally reflected:

2 / 2 /p U c IA c t 2

, 2 /r

IAF p I c

c

Lecture 21-Lecture 21-1515

Maxwell’s Rainbow

Light is an Electromagnetic Wave

f c


Physics 241 –Quiz 18b – March 27, 2008

An electromagnetic wave is traveling through a particular point in space where the direction of the electric field is along the +z direction and that of the magnetic field is along +y direction at a certain instant in time. Which direction is this wave traveling in?

a) +x

b x

c y

d z

e) None of the above


Physics 241 –Quiz 18c – March 27, 2008

An electromagnetic wave is traveling in +y direction and the magnetic field at a particular point on the y-axis points in the +z direction at a certain instant in time. At this same point and instant, what is the direction of the electric field?

a z

b x

c y

d) +x

e) None of the above

Documents

Lecture 21-1 Resonance For given peak, R, L, and C, the current amplitude I peak will be at the maximum when the impedance Z is at the minimum. Resonance