1P28-

Class 28: Outline

Hour 1: Displacement Current Maxwell’s Equations

Hour 2: Electromagnetic waves

2P28-

Finally: Bringing it All Together

3P28-

Displacement Current

4P28-

Ampere’s Law: Capacitor

Consider a charging capacitor: I Use Ampere’s Law to calculate the

magnetic field just above the top plate

1) Red Amperian Area, Ienc= I 2) Green Amperian Area, I = 0

What’s Going On?

0Ampere's law: enc d Iµ⋅ =∫ B s

5P28-

Displacement Current

0 E

d ddQ I

dt dtε Φ = ≡

We don’t have current between the capacitor plates but we do have a changing E field. Can we “make” a current out of that?

0 0 0

E

QE Q EA A

ε εε

= ⇒ = = Φ

This is called (for historic reasons) the Displacement Current

6P28-

Maxwell-Ampere’s Law

0

0 0 0

( )encl d C

E encl

d I I

dI dt

µ

µ µ ε

⋅ = +

Φ = +

∫ B s

7P28-

PRS Questions: Capacitor

8P28-

Maxwell’s Equations

9P28-

Electromagnetism Review

• E fields are created by: (1) electric charges (2) time changing B fields

• B fields are created by (1) moving electric charges

(NOT magnetic charges) (2) time changing E fields

• E (B) fields exert forces on (moving) electric charges

Gauss’s Law Faraday’s Law

Ampere’s Law

Maxwell’s Addition

Lorentz Force

10P28-

Maxwell’s Equations

0

0 0 0

(Gauss's Law)

(Faraday's Law)

0 (Magnetic Gauss's Law)

(Ampere-Maxwell Law)

( (Lorentz force Law)

in

S

B

C

S

E enc

C

Qd

dd dt

d

dd I dt

q

ε

µ µ ε

⋅ =

Φ ⋅ = −

⋅ =

Φ ⋅ = +

= + ×

∫∫

∫

∫∫

∫

E A

E s

B A

B s

F E v B)

11P28-

Electromagnetic Radiation

12P28-

A Question of Time…

http://ocw.mit.edu/ans7870/8/ 8.02T/f04/visualizations/light/ 05-CreatingRadiation/05-pith_f220_320.html

13P28-

14P28-

Electromagnetic Radiation: Plane Waves

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html

15P28-

Traveling Waves

Consider f(x) =

x=0

What is g(x,t) = f(x-vt)?

x=0

t=0

x=vt0

t=t0

x=2vt0

t=2t0

f(x-vt) is traveling wave moving to the right!

16P28-

Traveling Sine Wave Now consider f(x) = y = y0sin(kx):

x

Amplitude (y0) 2Wavelength ( )

wavenumber ( )k πλ =

What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y0sin(k(x+vt)) = y0sin(kx+kvt)

17P28-

Traveling Sine Wave

Amplitude (y0)

1Period ( ) frequency ( )

2 angular frequency ( )

T f π

ω

=

=

( )0 siny y kx kvt= +

0 0sin( ) sin( )y y kvt y tω= ≡At x=0, just a function of time:

18P28-

Traveling Sine Wave

0 sin( )y y kx tω= −Wavelength: Frequency :

2Wave Number:

Angular Frequency: 2 1 2Period:

Speed of Propagation:

Direction of Propagation:

f

k

f

T f

v fk x

λ

π λ ω π

π ω

ω λ

=

=

= =

= =

+

i i

i

i

i

i

i

19P28-

Electromagnetic Waves

Remember: f cλ =

Hz

20P28-

Electromagnetic Radiation: Plane Waves

Watch 2 Ways: 1) Sine wave

traveling to right (+x)

2) Collection of out of phase oscillators (watch one position)

Don’t confuse vectors with heights – they are magnitudes of E (gold) and B (blue)

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html

21P28-

PRS Question: Wave

22P28-

Group Work: Do Problem 1

23P28-

Properties of EM Waves

8

0 0

1 3 10 m v c sµ ε

= = = ×

0

0

EE cB B

= =

Travel (through vacuum) with speed of light

At every point in the wave and any instant of time, E and B are in phase with one another, with

E and B fields perpendicular to one another, and to the direction of propagation (they are transverse):

Direction of propagation = Direction of ×E B

Direction of Propagation E E= E sin( k p̂ ⋅r −ωt); B B = B sin( k p̂ ⋅r −ωt)ˆ

0 ( ) ˆ0 ( )

P28-24

( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ

ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ ˆˆ

z x y z x y

⋅

− − − − − −

E B p p r i j k

j k i k i j j i k

k j i i k j

ˆ ˆ ̂× =E B p

25P28-

PRS Question: Direction of Propagation

26P28-

In Class Problem: Plane EM Waves

27P28-

Energy & the Poynting Vector

28P28-

Energy in EM Waves 2 2

0 0

1 1 ,2 2E Bu E u Bε

µ = =Energy densities:

Consider cylinder: 2 2

0 0

1( ) 2E B

BdU u u Adz E Acdtε µ

⎛ ⎞ = + = +⎜ ⎟

⎝ ⎠

What is rate of energy flow per unit area?

0 02

c EB cEB c

ε µ

⎛ ⎞ = +⎜ ⎟

⎝ ⎠

( )2 0 0

0

1 2 EB cε µµ

= + 0

EB

µ =

1 dUS A dt

= 2

2 0

02 c BEε

µ ⎛ ⎞

= +⎜ ⎟ ⎝ ⎠

29P28-

Poynting Vector and Intensity

0

: Poynting vector µ × = E BS

units: Joules per square meter per sec

Direction of energy flow = direction of wave propagation

Intensity I: 2 2 0 0 0 0

0 0 02 2 2 E B E cB I S

cµ µ µ ≡< > = = =

30P28-

Energy Flow: Resistor

0µ × = E BS

On surface of resistor is INWARD

31P28-

PRS Questions: Poynting Vector

32P28-

Energy Flow: Inductor On surface of inductor with increasing current is INWARD

0µ × = E BS

33P28-

Energy Flow: Inductor On surface of inductor with decreasing current is OUTWARD

0µ × = E BS