Chapter 29: Electromagnetic Waves Thursday November 10th

• Transformers - demo • Maxwell’s equations • Electromagnetic waves

• Wave equations • Speed of light • Relations between quantities • Energy flux and intensity

Reading: up to page 515 in the text book (Ch. 28/29)

Mini-exam 5 next Thursday (AC circuits and EM waves) 55 unregistered iClickers – any takers?

Transformers

Soft iron

• Flux the same on both sides, but number of turns, N, is different • Total flux through primary and secondary coils depends on N1 and N2

V1= N

1!; V

2= N

2!; "

V2

V1

=N

2

N1

The basic equations of electromagnetism so far.....

Gauss’ law for B (no magnetic equivalent of charge):

!B =!B "d

!A"# = 0

!E =

!E "d

!A"# =

qenc

$o

Gauss’ law:

!B !d

!l = µ

o!! Ienc

Ampère’s law:

Faraday’s law:

! =

!E !d!l ="" #

d$B

dt

The basic equations of electromagnetism so far.....

In vacuum:

} Symmetry

Is it possible that a time-varying electric field could produce a magnetic field, thereby restoring symmetry?

!E =!E !d

!A!! = 0

!B =!B "d

!A"# = 0

} No Symm- etry

!B !d

!l = 0!"

!E !d

!l =!! "

d#B

dt

I I

I I

Maxwell’s displacement current

!B !d

!l = µ

oI

enc"" = µo#

J

= µo

!J !d

!A"

Stokes’ theorem: The choice of surface should not matter

I I

I I

Maxwell’s displacement current

!B !d!s = µ

oI + I

d( )""

I

d= !

o

d!E

dt

!B !d

!l = µ

oI + µ

o"" !o

d#E

dt

Stokes’ theorem: The choice of surface should not matter

Maxwell’s equations

The main thing to note here is the symmetry in the last two equations: a time varying magnetic field produces an electric field; similarly, a time varying electric field produces a magnetic field.

Maxwell’s equations in vacuum

!E =!E !d

!A!! = 0

!B =!B "d

!A"# = 0

!B !d

!l = µ

o!

o

d"E

dt"#

!E !d!l ="" #

d$B

dtThe main thing to note here is the symmetry in the last two equations: a time varying magnetic field produces an electric field; similarly, a time varying electric field produces a magnetic field.

Electromagnetic waves

Propagation Electromagnetic perturbation breaks completely free from the charge/current

Dipole radiation

Field does not appear instantaneously

Maxwell’s equations guarantee that electric and magnetic fields are perpendicular to each other and perpendicular to the direction of propagation; they are polarized.

Spherical waves

Plane waves

!E!

!B

Electromagnetic waves

Maxwell’s equations in vacuum

!E =!E !d

!A!! = 0

!B =!B "d

!A"# = 0

!B !d

!l = µ

o!

o

d"E

dt"#

!E !d!l ="" #

d$B

dt

!"!B = µ

o!o

d!Edt

#

$%%%

&

'(((

!"

!E =#

d!Bdt

$

%&&&

'

()))

Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns

Let there be light!!

!"

!B = µ

o!

o

d!Edt

!"

!E =#

d!Bdt

Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns

Maxwell’s equations in vacuum can be solved simultaneously to give identical differential equations for E and B:

!2!E = µ

o!

o

"2!E"t 2

and !2!B = µ

o!

o

"2!B"t 2

!2 ="2

"x 2+!2

!y2+!2

!z 2The Laplacian differential operator

The Electromagnetic Wave Equations

Let there be light!!

!"

!B = µ

o!

o

d!Edt

!"

!E =#

d!Bdt

Stokes’ Theorem: Gives differential form of Maxwell’s equ’ns

Maxwell’s equations in vacuum can be solved simultaneously to give identical differential equations for E and B:

!2!E!x 2= µ

o!o

!2!E!t 2

and!2!B!x 2= µ

o!o

!2!B!t 2

The Electromagnetic Wave Equations

Waves in one-dimension (traveling along x)

Review of waves (PHY2048)

2k πλ

= k is the angular wavenumber.

Tran

sver

se w

ave

2Tπω = w is the angular frequency.

12

fT

ωπ

= =

v = !!k= ! "T

= ! f "

frequency

velocity

my(x,t) = Asin(kx ±! t +")

Amplitude Displacement Phase }

Phase shift

angular wavenumber angular frequency

Electromagnetic waves

x

Direction of motion

!E x,t( )= Ep sin kx !!t( ) j!B x,t( )= Bp sin kx !!t( )k

• The E and B fields are still related via Ampère’s and Faraday’s laws.

• For a plane wave traveling in the x direction (see text):

Electromagnetic waves • The E and B fields are still related via Ampère’s and Faraday’s laws.

• For a plane wave traveling in the x direction (see text):

!Ez

!x=!B

y

!t,

!Ey

!x=!"B

z

!t, E

x= B

x= 0

x

Direction of motion

Electromagnetic waves • Plugging these wave solutions into the wave equation:

!2Ey=!k 2E

y= µ

o!o

!2Ey

!t 2=!!2µ

o!oEy

!!2

k 2= c2 =

1µ

o!

o

, or c =1µ

o!

o

• Plugging these wave solutions into Faraday’s law: !E

y

!x= kE

pcos kx !!t( )=!"Bz

!t= !B

pcos kx !!t( )

!Ep

Bp

=!k= c