Slide 29-1

Chapter 29: Maxwell’s Equation and EM Waves

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Equations of electromagnetism: a review •  We’ve now seen the four fundamental equations of electromagnetism,

here listed together for the first time. •  But one is incomplete: Ampère’s law needs refining

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Clicker Question

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Maxwell’s Adjustment to Ampere’s Law

•  The result of Ampere’s law depends on which surface is used to determine the encircled current! •  Can’t have contradictory results –

either there is a B field or there isn’t!

•  Notice that electric field is changing inside conductor.

•  Ampere postulated that a changing electric flux induces a magnetic field (similar to how a changing magnetic flux induces an electric field)

��B · �dr = µ0Ienc

Slide 29-5

Maxwell’s Adjustment to Ampere’s Law •  Need to add a term to the right side of

Ampere’s law to account for the changing electric flux. This is the displacement current.

•  Notice, for a parallel plate capacitor, the electric flux is

•  The the rate of change of the flux is •  So Maxwell added a displacement current

term to Ampere’s law:

•  Corresponding displacement current density:

��B · d�� = µ0(I + Id)enc = µ0

�I + �0

dΦE

dt

�

enc

ΦE = EA =σ

�0=

Q

�0dΦE

dt=

I

�0

(Ampere-Maxwell Law)

�Jd = �0d �E

dt

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Slide 29-7

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•  Determine the electric field between the plates:

��B · d�� = µ0�0

dΦE

dt

2πrB = µ0�0πr2 I

�0πR2

B = µ0Ir

2πR2(r < R)

Slide 29-9

Consider a large parallel plate capacitor as shown, charging so that Q = Q0+βt on the positively charged plate. Assuming the edges of the capacitor and the wire connections to the plates can be ignored, what is the magnitude of the magnetic field B halfway between the plates, at a radius r?

Q -Q

a

d

r I I

z

s

µ0!2"r

µ0!r2d2

µ0!d2a2

µ0!a2"r2

A.

B.

C.

D.

E. None of the above

Slide 29-10

Maxwell’s equations •  The four complete laws of electromagnetism are

collectively called Maxwell’s equations. •  They describe all electromagnetic fields in the universe, outside

the realm of quantum physics.

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Maxwell’s equations in vacuum •  In a vacuum there’s no electric charge and therefore also

no electric current. Maxwell’s equations in vacuum

A changing electric field is a source for a magnetic field, and a changing magnetic field is a source for an electric field. These equations infer the possibility of electromagnetic waves!

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Wave Equation •  Maxwell’s equations can be manipulated to derive the

following wave equations:

•  Very similar to wave equation for sound waves! Except here it is the electric and magnetic field that is “wiggling.”

•  For plane waves traveling in one direction (say the x direction):

∇2 �E = µ0�0∂2 �E

∂t2∇2 �B = µ0�0

∂2 �B

∂t2

∇2 =∂2

∂x2x +

∂2

∂y2y +

∂2

∂z2z

∂2 �E

∂x2= µ0�0

∂2 �E

∂t2∂2 �B

∂x2= µ0�0

∂2 �B

∂t2

Slide 29-13

Plane electromagnetic waves •  A plane electromagnetic wave are waves propagating in

one direction with one wavelength. (E and B do not vary with respect to the other two dimensions)

•  The fields are perpendicular to each other and to the direction of propagation.

•  Mathematically

( ) ( )

( ) ( )p

p

ˆ, sinˆ, sin

E x t E kx t j

B x t B kx t k

ω

ω

= −

= −

r

r

k =2π

λω =

2π

T

( �E × �B gives direction of propagation)

Slide 29-14

y x t x ty x t x t1

2

2 24 2

( , ) sin( )( , ) sin( )

= !

= !

Two traveling waves 1 and 2 are described by the equations.

All the numbers are in the appropriate SI (mks) units. Which wave has the higher speed? A) Wave 1 B) Wave 2 C) Both have the same speed.

Clicker Question

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Plane Waves as Solutions

•  plane wave expression is a solution if

•  This also implies that

ωk=

1ε0µ0

��B · d�� = µ0�0

d

dt

�

S

�E · d �A�

�E · d�� = − d

dt

�

S

�B · d �A

= c = 3.0× 108 m/s

=⇒ ∂E

∂x= −∂B

∂t=⇒ ∂B

∂x= −�0µ0

∂E

∂t

fλ = c

kEp = ωBp kBp = �0µ0ωEp

Bp =Ep

c

⇒ ∂

∂x

∂E

∂x= − ∂

∂t(−�0µ0

∂E

∂t)

∂2E

∂x2− �0µ0

∂2E

∂t2= 0

E(x, t) = E0 sin(kx− ωt)

B(x, t) = B0 sin(kx− ωt)

Slide 29-16

General Results for EM Radiation •  Transverse waves (E and B perpendicular to direction of

propagation): direction of propagation: •  E and B perpendicular to each other. •  E = cB; E and B oscillate in phase •  Propagation speed is the speed of light in a vacuum

•  independent of wavelength: •  radio waves, light, infrared radiation, X-rays are all the same

phenomena! •  In matter, the speed of light is

•  No medium required for propagation (no ether)

c = λ f = ω/k

v =1√

�µ=

c

n

�E × �B ∝ k

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Clicker question •  At a particular point, the electric field of an

electromagnetic wave points in the direction, while the magnetic field points in the direction. Which of the following describes the propagation direction?

A.  B.  C.  either or but you can’t tell which D. 

x+x−

y−x+ x−

y+z−

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Clicker question •  A planar electromagnetic wave is propagating through

space. Its electric field vector is given by

Its magnetic field vector is

�E = Ep cos(kz − ωt)i

1) �B = Bp cos(kz − ωt)j

2) �B = Bp cos(ky − ωt)k

3) �B = Bp cos(ky − ωt)j

4) �B = Bp cos(kz − ωt)k

5) �B = Bp sin(kz − ωt)i

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The Electromagnetic Spectrum

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•  Which type of radiation travels with the highest speed? 1.  visible light 2.  X-rays 3.  Gamma-rays 4.  radio waves 5.  they all have the same speed

Clicker question

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Producing electromagnetic waves •  Electromagnetic waves are generated ultimately by accelerated

electric charge. •  Details of emitting systems depend on wavelength, with most efficient

emitters being roughly a wavelength in size. •  Radio waves are generated by alternating currents in metal antennas. •  Molecular vibration and rotation produce infrared waves. •  Visible light arises largely from atomic-scale processes. •  X rays are produced in the rapid deceleration of electric charge. •  Gamma rays result from nuclear processes.

A radio transmitter and antenna Electric fields of an oscillating electric dipole

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Antennae

An electric field parallel to an antenna (electric dipole) will “shake” electrons and produce an AC current.

A magnetic dipole antenna (for AM radios) should be oriented so that the B-field passes into and out of the plane of a loop, inducing a current in the loop.

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Energy in EM waves •  Electromagnetic waves transport energy

•  The Poynting vector describes the rate of energy flow per unit area (W/m2 in SI):

•  For plane waves (traveling in x direction with E oriented in z direction):

•  Averaging over the time variations of the oscillating fields gives the average value, also called the average intensity:

•  Far from a localized source of radiation, electric field decreases as 1/r. Thus, (as required by conservation of energy)

�S =1µ0

EpBp cos2(kx− ωt)i

S = uc = (uE + uB)c = �E2c

I =< S >=12

1µ0

EpBp =12

E2p

2µ0c=

12�0E

2pc

I ∝ 1r2

�S =1

µ0

�E × �B

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Two radio dishes are receiving signals from a radio station which is sending out radio waves in all directions with power P. Dish 2 is twice as far away as Dish 1, but has twice the diameter. Which dish receives more power? A: Dish 1 B: Dish 2 C: Both receive the same power

Dish 1 Dish 2

Clicker Question

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Example: The intensity of the sunlight that reaches Earth’s upper atmosphere is 1400 W/m2.

(a) What is the total average power output of the Sun, assuming it to be an isotropic source?

( )( )( )

W100.4m 1050.1 W/m14004

4

26

2112

2

×=

×=

==

π

πRIIAPav

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(b) What is the intensity of sunlight incident on Mercury, which is 5.8x1010 m from the Sun?

(c) What is the maximum electric field (if monochromatic light)

( )2

210

26

2

W/m9460m 108.54

W100.44

=

×

×=

==

π

πrP

API avav

Example continued:

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Flux and Solar Heating

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Polarization

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Light passed through a polarizing filter has an intensity of 2.0 W/m2. How should a second polarizing filter be arranged to decrease the intensity to 1.0 W/m2?

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An unpolarized beam of light passes through 2 Polaroid filters oriented at 45o with respect to each other. The intensity of the original beam is Io. What is the intensity of the light coming through both filters?

A: (1/1.4)Io B: (1/2)Io C: (1/4)Io D: (1/8)Io E: None

Io

c

Clicker Question

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Clicker question •  Two polarizers are oriented at right angles, so no light

gets through the combination. A third polarizer is inserted between the two, with its preferred direction at 45° to the others. How will this “sandwich” of polarizers affect a beam of initially unpolarized light?

A.  All of the initial light will be blocked. B.  Half of the initial light is blocked. C.  One-quarter of the initial light is blocked. D.  None of the initial light will be blocked.

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