24.4 The Energy Carried by Electromagnetic Waves

A related quantity: Intensity of EM waves (How bright is the light hitting you?) Analogous to sound intensity (how loud is the sound wave arriving at you)?

1

y

x

z area A

tS

∆==

(Area)energy EM Total

AreaPower

t∆=

(Area)olume)density)(Venergy (EM

We compute the intensity by finding the energy contained the gray box (of length c∆t, and area A) passing through the yellow frame (area A) in time ∆t.

tcx ∆=∆ ( ) uctA

AtcuS ⋅=∆⋅∆⋅

=

Intensity: S (unit: W/m2)

ucS ⋅=→

202

1 EcS oε⋅=2

021 BcS

oµ⋅=and

Average intensity of an electromagnetic wave:

20

20 2

121 BEu

oo µε ==

2

Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS value is Erms = 720 N/C. Find

(a) the average intensity of this electromagnetic wave and (b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u. (c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u. *** a.u. = astronomical unit.

3

Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS value is Erms = 720 N/C. Find

(a) the average intensity of this electromagnetic wave and (b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u. (c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u. *** a.u. = astronomical unit.

4

Example: Sunlight enters the top of the earth's atmosphere with an electric field whose RMS value is Erms = 720 N/C. Find

(a) the average intensity of this electromagnetic wave and (b) the rms value of the sunlight's magnetic field At the Earth, orbiting the Sun at 1.00 a.u. (c) Repeat (a) and (b) for Venus, which orbits the Sun at a radius of 0.723 a.u. *** a.u. = astronomical unit.

23

2

2

212822

0 mW 10 38.1

CN 720

mNC 1085.8

sm 1000.3

21 )a( ×=

⋅

×

×=== −

rmsoo EcEcS εε

T 1040.2 m/s 1000.3

N/C 720 )b( 68

00

−×=×

==→=c

EBc

EB rmsrms

723.0a.u. 00.1a.u. 723.0 ,

4 and

4 )c(

2

2

2

2

2

==

==→==

−

E

V

E

V

V

E

E

V

E

SUNE

V

SUNV r

rrr

rr

SS

rPS

rPS

ππ

( )28

2

8

2

2

W/m 1063.2 (0.723)

W/m 1038.1/

×=×

==

=

−

EV

EE

E

VV rr

SSrrS

2202

1rms

oo

BcBcSµµ

=⋅=

T 1032.3 m/s 1000.3

)W/m 10m/A)(2.63T 104( 6-8

237

×=×

×⋅×==→

−πµcSB o

rms

24.6 Polarization

In (linearly) polarized light, the electric field oscillates along a single transverse direction.

5

POLARIZED ELECTROMAGNETIC WAVES

In unpolarized light, the electric field oscillates along random transverse direction at a given moment in time.

θ E0

E0’=E0cosθ

24.6 Polarization

Polarized light may be produced from unpolarized light with the aid of polarizing material.

6

2002

1 EcS ε=

( ) ( )2002

00 cos21'

21' θεε EcEcS ==

θcos' 00 EE =

Average Intensity of light:

Absorption and retransmission by antennae in polarizer

θθε 22200 coscos

21' SEcS =

=

Malus’ Law For unpolarized incident light Random Input polarization:

21coscos 22 =→ θθ

( ) SSS21cos' 2 == θ

Primed = after polarizer Unprimed = before polarizer

24.6 Polarization

MALUS’ LAW (general case)

θ2cosoSS =

intensity before analyzer

intensity after analyzer 7 The textbook’s

notational convention

E0

E0cosθ

θ = angle between polarization before and after analyzer

24.6 Polarization

Example Using Polarizers and Analyzers What value of θ should be used so the average intensity of the polarized light reaching the photocell is one-tenth the average intensity of the unpolarized light?

8

( ) θ221

101 cosoo SS =

θ251 cos=

51cos =θ

4.63=θ

24.6 Polarization

THE OCCURANCE OF POLARIZED LIGHT IN NATURE

9

Chapter 25

The Reflection of Light: Mirrors

10

We will start to treat light in terms of “rays” -- technical for the poetic “beam” of light. This works as long as the “width” of our rays are significantly larger than the wavelength of the light

25.1 Wave Fronts and Rays A hemispherical view of a sound wave emitted by a small pulsating sphere (i.e. a point-like source).

11

At large distances from the source, the wave fronts become less and less curved.

−=

λππψψ xfttx 22sin),( 0

The rays are perpendicular to the wave fronts (locations where the wave is at the same phase angle, e.g. at the maxima or compressions.

They become plane-waves described by:

25.2 The Reflection of Light

LAW OF REFLECTION

The incident ray, the reflected ray, and the normal to the surface all lie in the same plane, and the angle of incidence equals the angle of reflection.

12

2 types of reflecting surfaces: (a) smooth surface

Specular reflection: the reflected rays are parallel to each other.

(b) rough surface

Diffuse reflection

Plane mirror

Law of reflection still applies locally, where the normal is the direction perpendicular to the tangent plane

B

B

25.4 Spherical Mirrors

Just as it does for a plane mirror, the Law of reflection still applies locally, where the normal is the direction perpendicular to the tangent plane

13

Mirrors cut from segments of a sphere (Spherical mirrors) are used in many optical instruments.

If the inside surface of the spherical mirror is silvered or polished, it is a concave mirror.

If the outside surface is silvered or polished, it is a convex mirror.

The principal axis of the mirror is a straight line drawn through the center (C) and the midpoint of the mirror (B).

14

Applications of spherical mirrors

Wide-angle rear-view convex mirror found on many vehicles.

8.2 m diameter (largest single piece) reflector (concave mirror***) of the SUBARU Telescope National Astronomical Observatory of Japan (Mauna Kea Hawaii)

*** actually the SUBARU main mirror is closer to an hyperboloid

Large radius-of-curvature (R) concave mirrors to enlarge nearby object.

e.g. compact mirror