ire22512---eksamensoppgave---statistikk-og-elektrofysikk-deleksamen-i-elektrofysikk-17.12.2018

NY EKSAMEN

Emnekode: Emnenavn:

IRE 22512 Statistikk og elektrofysikk

(deleksamen elektrofysikk)

Dato: 17.des 2018 Eksamenstid: 0900-1200 Sensurfrist: 07.jan 2019

Antall oppgavesider: 4 Faglærer: Per Erik Skogh Nilsen

Telefon: 47 28 85 23

Antall vedleggsider: 10 Oppgaven er kontrollert: Ja

Hjelpemidler:

Kalkulator og matematiske tabeller.

Kopier av oppsummeringsark fra læreboka.

Om eksamensoppgaven:

Alle deloppgaver har lik vekt

Oppgave 1

a) Figuren under viser et tynt metallisk kuleskall som netto har en uniformt fordelt positiv ladning på overflaten. En ny positiv ladning (+Q) plasseres deretter på utsiden som vist på figuren.

+Q

Hvilket av følgende utsagn er mest korrekt for det elektriske feltet i sentrum? Velg og forklar.

I: Feltet er null.

II: Retningen er rett mot høyre.

III: Retningen er rett oppover.

IV: Retningen er rett nedover.

V: Retningen er rett mot venstre.

b) En kule har masse 100 kg og netto ladning +1,00 C.En annen kule har masse 150 kg og elektrisk ladning q

. Når du plasserer dem nær hverandre finner du at de er i likevekt.

Bestem ladningen q

. Oppgi svaret i antall elementærladninger (e).

c) En nøytral metallplate beveger seg med konstant hastighet v som på figuren.

Den beveger seg gjennom et uniformt magnetfelt som går ut av arket.

v

Hvilket av alternativene under beskriver ladningsfordelingen på overflaten av metallplaten best? Velg og forklar.

I

+ + + + +

+ - - - - -

II

- - - - -

- +

+ + + +

III IV

+ +

+ + + +

-

- -

- - -

V

- -

- - - -

+

+ +

+ + + .

1

Oppgave 2

a) En platekonsensator har to plater som hver har areal 1,00 cm

. Avstanden mellom platene er 0,25 mm. Mellom platene kan det være et materiale med dielektrisitetskonstant κ = 3,3.

Kondensatoren blir ladet opp av et 12,0 V batteri.

Hvor stor energi kan lagres i kondensatoren med og uten dielektrikum?

b) Tre kondensatorer er koplet som på figuren under. De har kapasitansene C

= 6, 00 µ F, C

= 3, 00 µ F og C

= 5, 00 µ F. Nettverket gis spenningen U

. Når oppladingen er maksimal, er ladningen på C

lik 40,0 µ C.

Hva blir U

?

a b

C

C

C

c) En platekondensator skal fylles delvis med et dielektrikum. Det kan gjøres enten ved fylle hele arealet i halve bredden (figuren til venstre) eller ved å fylle hele bredden for deler av arealet (figuren til høyre).

Andelen f er arealet med diekektrikum delt på hele arealet.

Ved hvilken andel (f ) blir kapasitansen den samme i begge tilfeller (jf. figurene under)?

Oppgave 3

a) Et elektron beveger seg i papirplanet mot en skjerm i en bane som vist på figuren under.

I det grå feltet blir det utsatt for et uniformt magnetisk felt med størrelse | B|. ~ Hvilket alternativ angir den korrrekte retningen til B? Velg og forklar. ~

I: Retningen er oppover på arket.

II: Retningen er nedover på arket.

III: Retningen er langs banen.

IV: Retningen er inn i arket.

V: Retningen er ut av arket

b) Et ion med masse m og ladning q går rett fram. Vinkelrett på hastigheten er et elektrisk felt med størrelse 0, 25 kN

C . Vinkelrett på både hastigheten og det elektriske feltet er et magnetisk felt som har størrelse 2, 0mT. Regn ut størrelsen på hastigheten og tegn en figur som viser et tilfelle med mulige retninger på feltene.

c) Gauss lov for magnetisme kan formuleres som H B ~ · d ~ A = 0 og Gauss lov for elektrisitet som H ~ E · d ~ A = q

²

.

Forklar kort det fysiske innholdet i uttrykkene.

3

Oppgave 4

a) To parallelle ledninger A og B ligger i næreheten av hverandre . I A går strømmen i og i B går strømmen 3i, begge i samme retning. Se figuren

A

i 3i B

Sammenlign de magnetiske kreftene mellom lederene og forklar hvilken beskrivelse som er korrekt.

I: A virker med en sterkere kraft på B enn B på A.

II: A virker med en svakere kraft på B enn B på A.

III: Kreftene er like store og frastøtende.

IV: Kreftene er like store og tiltrekkende.

V: Det virker ikke magnetiske krefter mellom ledningene.

b) 3 veldig lange rette ledere fører hver en konstant strøm i.

3 rektangulære metalløkker beveges med hver sin ~ v.

A

i

~ v

B

i

~ v

C

i ~ v

Ett av alternativene angir korrekt hvilke sløyfer det blir indusert strøm i.

Velg hvilket og forklar.

I: Bare A og B.

II: Bare A og C.

III: Bare B og C.

IV: Alle tre.

V: Ingen av dem

c) En sirkulær loop av jern starter med en radius på 164 cm, men radius avtar med den konstante farten 14,0 cm/s. Loopen er i et uniformt magnetfelt på 0,900 T som står vinkelrett på planet loopen ligger i.

Anta du ser mot loopen og at magnetfeltet går fra deg inn mot loopen.

Bestem den elektromotoriske spenningen etter 6,00 s og forklar om strømretningen er med

eller mot urviserene (sett fra deg).

Formler og kapitteloppsummeringer

• Overflateareal kule: 4πr

• Overflateareal sylinder 2 π r h + 2 π r

• Areal sirkel πr

• Volum kule: 4 3 π r

• Størrelse kryssprodukt | ~ A × B ~ | = | ~ A | · | B ~ | · sin( θ )

• Newtons 2.lov: Σ~ F = m ~ a

• Kraftmoment: ~ τ =~ r × ~ F

• Newtons gravitasjonslov: ~ F

= G · m · M r

r ˆ

• Coulombs konstant k = 9, 00 · 10

Nm

C

PHYSIC AL CONSTANTS

Conversion Factors (more conversion factors in Appendix C)

SI PREFIXES

POWER PREFIX SYMBOL

yotta Y

zetta Z

exa E

peta P

tera T

giga G

mega M

kilo k

hecto h

deca da

— —

deci d

centi c

milli m

micro

nano n

pico p

femto f

atto a

zepto z

yocto y

10²²⁴ 10²²¹ 10²¹⁸ 10²¹⁵ 10²¹² 10²⁹

μ 10²⁶

10²³ 10²² 10²¹ 10⁰ 10¹ 10² 10³ 10⁶ 10⁹ 10¹² 10¹⁵ 10¹⁸ 10²¹ 10²⁴

THE GREEK ALPHABET

UPPERCASE LOWERCASE Alpha

Beta Gamma Delta Epsilon Zeta

Eta H

Theta Iota Kappa Lambda Mu Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi

Omega V v

c C

x X

f F

y Y

t T

s S

r R

p P

o O

j J

n N

m M

l L

k K

i I

u U

h z Z

P E

d D

g G

b B

a A

CONSTANT SYMBOL THREE-FIGURE VALUE BEST KNOWN VALUE*

Speed of light c 299 792 458 (exact)

Elementary charge e

Electron mass Proton mass

Gravitational constant G

Permeability constant (exact)

Permittivity constant (exact)

Boltzmann’s constant k

Universal gas constant R

Stefan–Boltzmann constant Planck’s constant

Avogadro’s number Bohr radius

*Parentheses indicate uncertainties in last decimal places. Source: U.S. National Institute of Standards and Technology, 2007 values

5.291 772 08591362310²¹¹m 5.29310²¹¹m

a0

6.022 141 791302310²³mol²¹ 6.02310²³mol²¹

NA

6.626 068 961332310²³⁴J#s 6.63310²³⁴J#s

h15 2p"s 2 5.67310²⁸W/m²#K^{4 5.670 400}1402310²⁸W/m²#K⁴ 8.314 4721152 J/K#mol 8.31 J/K#mol

1.380 65041242310²²³J/K 1.38310²²³J/K

1/m0c² 1F/m2

8.85310²¹²C²/N#m²

P0

4p310²⁷ 1^H/m2

1.26310²⁶N/A² m0

6.674 281⁶⁷2^{310211N#m2/kg2} 6.67310²¹¹N#m²/kg²

1.672 621 6371832310²²⁷kg 1.67310²²⁷kg

m_p 9.11310²³¹kg 9.109 382 151452310²³¹kg me

1.602 176 4871402310²¹⁹C 1.60310²¹⁹C

m/s 3.00310⁸m/s

Length

Velocity

Mass, energy, force

Time

Pressure

Rotation and angle

Magnetic field 1 gauss510²⁴T 1 rev/s560 rpm 1 rev5360°52prad 1 rad5180°/p 557.3°

1 atm514.7 lb/in²

1 atm5101.3 kPa5760 mm Hg

1 year53.16310⁷s 1 day586,400 s

5weight of 0.454 kg 1 pound 1lb254.448 N 1 eV51.602310²¹⁹J 1 kWh53.6 MJ 1 Btu51.054 kJ 1 cal54.184 J 1 u51.661310²²⁷kg

1 m/s52.24 mi/h53.28 ft/s 1 mi/h50.447 m/s

1 light year59.46310¹⁵m 1 ft50.3048 m

1 mi51.609 km 1 in52.54 cm

CHAPTER 20 SUMMARY

Big Picture

Applications

Key Concepts and Equations

This chapter introduces several big ideas. First is electric charge,a fundamental property of matter that comes in positive and negative forms. Like charges repel and opposites attract; this is the electric force.It’s convenient to define the electric fieldas the force per unit charge that a charge would experience if placed in the vicinity of other charges. Both force and field obey the superposition principle, meaning that the effects of several charges add vectorially.

Coulomb’s lawdescribes the electric force between point charges:

The electric field is the force per unit charge, and therefore the force a given charge qexperiences in a field is F!

5qE! . E!

5F! /q,

The field of a point charge follows from Coulomb’s law:

E! 5kq

r²r^

Fields of charge distributions are found by summing fields of individual point charges, or by integrating in the case of continuously distributed charge:

Adipoleconsists of equal but opposite charges a distance dapart.

For distances large compared with d, the dipole field drops as and the dipole is completely characterized by its dipole momentp5qd.

1/r³,

6q The field of an infi-

nite line drops as with the charge per unit length.

This is a good approxi- mation to the field near an elongated structure like a wire.

E52kl/r, l

1/r:

Point charges respond to electric fields with accel- eration proportional to the charge-to-mass ratio q/m.

A dipole in an electric field experiences a torque that tends to align it with the field: t!

5p! 3E!

. F^r125

F12 r

r

q₁ q₂

r² rˆ rˆ kq₁q₂

E1 r

E2 E₃ r

r₁ q₁

q₂ q3

r₂ r₃

r

r

rˆ rˆ rˆ1

rˆ₂

rˆ₃ rˆ

dEr

dE dEr

P P

dq dq

dq r

r r

S

E(P)^r 5E^r₁1E^r₂1E^r₃5 rˆ_i k dq r² E(P)^r 53dE^r53 rˆ

Electric field atP

E P

q

r

F^r Put a charge q at P, and r

the force on q is F5qE.^r

Field is stronger closer to the charge.

Field weakens with increasing distance from the charge.

2 1

p d

r 1q 2q

F+ r

F^r_– E^r

p^r Torque rotates dipole clockwise.

2 1

Dielectricsare insulating materials whose molecules act like electric dipoles.

If the field is nonuniform, there’s also a net force on the dipole.

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

CHAPTER 21 SUMMARY

Big Picture

Key Concepts and Equations

The big idea here is Gauss’s law—a universal statement about electric fields that’s closely related to Coulomb’s inverse-square law but expressed in terms of the global behavior of the field over any closed surface. Using the electric field-linepicture, Gauss’s law says that the number of field lines emerging from a closed surface depends only on the net charge enclosed; more rigorously, it says that the electric fluxthrough the surface is proportional to the enclosed charge.

Electric flux describes the amount of electric field crossing an area.

for a flat area perpendicular to a uniform field F 5EA

F

F 53E!#dA!

, in general

Gauss’s law gives the fields of symmetric charge distributions:

Spherical symmetry: Plane symmetry:

Gauss’s law and conductors:

● The field is zero inside a conductor in electrostatic equilibrium.

● Any net charge resides on a conductor’s surface.

● The field at the surface is perpendicular to the surface and has magnitude s/P0.

Eight lines pass through any closed surface surrounding q.

The number of lines through a surface depends on the net charge enclosed.

Point charge q Point charges 1q,2q/2

E^r

dA^r E^r In terms of flux, Gauss’s law reads

Here P051/4pk58.85310²¹²C²/N#m²is another way of expressing the Coulomb constant k59.0310⁹N#m²/C².

$E!

#dA!

5qenclosed/P0.

Applications

Q R

Outside:

Inside uniformly charged sphere:

Inside hollow sphere: E50 E5kQr

R³ E5 Q

4pP0r²5kQ r²

Line symmetry:

lC/m

Outside:

Inside hollow pipe:

E50 E5 l

2pP0r

Outside charged slab: E5 s 2P0

sC/m²

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1

11 11 11 E^r

1 11 1 1 1 11 11 1 1 11 1 1 E^r50 E5 s/e₀

Surface charge density s

CHAPTER 22 SUMMARY

Big Picture

Key Concepts and Equations

Applications

The big idea here is electric potential difference—a measure of the energy per unit charge involved in moving charge between two points in an electric field. Because the electric field is conservative, potential difference is path independent and thus depends only on the two points in question.

The dipole potential is

where is the dipole moment and the angle is measured from the dipole axis.

u p5qd

V5kpcosu r²

In charged conductors, the charge density is generally highest, and the field strongest, where a con- ductor curves sharply.

Electric potential difference between points AandBis the negative of the line integral of the electric field over any path from AtoB:

When a charge “falls” through a potential difference it gains energy qDV.

DV,

DVAB5DUAB

q 5 23

B AE!#dr!

Potentials of charge distributions follow by summing or integrating the fields of pointlike charge elements:

V53k dq

r 1continuous charge distribution2 V5akqi

ri 1discrete charges2 Equipotentialsare surfaces of constant potential perpendicular to the electric field. Where equipotentials are close, the field is strong. The field component in a given direction depends on the rate at which po- tential changes with position; thus,

Ex5 2dV dx The potential in the field of a

point charge is

where the zero of potential is taken at infinity and ris the distance from the point charge.

V1r25kq/r,

In a uniform field, the potential difference becomes DVAB5 2E!#Dr!

A B

E^r Dr^r

E

dr Er

r

dV5 2E^r#dr^r

r

B

A

e E

q P

r

r V(r) is energy per unit charge to get from` to P.

P r1

q₁

q₂ q₃

r2 r3

P r dq

dV5k dq r

Steep hill, close contours, strong field

Circles are equipotentials.

E^r

Field and equipotentials are perpendicular.

E V,0 on the

negative side.

V.0 on the positive side.

V50 on the bisector.

r

Strongest field 111 11

CHAPTER 23 SUMMARY

Big Picture

Applications

Key Concepts and Equations

Theenergy densityin an electric field Eis

Integrating over volume gives the total electric energy Ustored in the field:

U53uEdV.

uE5¹₂P0E².

The big idea here is that allelectric fields represent stored energy. This energy is associated with the work needed to assemble a distribution of electric charge, and may be negative or positive.

You do positive work to assemble this charge distribution . . . . . . and therefore the stored electric energy Uis positive.

The energy stored in a capacitor is U5¹₂CV².

Acapacitoris a pair of insulated conductors used to store electric energy. Capacitanceis the ratio of charge to potential difference:

For a parallel-plate capacitor:

C5P0A/d C5Q/V

Capacitors in parallel add: C5C11C2.

Capacitors in series add reciprocally: 1 C5 1

C1

1 1 C2

. 1q

1q

U.0 You do negative work to assemble this charge distribution . . . . . . and therefore the stored electric

energy Uis negative. ^{1q 2q}

U,0

The electric energy U stored in the shaded volume is

U53^{uEdV5 E0}3^E2dV.

E^r 1

2

1 2 At this point

the field strength isE, so there’s electric energy whose density isuE5 e0E².

Q Potential

difference V

Spacingd 2Q

A

C1 C2 C5C11C2

Capacitors in parallel have the same voltage.

C1 C₂

5

C 1 C1

1 C2

Capacitors in series have the same charge.

Complicated circuits are analyzed by breaking them into parallel and series combinations:

C1

C2 C3

A

B

(a) (b) (c)

C1 A

B

C23 A

B

C₁₂₃

Adielectricbetween capacitor plates increases the capacitance, as determined by the dielectric constantkof the material: CSkC0.

CHAPTER 24 SUMMARY

Big Picture

Applications

Key Concepts and Equations

The big idea here is electric current—the flow of electric charge—and its microscopic cousin, current density. With current we don’t have elec- trostatic equilibrium, and there’s usually an electric field in a current-carrying conductor. Ohm’s lawis an empirical statement—not a fundamental law of physics—that relates current and voltage, or current density and electric field.

The microscopic version of Ohm’s lawrelates electric field, current density, and conductivity (or its inverse, resistivity ):

The macroscopic version relates voltage, current, and resistance:

I5V/R J!

5 sE! r s

Electric power is the product of voltage and current:

Using Ohm’s law, this can also be written

P5V² R P5I²R

P5IV

Different types of conductors have different conduction mechanisms. In metals, free electrons carry the current; in ionic solutions, both positive and negative ions are involved; in plasmas, the charge carriers are free electrons and ions; and in semiconductors, both electrons and posi- tive holes carry current, with semiconductor conduction properties readily adjustable.

Superconductorsare materials that exhibit zero resistance at sufficiently low temperatures.

Electrical safetyis a matter of avoiding currents high enough to cause biological harm, and that means avoiding voltages high enough to drive such currents.

Quantitatively, currentis defined as the rate of charge flow:

Current densityis the current per unit area. Its magnitude is J5I

A I5DQ

Dt

Microscopically, current depends on the density of charge carriers, their charge, and the drift velocity:

J! 5nqv!

d

I5nqAvd and

Charge DQ crosses this area in time Dt.

There are ncharge carriers per unit volume, with charge q and drift velocity vd.

A

Conductor of material with conductivity s and resistivity r 51

/

CurrentI through conductor

Electric field and current density are vectors defined at each point; they're related byJ5 sE.

resistanceR5 rL A

VoltageVacross conductor

A E^r

r r

J^r I5JA

E^r

E^r A bound electron jumps leftward, moving the hole to the right.

Electron and hole move oppositely in an electric field.

Free electron

Hole Electrons Holes

CHAPTER 26 SUMMARY

Big Picture

Key Concepts and Equations

The big new idea here is magnetism—an interaction that fundamentally involves moving electric charge. Moving charge produces magnetic fields, and moving charges respond to magnetic fields by experiencing a magnetic force.

Themagnetic forceon a charge qmoving with velocity in a magnetic field is The force acts at right angles to both and B! and therefore it does no work.

, v! F!

5qv! 3B!

1magnetic force2

B! v!

TheBiot–Savart lawdescribes the magnetic field arising from a small element of steady current:

Here is the permeability constant, with value 4p310²⁷N/A².

m0

dB! 5 m0

4p Idl!

3r^ r²

dB! Ampère’s lawprovides a more global description of how

magnetic fields arise from currents, relating the line inte- gral around any closed loop to the encircled current:

Ampère’s law in this form applies only to steady currents.

CB!#dr!5 m0Iencircled

Gauss’s law for magnetismexpresses the fact that there are no magnetic monopoles—

magnetic analogs of electric charge—and that magnetic field lines therefore do not begin or end:

Static electric fields, in contrast, always begin or end on electric charges.

CB!#dA! 50

Fields of simple current distributions:

Line current: B5m0I Current sheet: B5¹₂m0Js Solenoid: B5 m0nI 2pr

Magnetism in matter arises from the interactions of atomic-scale current loops.Ferromagneticmaterials have strong interactions and exhibit the bulk magnetism associated with per- manent magnets and with magnetic materials like iron. Paramagnetism anddiamagnetismare weaker mani- festations of magnetism in matter.

F^r B^r q v^r Force is

perpendicular to both v and B.^{r r}

rˆ dl dl is a small piece of the wire.

r

dB is into page.^r ris a unit vectorr

fromdl toward P.

ˆ

r

I

e P

r

Electric field Magnetic field

A charged particle moving perpendicular to a uniform magnetic field undergoes circular motion with the cyclotron frequency More generally, charged par- ticles in magnetic fields follow spiral paths,

“trapped” on the field lines.

f5qB/2pm.

The magnetic force on a straight wire of lengthlcarrying current Iin a uniform mag- netic field is Parallel wires a distancedapart experience forces from each

other’s magnetic field: The

force is attractive for currents in the same di- rection, repulsive for currents in opposite directions.

F5m0I1I2l 2pd . F!

5Il! 3B!

.

A current loop gives rise to a magnetic field that, at distances large compared with the loop’s size, is a dipole field. The loop’s mag- netic dipole moment has magnitude withAthe loop area, and the loop responds to an external magnetic field by experiencing the torque typical of a dipole: t!

5 m! 3B!

. m 5IA,

Applications

I₂ I₁ F^r

F^r

B

I

r

Br l

I_out

CHAPTER 27 SUMMARY

Big Picture

Applications

Key Concepts and Equations

The big idea here is electromagnetic induction, a phenomenon in which a changing magnetic field produces an electric field. Applied to circuits, induction results in induced emfs that drive induced currents.

Faraday’s lawdescribes induction quantitatively, relating the line inte- gral of the induced electric field to changing magnetic flux:

$E!#dr!5 2dFB/dt

Lenz’s lawshows that electromagnetic induction is consistent with conservation of energy, which requires that induced effects act to op- pose the changes that give rise to them.

Magnetic fields contain stored energy, as do electric fields. The magnetic-energy densityis uB5 B²

2m₀

Diamagnetismoccurs when electromagnetic induction results in atoms acquiring net magnetic moments; the result is a repulsive interaction.

Electric generatorsconvert mechanical to electrical energy by moving conductors in magnetic fields to induce emfs that drive currents.

Inductorsare wire coils that encircle their own magnetic flux, giving self- inductance An inductor opposes changes in current, producing an emf given by Circuit quantities in a simple RLcircuit change with inductive time constantL/R.

E5 2L1dI/dt2. L5 F_B/I.

S S N

I

0

2 1

I

0

2 1

Here a moving magnet produces the changing magnetic field. Here a change in current produces the changing magnetic field.

E^r B^r Region of

increasingB^r Induced

electric field

I Increasingr

B into page

Counterclockwise current makes loop magnetic moment out of page, opposing increase in B.^r

Rotation of loop changes the magnetic flux, inducing an emf.

N

Stationary brushes Rotating

conducting loop Rotating

slip rings

Electric load

S

E0

R I

L

L/R Time I

E0/R

I5E0

R112e^2Rt/L2 In conductors, Faraday’s law gives the induced emf: E5 2dF_B/dt.

CHAPTER 29 SUMMARY

Big Picture

Applications

Key Concepts and Equations

The big idea here—and one of the biggest ideas in physics—is that electric and mag- netic fields together form self-regenerating structures that propagate through space as electromagnetic waves. What makes these waves possible is that changing magnetic fields induce electric fields (Faraday’s law), and changing electric fields induce mag- netic fields (Ampère’s law, with Maxwell’s modification). Electromagnetic (EM) waves in vacuum consist of electric and magnetic fields perpendicular to each other and to the direction of wave propagation, and in phase.

Maxwell’s equationsdescribe completely the behavior of electric and magnetic fields in classical physics:

Maxwell’s equations show that electromagnetic waves are possible and that their speed in vacuum, the speed of light c, is related to the electric and magnetic constants and

The value of cis very nearly Its exact value, used in defining the meter, is 299,792,458 m/s.

3.00310⁸m/s.

c5 1 1P0m0

m0: P0

In vacuum, the electric and magnetic fields of a wave are related by

The wave’s frequency and wavelength are re- lated by

fl 5c E5cB

EM waves can have any wavelength; the whole range constitutes the electromagnetic spectrum.

Polarizationdescribes the direction of an EM wave’s electric field and is a property widely used in scientific research and in technological de- vices including the ubiquitous liquid crystal displays. When polarized light of intensity is incident on a polarizer with its transmission axis at angle to the polarization, the light emerges with intensity

S5S0cos²u u

S0

EM waves carry both energy and momentum. The Poynting vector

describes the rate of energy flow per unit area, while the momentum flow results in a radiation pressure:

Prad5S c S!

5E! 3B! m0

10⁶ 10³ 1 10²³ 10²⁶ 10²⁹ 10²¹²

10³

1 10⁶ 10⁹ 10¹² 10¹⁵ 10¹⁸ 10²¹

Radio Micro-

wave

Infrared Ultra- violet

X rays Gamma rays Wavelength (m)

Visible light Frequency (Hz)

700 nm 650 600 550 500 450 400 nm

Wavelength Red Violet

Law Mathematical Statement What It Says

Gauss for E!

$E!#dA! 5 q

P0

How charges produce electric fields; field lines begin and end on charges.

Gauss for B!

$B!#dA!

50 No magnetic charge; magnetic

field lines don’t begin or end.

Faraday $E!#dr! 5 2dFB

dt

Changing magnetic flux produces electric fields.

Ampère $B!#dr!5 m0I1 m0P0

dF_E dt

Electric current and changing elec- tric flux produce magnetic fields.

Direction of wave propagation E

B y

z c

x

r

E^r

E^r

r B^r

B^r