generac i on de ondas em

8/19/2019 Generac i on de Ondas Em

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13-1

Chapter 13

Maxwell’s Equations and Electromagnetic Waves

13.1 The Displacement Current ................................................................................ 13-3

13.2 Gauss’s Law for Magnetism .............................................................................. 13-5

13.3 Ma well’s !"uations ......................................................................................... 13-5

13.# $lane !lectromagnetic %a&es ........................................................................... 13-'

13.#.1 (ne-Dimensional %a&e !"uation ........................................................................ 13-1)

13.5 *tan+ing !lectromagnetic %a&es .................................................................... 13-13

13., $o nting ector ............................................................................................... 13-15

! ample 13.1/ *olar Constant ............................................................................. 13-1'! ample 13.2/ 0ntensit of a *tan+ing %a&e ...................................................... 13-1

13.,.1 !nerg Transport .................................................................................................. 13-1

13.' Momentum an+ a+iation $ressure ................................................................ 13-22

13. $ro+uction of !lectromagnetic %a&es ............................................................ 13-23

Animation 13.1 / !lectric Dipole a+iation 1 .................................................... 13-25 Animation 13.2 / !lectric Dipole a+iation 2 .................................................... 13-25 Animation 13.3 / a+iation 4rom a uarter-%a&e 6ntenna .............................. 13-2,

13. .1 $lane %a&es .......................................................................................................... 13-2,13. .2 *inusoi+al !lectromagnetic %a&e ........................................................................ 13-31

13. *ummar ......................................................................................................... 13-33

13.1) 6ppen+i / eflection of !lectromagnetic %a&es at Con+ucting *urfaces .. 13-35

13.11 $ro7lem-*ol&ing *trateg / Tra&eling !lectromagnetic %a&es ..................... 13-3

13.12 *ol&e+ $ro7lems ............................................................................................ 13-#1

13.12.1 $lane !lectromagnetic %a&e .................................................................. 13-#113.12.2 (ne-Dimensional %a&e !"uation .......................................................... 13-#213.12.3 $o nting ector of a Charging Capacitor ............................................... 13-#313.12.# $o nting ector of a Con+uctor ............................................................. 13-#5

13.13 Conceptual uestions .................................................................................... 13-#,

13.1# 6++itional $ro7lems ...................................................................................... 13-#'

13.1#.1 *olar *ailing ........................................................................................... 13-#'


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13.1#.2 eflections of True Lo&e ........................................................................ 13-#'13.1#.3 Coa ial Ca7le an+ $ower 4low .............................................................. 13-#'13.1#.# *uperposition of !lectromagnetic %a&es ............................................... 13-#13.1#.5 *inusoi+al !lectromagnetic %a&e .......................................................... 13-#13.1#., a+iation $ressure of !lectromagnetic %a&e ........................................ 13-#

13.1#.' !nerg of !lectromagnetic %a&es ......................................................... 13-#13.1#. %a&e !"uation ........................................................................................ 13-5)13.1#. !lectromagnetic $lane %a&e .................................................................. 13-5)

13.1#.1)................................................................ *inusoi+al !lectromagnetic %a&e13-5)


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Maxwell’s Equations and Electromagnetic Waves

13.1 The Displacement Current

0n Chapter 8 we learne+ that if a current-carr ing wire possesses certain s mmetr 8 themagnetic fiel+ can 7e o7taine+ 7 using 6mpere’s law/

. .

° ∫ ⋅ d s = µ ) I enc913.1.1:

The e"uation states that the line integral of a magnetic fiel+ aroun+ an ar7itrar close+

loop is e"ual to µ ) I enc 8 where I enc is the con+uction current passing through the surface

7oun+ 7 the close+ path. 0n a++ition8 we also learne+ in Chapter 1) that8 as aconse"uence of the 4ara+a ’s law of in+uction8 a changing magnetic fiel+ can pro+uce anelectric fiel+8 accor+ing to

. . . .d

E ⋅ d s =− dt ∫∫ ⋅

d !S

913.1.2:

(ne might then won+er whether or not the con&erse coul+ 7e true8 namel 8 a changing

electric fiel+ pro+uces a magnetic fiel+. 0f so8 then the right-han+ si+e of !". 913.1.1: will. .ha&e to 7e mo+ifie+ to reflect such ;s mmetr < 7etween E an+ .

To see how magnetic fiel+s can 7e create+ 7 a time-&ar ing electric fiel+8 consi+er acapacitor which is 7eing charge+. During the charging process8 the electric fiel+ strengthincreases with time as more charge is accumulate+ on the plates. The con+uction currentthat carries the charges also pro+uces a magnetic fiel+. 0n or+er to appl 6mpere’s law tocalculate this fiel+8 let us choose cur&e C shown in 4igure 13.1.1 to 7e the 6mperian loop.


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"igure 13.1.1 *urfaces S 1 an+ S 2 7oun+ 7 cur&e C .

0f the surface 7oun+e+ 7 the path is the flat surface S 1 8 then the enclose+ current

is I enc = I . (n the other han+8 if we choose S 2 to 7e the surface 7oun+e+ 7 the cur&e8then I

enc= ) since no current passes through S

2. Thus8 we see that there e ists an

am7iguit in choosing the appropriate surface 7oun+e+ 7 the cur&e C .

Ma well showe+ that the am7iguit can 7e resol&e+ 7 a++ing to the right-han+ si+e ofthe 6mpere’s law an e tra term

I d = ε ) d Φ E dt

913.1.3:

which he calle+ the ; displacement current .< The term in&ol&es a change in electric flu .

The generali=e+ 6mpere’s 9or the 6mpere-Ma well: law now rea+s. .

° ∫ ⋅ d s = µ ) I + µ ) ε )

d Φ E dt

= µ ) 9 I + I d : 913.1.#:

The origin of the +isplacement current can 7e un+erstoo+ as follows/

"igure 13.1.# Displacement through S 2

0n 4igure 13.1.28 the electric flu which passes through S 2 is gi&en 7

. .

Φ E = º

∫∫ S

E ⋅ d ! = EA =Q

ε )

913.1.5:

where A is the area of the capacitor plates. 4rom !". 913.1.3:8 we rea+il see that the


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.

B º B d A 0S

+isplacement current I d is relate+ to the rate of increase of charge on the plate 7

I = ε d Φ E = dQ913.1.,:

d )

dt dt

>owe&er8 the right-han+-si+e of the e pression8 dQ ? dt 8 is simpl e"ual to the con+uction

current8 I . Thus8 we conclu+e that the con+uction current that passes through S 1 is

precisel e"ual to the +isplacement current that passes through S 28 namel I = I d . %iththe 6mpere-Ma well law8 the am7iguit in choosing the surface 7oun+ 7 the 6mperianloop is remo&e+.

13.# $auss’s %aw &or Magnetism

%e ha&e seen that Gauss’s law for electrostatics states that the electric flu through aclose+ surface is proportional to the charge enclose+ 94igure 13.2.1a:. The electric fiel+lines originate from the positi&e charge 9source: an+ terminate at the negati&e charge9sin@:. (ne woul+ then 7e tempte+ to write +own the magnetic e"ui&alent as

. .

Φ B = º

∫∫ S

⋅ d ! =Qm

µ )

913.2.1:

where Qm is the magnetic charge 9monopole: enclose+ 7 the Gaussian surface. >owe&er8+espite intense search effort8 no isolate+ magnetic monopole has e&er 7een o7ser&e+.>ence8 Qm = ) an+ Gauss’s law for magnetism 7ecomes

913.2.2:


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"igure 13.#.1 Gauss’s law for 9a: electrostatics8 an+ 97: magnetism.

This implies that the num7er of magnetic fiel+ lines entering a close+ surface is e"ual tothe num7er of fiel+ lines lea&ing the surface. That is8 there is no source or sin@. 0na++ition8 the lines must 7e continuous with no starting or en+ points. 0n fact8 as shown in4igure 13.2.197: for a 7ar magnet8 the fiel+ lines that emanate from the north pole to thesouth pole outsi+e the magnet return within the magnet an+ form a close+ loop.

13.3 Maxwell’s Equations

%e now ha&e four e"uations which form the foun+ation of electromagnetic phenomena/


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0 0

Law !"uation $h sical 0nterpretation

.GaussAs law for

E

. . Qº ∫∫ E ⋅ d ! = ε S )

!lectric flu through a close+surface is proportional to the

4ara+a As law. . d Φ

° ∫ E ⋅ d s = − BChanging magnetic flu pro+uces anelectric fiel+

.GaussAs aw or

.

º ∫∫ ⋅ d ! = )S The total magnetic flu through aclose+ surface is =ero

6mpere − Ma welllaw

. . d Φ° ⋅ d s = µ I + µ ε

!lectric current an+ changingelectric flu pro+uces a magnetic

Collecti&el the are @nown as Ma well’s e"uations. The a7o&e e"uations ma also 7ewritten in +ifferential forms as

. ρ ∇⋅ E =

ε ).

.

∇× E = − ∂

∂t 913.3.1:.

∇⋅ = ). . ∂

.

∇× = µ ) ' + µ ) ε ) ∂t

.where ρ an+ ' are the free charge an+ the con+uction current +ensities8 respecti&el . 0nthe a7sence of sources where Q = )8 I = ) 8 the a7o&e e"uations 7ecome

. .

º ∫∫ E ⋅ d ! = )S . . d Φ

° ∫ E ⋅ d s = − B. .

dt 913.3.2:º ∫∫ ⋅ d ! = )S

. . d Φ⋅ d s = µ ε E ° ∫ ) )

dt

6n important conse"uence of Ma well’s e"uations8 as we shall see 7elow8 is the pre+iction of the e istence of electromagnetic wa&es that tra&el with spee+ of light

c =1? . The reason is +ue to the fact that a changing electric fiel+ pro+uces a

E


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magnetic fiel+ an+ &ice &ersa8 an+ the coupling 7etween the two fiel+s lea+s to thegeneration of electromagnetic wa&es. The pre+iction was confirme+ 7 >. >ert= in 1 '.


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13.( )lane Electromagnetic Waves

To e amine the properties of the electromagnetic wa&es8 let’s consi+er for simplicit an.

electromagnetic wa&e propagating in the B x-+irection8 with the electric fiel+ E pointing.

in the B y-+irection an+ the magnetic fiel+ in the B z -+irection8 as shown in 4igure 13.#.1 7elow.

"igure 13.(.1 6 plane electromagnetic wa&e. .

%hat we ha&e here is an e ample of a plane wa&e since at an instant 7oth E an+ areuniform o&er an plane perpen+icular to the +irection of propagation. 0n a++ition8 thewa&e is transverse 7ecause 7oth fiel+s are perpen+icular to the +irection of propagation8

. .which points in the +irection of the cross pro+uct E × .

sing Ma well’s e"uations8 we ma o7tain the relationship 7etween the magnitu+es ofthe fiel+s. To see this8 consi+er a rectangular loop which lies in the xy plane8 with the left

si+e of the loop at x an+ the right at x + ∆ x . The 7ottom si+e of the loop is locate+ at y 8an+ the top si+e of the loop is locate+ at y + ∆ y 8 as shown in 4igure 13.#.2. Let the unit

&ector normal to the loop 7e in the positi&e z -+irection8 n D = * D .

."igure 13.(.# *patial &ariation of the electric fiel+ E

sing 4ara+a ’s law . . . .

d ° ∫ E ⋅ d s =− dt


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∫∫ ⋅ d ! 913.#.1:


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the left-han+-si+e can 7e written as

. .E ⋅ d s = E 9 x + ∆ x:∆ y − E 9 x:∆ y = E E 9 x + ∆ x: − E 9 x:F∆ y= ∂ E y (∆ x ∆ y)

913.#.2:

° ∫ y y y y ∂ xwhere we ha&e ma+e the e pansion

E y 9 x + ∆ x: = E y

∂ E 9 x: + y ∆ x +… 913.#.3:

∂ x

(n the other han+8 the rate of change of magnetic flu on the right-han+-si+e is gi&en 7

. .d − ∫∫ ⋅ d ! = −⎛ ∂ B z ⎞ (∆ x ∆

y )⎝ ⎠

913.#.#:

dt ⎜ ∂t ⎟

!"uating the two e pressions an+ +i&i+ing through 7 the area ∆ x∆ y iel+s

913.#.5:

The secon+ con+ition on the relationship 7etween the electric an+ magnetic fiel+s ma 7e+e+uce+ 7 using the 6mpere-Ma well e"uation/

. . . .⋅ d s = µ ε

d E ⋅ d ! 913.#.,:

° ∫ ) ) dt ∫∫

Consi+er a rectangular loop in the xz plane +epicte+ in 4igure 13.#.38 with a unit normal

n = + .

∂ E ∂

∂ ∂


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."igure 13.(.3 *patial &ariation of the magnetic fiel+

The line integral of the magnetic fiel+ is. .

° ∫ ⋅ d s = B z 9 x:∆ z − B z 9 x + ∆ x:∆ z = E B z 9 x: − B z 9 x + ∆ x:F∆ z

913.#.':

⎛ ∂ B z⎞ ( x z )= −⎜ ⎟ ∆ ∆⎝ ∂ x ⎠

(n the other han+8 the time +eri&ati&e of the electric flu is

. . d

) ) ∫∫ E ⋅ d ! = µ ε ⎛ ∂ E y ⎞ (

∆ x ∆ z)

913.#. :

dt ⎝ ∂t ⎠

!"uating the two e"uations an+ +i&i+ing 7 ∆ x∆ z 8 we ha&e

∂ B z ⎛ ∂ E y⎞−

∂ x= µ ) ε ) ⎜

∂t⎟

913.#. : ⎝ ⎠

The result in+icates that a time-&ar ing electric fiel+ is generate+ 7 a spatiall &ar ingmagnetic fiel+.

sing !"s. 913.#.#: an+ 913.#. :8 one ma &erif that 7oth the electric an+ magneticfiel+s satisf the one-+imensional wa&e e"uation.

To show this8 we first ta@e another partial +eri&ati&e of !". 913.#.5: with respect to x8

an+ then another partial +eri&ati&e of !". 913.#. : with respect to t /

2 2∂ E y ∂ ⎛ ∂ B z⎞ ∂ ⎛ ∂ B z⎞ ∂ ⎛ ∂ E y⎞ ∂ E y 913.#.1):∂ x2 =− x ⎜ ∂t ⎟ =− t

⎜ ∂ x ⎟ =− t⎜ − µ ) ε ) ∂t ⎟

= µ ) ε )∂t 2

∂ ⎝ ⎠ ∂ ⎝ ⎠ ∂ ⎝ ⎠

noting the interchangea7ilit of the partial +ifferentiations/

µ ε

) ) ⎜ ⎟


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∂ ⎛ ∂ B z⎞ = ∂ ⎛ ∂ B z⎞ 913.#.11:

∂ x ⎜ ∂t ⎟ ∂t ⎜

∂ x ⎟⎝ ⎠ ⎝ ⎠

*imilarl 8 ta@ing another partial +eri&ati&e of !". 913.#. : with respect to x iel+s8 an+then another partial +eri&ati&e of !". 913.#.5: with respect to t gi&es

∂ B z ∂ ⎛ ∂ E y⎞ ∂ ⎛ ∂ E y⎞ ∂ ⎛ ∂ B z⎞ ∂ B z 2 2∂ x2 =−

x⎜ µ ) ε ) ∂t ⎟

= − µ ) ε )∂t ⎜

∂ x⎟ = − µ ) ε ) ∂t ⎜

− ∂t ⎟ = µ ) ε ) ∂t2 913.#.12:∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

The results ma 7e summari=e+ as/

⎛ ∂2

∂2

⎞ ⎧ E y 9 x8t :⎫⎜ ∂ x2 −

µ

)ε

) ∂t 2 ⎟ ⎨ B 9 x8t : ⎬ =

) 913.#.13:

⎝ ⎠ ⎩ z ⎭


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0 0

ecall that the general form of a one-+imensional wa&e e"uation is gi&en 7

⎛ ∂2 1− ∂2⎞ ψ 9 x8t : = )913.#.1#:

⎜ ∂ x2 v2 ∂t 2⎟⎝ ⎠

where v is the spee+ of propagation an+ ψ 9 x8t : is the wa&e function8 we see clearl that

7oth E y an+ B z satisf the wa&e e"uation an+ propagate with the spee+ of light/

v =1

= 19#π ×1) T ⋅m?6:9 . 5 ×1) C ? ⋅m : = 2. ' ×1) m?s = c

913.#.15:

−' −12 2 2

Thus8 we conclu+e that light is an electromagnetic wa&e. The spectrum of electromagnetic wa&es is shown in 4igure 13.#.#.

"igure 13.(.( !lectromagnetic spectrum

13.(.1 ,ne-Dimensional Wave Equation

0t is straightforwar+ to &erif that an function of the form ψ 9 x ± vt : satisfies the one-+imensional wa&e e"uation shown in !". 913.#.1#:. The proof procee+s as follows/

Let x′ = x ± vt which iel+s ∂ x′ ?∂ x = 1 an+ ∂ x′ ?∂t = ±v . sing chain rule8 the first two partial +eri&ati&es with respect to x are


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∂ψ 9 x′: = ∂ψ ∂ x′ =

∂ψ 913.#.1,:

∂ x ∂ x′ ∂ x ∂ x′

∂2

ψ ∂ ⎛ ∂ψ ⎞ ∂2

ψ ∂ x′ ∂2

ψ = ⎜ ⎟ = = 913.#.1':∂ x2 ∂ x ⎝ ∂ x′ ⎠ ∂ x′2

∂ x∂ x′2

*imilarl 8 the partial +eri&ati&es in t are gi&en 7

∂ψ = ∂ψ ∂ x′ = ±v

∂ψ

913.#.1 :

∂t ∂ x′ ∂t ∂ x′

2 2 2∂ ψ = ∂ ⎛ ±v

∂ψ ⎞ = ±v∂ ψ ∂ x′ = v2

∂ ψ ∂t 2 ∂t ⎜ ∂ x′ ⎟ ∂ x′2 ∂t ∂ x′2 913.#.1 :

⎝ ⎠

Comparing !". 913.#.1': with !". 913.#.1 :8 we ha&e

∂2ψ ∂ x A2

∂2ψ 1= =∂ x2 v2

∂2

ψ ∂t 2

913.#.2):

which shows that ψ 9 x ± vt : satisfies the one-+imensional wa&e e"uation. The wa&e

e"uation is an e ample of a linear +ifferential e"uation8 which means that if ψ 1 9 x8t : an+

ψ 2 9 x8t : are solutions to the wa&e e"uation8 then ψ 1 9 x8t : ±ψ 2 9 x8t : is also a solution. Theimplication is that electromagnetic wa&es o7e the superposition principle.

(ne possi7le solution to the wa&e e"uations is.E = E 9 x8t : += E cos k 9 x − vt : += E cos9kx − ω t : +

y ) ). 913.#.21:= B 9 x8t :* = B cos k 9 x − vt :*

= Bcos9kx − ω t :*

z ) )

where the fiel+s are sinusoi+al8 with amplitu+es E ) an+relate+ to the wa&elength λ 7

B) . The angular wa&e num7er k is


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k = 2π λ 913.#.22:

an+ the angular fre"uenc ω is

ω = kv = 2π

v

= 2π f λ 913.#.23:

where f is the linear fre"uenc . 0n empt space the wa&e propagates at the spee+ of light8v = c . The characteristic 7eha&ior of the sinusoi+al electromagnetic wa&e is illustrate+ in4igure 13.#.5.


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E0 cB0k

"igure 13.(. $lane electromagnetic wa&e propagating in the B x +irection.. .

%e see that the E an+ fiel+s are alwa s in phase 9attaining ma ima an+ minima at thesame time.: To o7tain the relationship 7etween the fiel+ amplitu+es E ) an+use of !"s. 913.#.#: an+ 913.#. :. Ta@ing the partial +eri&ati&es lea+s to

B) 8 we ma@e

∂ E y∂ x = −kE ) sin9kx − ω t :

913.#.2#:

an+

∂ B z ∂t = ω B) sin9kx −

ω t :

913.#.25:

which implies E ) k = ω B) 8or

913.#.2,:

4rom !"s. 913.#.2): an+ 913.#.21:8 one ma easil show that the magnitu+es of the fiel+sat an instant are relate+ 7

E = c B913.#.2':

Let us summari=e the important features of electromagnetic wa&es +escri7e+ in !".913.#.21:/

. .

1. The wa&e is trans&erse since 7oth E an+ fiel+s are perpen+icular to the +irection. .of propagation8 which points in the +irection of the cross pro+uct E × .


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. .#. The E an+

fiel+s are perpen+icular to each other. Therefore8 their +ot pro+uct

. .&anishes8E ⋅ = ) .


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0 0

3. The ratio of the magnitu+es an+ the amplitu+es of the fiel+s is

E = E ) =

ω = c B B) k

#. The spee+ of propagation in &acuum is e"ual to the spee+ of light8 c = 1? .

5. !lectromagnetic wa&es o7e the superposition principle.

13. /tanding Electromagnetic Waves

Let us e amine the situation where there are two sinusoi+al plane electromagnetic wa&es8one tra&eling in the B x-+irection8 with

E 1 y 9 x8t : = E 1) cos9k 1 x − ω 1t :8

B1 z 9 x8t : = B1) cos9k 1 x − ω 1t :

913.5.1:

an+ the other tra&eling in the − x-+irection8 with

E 2 y 9 x8t : = − E 2) cos9k 2 x + ω 2t :8

B2 z 9 x8t : = B2) cos9k 2 x + ω 2t :

913.5.2:

4or simplicit 8 we assume that these electromagnetic wa&es ha&e the same amplitu+es9 E 1) = E 2) = E )8

B1) = B2) = B) : an+ wa&elengths 9 k 1 = k 2 = k 8 ω 1 = ω 2 = ω :. sing the

superposition principle8 the electric fiel+ an+ the magnetic fiel+s can 7e written as

E y 9 x8t : = E 1 y 9 x8t : + E 2 y 9 x8t : = E ) [cos9kx − ω t : − cos9kx + ω t :]913.5.3:

an+ B z 9 x8t : = B1 z 9 x8t : + B2 z 9 x8t : = B) [cos9kx − ω t : + cos9kx + ω t :] 913.5.#:

sing the i+entities

cos9α ± β : = cos α cos β ∓ sin α sin β 913.5.5:


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The a7o&e e pressions ma 7e rewritten as

E y 9 x8t : = E ) [cos kx cos ω t + sin kx sin ω t − cos kx cos ω t + sin kxsin ω t ]

= 2 E ) sin kx sin ω t

913.5.,:

an+

B z 9 x8t : = B) [cos kx cos ω t + sin kx sin ω t + cos kx cos ω t − sin kxsin ω t ]

= 2 B) cos kx cosω t

913.5.':

(ne ma &erif that the total fiel+s E y 9 x8t : an+ B z 9 x8t : still satisf the wa&e e"uationstate+ in !". 913.#.13:8 e&en though the no longer ha&e the form of functions of kx ± ω t .The wa&es +escri7e+ 7 !"s. 913.5.,: an+ 913.5.': are standing waves 8 which +o not

propagate 7ut simpl oscillate in space an+ time.

Let’s first e amine the spatial +epen+ence of the fiel+s. !". 913.5.,: shows that the totalelectric fiel+ remains =ero at all times if sin kx = ) 8 or

x = nπ k =

nπ 2π ?λ

= nλ

82

n = )818 28

…

.9no+al planes ofE :

913.5. :

The planes that contain these points are calle+ the nodal planes of the electric fiel+. (nthe other han+8 when sin kx = ±18 or

x = ⎛

n + 1⎞

π = ⎛ n + 1 ⎞

π

= ⎛ n +

1 ⎞ λ 8

n = )818 28… .9anti-no+al planes of E :

⎜ 2 ⎟ k ⎜ 2

⎟ 2π ?λ ⎜

2 #

⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 913.5. :

the amplitu+e of the fiel+ is at its ma imum 2 E ) . The planes that contain these points arethe anti-nodal planes of the electric fiel+. ote that in 7etween two no+al planes8 there isan anti-no+al plane8 an+ &ice &ersa.

4or the magnetic fiel+8 the no+al planes must contain points which meets the con+itioncos kx = ) . This iel+s


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x = ⎛

n + 1 ⎞ π =

⎛ n + 1 ⎞

λ 8 n = )818 28… .9no+al planes of:913.5.1):

⎜2⎟

k ⎜

2 #⎟

⎝ ⎠ ⎝ ⎠

.*imilarl 8 the anti-no+al planes for contain points that satisf cos kx = ±1 8 or

x = nπ k =nπ

2π ?λ

= nλ 82.

n = )818 28…

.9anti-no+al planes of

:

913.5.11:

.

Thus8 we see that a no+al plane of E correspon+s to an anti-no+al plane of 8 an+ &ice&ersa.

4or the time +epen+ence8 !". 913.5.,: shows that the electric fiel+ is =ero e&er wherewhen sin ω t = ) 8 or

t = nπ ω =nπ

2π ?

= n 82n = )818 28… 913.5.12:

where = 1? f = 2π ?ω is the perio+. >owe&er8 this is precisel the ma imum con+itionfor the magnetic fiel+. Thus8 unli@e the tra&eling electromagnetic wa&e in which theelectric an+ the magnetic fiel+s are alwa s in phase8 in stan+ing electromagnetic wa&es8the two fiel+s are ) ° out of phase.

*tan+ing electromagnetic wa&es can 7e forme+ 7 confining the electromagnetic wa&eswithin two perfectl reflecting con+uctors8 as shown in 4igure 13.#.,.

"igure 13.(.0 4ormation of stan+ing electromagnetic wa&es using two perfectlreflecting con+uctors.

13.0 )o nting 2ector


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0 0

0n Chapters 5 an+ 11 we ha+ seen that electric an+ magnetic fiel+s store energ . Thus8energ can also 7e carrie+ 7 the electromagnetic wa&es which consist of 7oth fiel+s.Consi+er a plane electromagnetic wa&e passing through a small &olume element of area

A an+ thic@ness dx 8 as shown in 4igure 13.,.1.

"igure 13.0.1 !lectromagnetic wa&e passing through a &olume element

The total energ in the &olume element is gi&en 7

d! = uAdx = 9u E + u B

: Adx = 1 ⎛

ε E 2 + B ⎞

Adx⎟

913.,.1:

2 ⎝ µ ) ⎠

where

u = 1

ε E 2

8 B2

u B =913.,.2:

2 2 µ )

are the energ +ensities associate+ with the electric an+ magnetic fiel+s. *ince theelectromagnetic wa&e propagates with the spee+ of light c 8 the amount of time it ta@es

for the wa&e to mo&e through the &olume element is dt = dx ? c . Thus8 one ma o7tainthe rate of change of energ per unit area8 +enote+ with the s m7ol S 8 as

S = d! = c ⎛

ε E 2 + B ⎞ 913.,.3:

Adt 2⎝ µ ) ⎠

The *0 unit of S is %?m 2. oting that E = cB an+ c = 1?ma 7e rewritten as

8 the a7o&e e pression

S = c ⎛

ε E 2 + B 2 ⎞ cB 2

⎟ = = cε ) E 2 = EB

⎜ )

2

E )

2

⎜ )⎟

⎜ )


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E B

913.,.#:2 ⎝ µ ) ⎠ µ ) µ )

0n general8 the rate of the energ flow per unit area ma 7e +escri7e+ 7 the $o nting.

&ector / 9after the Hritish ph sicist Iohn $o nting:8 which is +efine+ as

. . .

/ = 1

µ ).

E × 913.,.5:

. .

with / pointing in the +irection of propagation. *ince the fiel+s E an+ are perpen+icular8 we ma rea+il &erif that the magnitu+e of / is

. .

J / J= = EB = S 913.,.,:

µ ) µ )

6s an e ample8 suppose the electric component of the plane electromagnetic wa&e is.

E = E ) cos9kx − ω t : + . The correspon+ing magnetic component is.

= B) cos9kx − ω t :* D8

an+ the +irection of propagation is B x. The $o nting &ector can 7e o7taine+ as

. 1/ =

E

cos9kx −ω t :D +)×( B cos9kx −ω t :* ) = E ) B) cos 2 9kx−ω t : i

913.,.':

µ ) )

µ ) )

.

(


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E0 B00

10

"igure 13.0.# $o nting &ector for a plane wa&e.

6s e pecte+8 / points in the +irection of wa&e propagation 9see 4igure 13.,.2:.

The intensit of the wa&e8 I 8 +efine+ as the time a&erage of S 8 is gi&en 7

2 2

I = S = cos 2 9kx −ω t : = E ) B) =2 µ ) E )2c µ )

= cB)2 µ )913.,. :

where we ha&e use+ cos 2 9kx −ω t : = 1

2 913.,. :

To relate intensit to the energ +ensit 8 we first note the e"ualit 7etween the electrican+ the magnetic energ +ensities/

B2 9 E ? c: 2 E 2 ε E 2u = = = = ) = u 913.,.1): B

2 µ 2 µ 2c2 µ 2 E ) ) )

The a&erage total energ +ensit then 7ecomes

u = u E + u B = ε )

E 2 = ε ) 2

2)

2

913.,.11:

= B2= B)2 µ )

Thus8 the intensit is relate+ to the a&erage energ +ensit 7

I = S = c u 913.,.12:

E


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Example 13.1 /olar Constant

6t the upper surface of the !arth’s atmosphere8 the time-a&erage+ magnitu+e of the

$o nting &ector8 S = 1.35 ×1) 3 % m 2 8 is referre+ to as the solar constant .


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2 Sc 0

2 1.35 10 3 Wm 2 3.0 10 m ! . 5 10 1 2 " 2 # m 2

1.01 10 3 $ m3.0 10 m !

%

9a: 6ssuming that the *un’s electromagnetic ra+iation is a plane sinusoi+al wa&e8 what arethe magnitu+es of the electric an+ magnetic fiel+sK

97: %hat is the total time-a&erage+ power ra+iate+ 7 the *unK The mean *un-!arth

+istance is " = 1.5) ×1) 11 m .

/olution

9a: The time-a&erage+ $o nting &ector is relate+ to the amplitu+e of the electric fiel+ 7

S = c ε 2

2) )

Thus8 the amplitu+e of the electric fiel+ is

E = = = 1.)1 ×1) 3 m .

The correspon+ing amplitu+e of the magnetic fiel+ is

B = E ) = = 3.# ×1) −, T .)

c

ote that the associate+ magnetic fiel+ is less than one-tenth the !arth’s magnetic fiel+.

97: The total time a&erage+ power ra+iate+ 7 the *un at the +istance " is

# = S A =S #π " 2 = (1.35 ×1) 3 % m 2 ) #π (1.5) ×1) 11 m )2 = 3. ×1) 2, %

The t pe of wa&e +iscusse+ in the e ample a7o&e is a spherical wa&e 94igure 13.,.3a:8

which originates from a ;point-li@e< source. The intensit at a +istance r from the sourceis

I = S = 913.,.13:

#π r 2

which +ecreases as 1? r 2 . (n the other han+8 the intensit of a plane wa&e 94igure 13.,.37:remains constant an+ there is no sprea+ing in its energ .

E .

)


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"igure 13.0.3 9a: a spherical wa&e8 an+ 97: plane wa&e.

Example 13.# 4ntensit o& a /tanding Wave

Compute the intensit of the stan+ing electromagnetic wa&e gi&en 7

E y 9 x8t : = 2 E ) cos kx cos ω t 8 B z 9 x8t : = 2 B) sin kx sin ω t

/olution

The $o nting &ector for the stan+ing wa&e is

. . .

/ = E× =

192 E cos kx cos ω t +:

×92 Bsin kx sin ω t * :

µ ) µ )=

# E ) B) 9sin kx cos kx sin ω tcosω t : i

µ )5

E ) B) 9sin 2kx sin 2 ω t : i µ )

913.,.1#:

The time a&erage of S is S 5 E ) B) sin

2kx µ )

sin 2 ω t = ) 913.,.15:

The result is to 7e e pecte+ since the stan+ing wa&e +oes not propagate. 6lternati&el 8 wema sa that the energ carrie+ 7 the two wa&es tra&eling in the opposite +irections toform the stan+ing wa&e e actl cancel each other8 with no net energ transfer.

13.0.1 Energ Transport.

*ince the $o nting &ector / represents the rate of the energ flow per unit area8 the rate

) )


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.where d ! = dAn 8 where n is a unit &ector in the outward normal +irection. Thea7o&ee pression allows us to interpret / as the energ flu +ensit 8 in analog to the current

.+ensit ' in

. .

I = dQ = ' ⋅ d ! dt 913.,.1':

0f energ flows out of the s stem8 then / = S nan+

d! ? dt < ) 8 showing an o&erall

+ecrease of energ in the s stem. (n the other han+8 if energ flows into the s stem8 then./ = S 9−n : an+ d! ? dt > ) 8 in+icating an o&erall increase of energ .

6s an e ample to eluci+ate the ph sical meaning of the a7o&e e"uation8 let’s consi+er anin+uctor ma+e up of a section of a &er long air-core solenoi+ of length l 8 ra+ius r an+ nturns per unit length. *uppose at some instant the current is changing at a rate dI ? dt > ) .

sing 6mpere’s law8 the magnetic fiel+ in the solenoi+ is. .

° ∫ ⋅ d s = Bl = µ ) 9 $I :C or

.

= µ nI * 913.,.1 :

Thus8 the rate of increase of the magnetic fiel+ is

dB = µ ndI

913.,.1 :

dt )

dt

6ccor+ing to 4ara+a ’s law/

ε =

° ∫ .⋅

d

.=

−

d Φ B

913.,.2):

C dt

changing magnetic flu results in an in+uce+ electric fiel+.8 which is gi&en 7

E (2π r ) = − µ n⎛ dI ⎞π r 2

) ⎜ ⎟

∫∫

)

E s

d


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⎝ ⎠or

. µ nr ⎛ dI ⎞E = − ) 6913.,.21:

⎜ ⎟2 ⎝ ⎠d


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.The +irection of E is cloc@wise8 the same as the in+uce+ current8 as shown in 4igure13.,.#.

"igure 13.0.( $o nting &ector for a solenoi+ with dI ? dt > )

The correspon+ing $o nting &ector can then 7e o7taine+ as

.. . 2

/ = E × = 1 (− µ ) nr ⎛ dI ⎞ 6 ) ×( µ nI * ) = − µ ) n rI ⎛ dI ⎞ r

913.,.22:

µ 0 ) * ⎜ ⎟ )2 ⎝ dt ⎠ +⎜ ⎟

2 ⎝ dt⎠

which points ra+iall inwar+8 i.e.8 along the −r +irection. The +irections of the fiel+s an+the $o nting &ector are shown in 4igure 13.,.#.

*ince the magnetic energ store+ in the in+uctor is

! = ⎛ B ⎞

9π r 2l : = 1 µ π n2 I 2 r 2l 913.,.23:

B ⎜ ⎟ )⎝ 2 µ ) ⎠ 2

the rate of change of ! B is

d! B 2 2 ⎛ dI ⎞

where

# = dt

= µ ) π n Ir l ⎜ dt⎟ = I Jε J

913.,.2#:

d Φ B ⎛ dB⎞ 2 2 2⎛ dI⎞ε = − $ dt = −9nl : ⎜ dt ⎟

π r

= − µ ) n l π r

⎜ dt⎟ 913.,.25:

⎝ ⎠ ⎝ ⎠

is the in+uce+ emf. (ne ma rea+il &erif that this is the same as

. . µ ) n rI ⎛ dI ⎞ 2 2⎛ dI ⎞

, - µ

2

⎝ ⎠

2


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1d/d'

1d&/cd'

d&d'

−° ∫ / ⋅ d != ⎜ ⎟ ⋅92π rl : = µ ) π n Ir l ⎜ ⎟ 913.,.2,:2 ⎝ dt ⎠ ⎝ dt⎠

Thus8 we ha&e

d! . . B = − / ⋅ d ! > )dt

913.,.2':

The energ in the s stem is increase+8 as e pecte+ when dI ? dt > ) . (n the other han+8 if dI ? dt < ) 8 the energ of the s stem woul+ +ecrease8 with d! B ? dt < ) .

13.7 Momentum and 8adiation )ressure

The electromagnetic wa&e transports not onl energ 7ut also momentum8 an+ hence cane ert a radiation pressure on a surface +ue to the a7sorption an+ reflection of themomentum. Ma well showe+ that if the plane electromagnetic wa&e is completela7sor7e+ 7 a surface8 the momentum transferre+ is relate+ to the energ a7sor7e+ 7

∆ p = ∆!

c

9completea7sorption:

913.'.1:

(n the other han+8 if the electromagnetic wa&e is completel reflecte+ 7 a surface suchas a mirror8 the result 7ecomes

∆ p = 2∆! c

9completereflection:

913.'.2:

4or the complete a7sorption case8 the a&erage ra+iation pressure 9force per unit area: isgi&en 7

% # = = = 913.'.3: A

*ince the rate of energ +eli&ere+ to the surface is

= S A = IA

we arri&e at


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# = I

c9completea7sorption:

913.'.#:

*imilarl 8 if the ra+iation is completel reflecte+8 the ra+iation pressure is twice as greatas the case of complete a7sorption/

# = 2 I c9completereflection:

913.'.5:


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13.9 )roduction o& Electromagnetic Waves

!lectromagnetic wa&es are pro+uce+ when electric charges are accelerate+. 0n other wor+s8 a charge must ra+iate energ when it un+ergoes acceleration. a+iation cannot 7e

pro+uce+ 7 stationar charges or stea+ currents. 4igure 13. .1 +epicts the electric fiel+lines pro+uce+ 7 an oscillating charge at some instant.

"igure 13.9.1 !lectric fiel+ lines of an oscillating point charge

6 common wa of pro+ucing electromagnetic wa&es is to appl a sinusoi+al &oltagesource to an antenna8 causing the charges to accumulate near the tips of the antenna. Theeffect is to pro+uce an oscillating electric +ipole. The pro+uction of electric-+ipolera+iation is +epicte+ in 4igure 13. .2.

"igure 13.9.# !lectric fiel+s pro+uce+ 7 an electric-+ipole antenna.

6t time t = ) the en+s of the ro+s are charge+ so that the upper ro+ has a ma imum positi&e charge an+ the lower ro+ has an e"ual amount of negati&e charge. 6t this instantthe electric fiel+ near the antenna points +ownwar+. The charges then 7egin to +ecrease.6fter one-fourth perio+8 t = ?# 8 the charges &anish momentaril an+ the electric fiel+strength is =ero. *u7se"uentl 8 the polarities of the ro+s are re&erse+ with negati&echarges continuing to accumulate on the upper ro+ an+ positi&e charges on the lower untilt = ?2 8 when the ma imum is attaine+. 6t this moment8 the electric fiel+ near the ro+

points upwar+. 6s the charges continue to oscillate 7etween the ro+s8 electric fiel+s are pro+uce+ an+ mo&e awa with spee+ of light. The motion of the charges also pro+uces acurrent which in turn sets up a magnetic fiel+ encircling the ro+s. >owe&er8 the 7eha&ior


35/83

of the fiel+s near the antenna is e pecte+ to 7e &er +ifferent from that far awa from theantenna.

Let us consi+er a half-wa&elength antenna8 in which the length of each ro+ is e"ual to one"uarter of the wa&elength of the emitte+ ra+iation. *ince charges are +ri&en to oscillate

7ac@ an+ forth 7etween the ro+s 7 the alternating &oltage8 the antenna ma 7eappro imate+ as an oscillating electric +ipole. 4igure 13. .3 +epicts the electric an+ themagnetic fiel+ lines at the instant the current is upwar+. otice that the $o nting &ectorsat the positions shown are +irecte+ outwar+.

"igure 13.9.3 !lectric an+ magnetic fiel+ lines pro+uce+ 7 an electric-+ipole antenna.

0n general8 the ra+iation pattern pro+uce+ is &er comple . >owe&er8 at a +istance whichis much greater than the +imensions of the s stem an+ the wa&elength of the ra+iation8the fiel+s e hi7it a &er +ifferent 7eha&ior. 0n this ;far region8< the ra+iation is cause+ 7the continuous in+uction of a magnetic fiel+ +ue to a time-&ar ing electric fiel+ an+ &ice&ersa. Hoth fiel+s oscillate in phase an+ &ar in amplitu+e as 1? r .

The intensit of the &ariation can 7e shown to &ar as sin 2 θ ? r 2 8 where θ is the anglemeasure+ from the a is of the antenna. The angular +epen+ence of the intensit I 9θ : isshown in 4igure 13. .#. 4rom the figure8 we see that the intensit is a ma imum in a

plane which passes through the mi+point of the antenna an+ is perpen+icular to it.

"igure 13.9.( 6ngular +epen+ence of the ra+iation intensit .


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Animation 13.1 Electric Dipole 8adiation 1

Consi+er an electric +ipole whose +ipole moment &aries in time accor+ing to

p9t : = p 1 2 π t )

,1+ cos ⎜ ⎟ * 913. .1:

. ( ⎛ ⎞)* 1) ⎝ +

4igure 13. .5 shows one frame of an animation of these fiel+s. Close to the +ipole8 thefiel+ line motion an+ thus the $o nting &ector is first outwar+ an+ then inwar+8correspon+ing to energ flow outwar+ as the "uasi-static +ipolar electric fiel+ energ is

7eing 7uilt up8 an+ energ flow inwar+ as the "uasi-static +ipole electric fiel+ energ is 7eing +estro e+.

"igure 13.9. a+iation from an electric +ipole whose +ipole moment &aries 7 1) .

!&en though the energ flow +irection changes sign in these regions8 there is still a smalltime-a&erage+ energ flow outwar+. This small energ flow outwar+ represents the smallamount of energ ra+iate+ awa to infinit . (utsi+e of the point at which the outer fiel+lines +etach from the +ipole an+ mo&e off to infinit 8 the &elocit of the fiel+ lines8 an+thus the +irection of the electromagnetic energ flow8 is alwa s outwar+. This is theregion +ominate+ 7 ra+iation fiel+s8 which consistentl carr energ outwar+ to infinit .

Animation 13.2 Electric Dipole 8adiation #

4igure 13. ., shows one frame of an animation of an electric +ipole characteri=e+ 7

.p9t : = p cos ⎛ 2π t ⎞ *

913. .2:

) ⎜ ⎟⎝ ⎠

-⎠


37/83

The e"uation shows that the +irection of the +ipole moment &aries 7etween +* an+ −* .


38/83


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To ma@e an electromagnetic plane wa&e8 we +o much the same thing we +o when wema@e wa&es on a string. %e gra7 the string somewhere an+ sha@e it8 an+ there7


40/83

generate a wa&e on the string. %e +o wor@ against the tension in the string when wesha@e it8 an+ that wor@ is carrie+ off as an energ flu in the wa&e. !lectromagneticwa&es are much the same proposition. The electric fiel+ line ser&es as the ;string.< 6swe will see 7elow8 there is a tension associate+ with an electric fiel+ line8 in that when wesha@e it 9tr to +isplace it from its initial position:8 there is a restoring force that resiststhe sha@e8 an+ a wa&e propagates along the fiel+ line as a result of the sha@e. Toun+erstan+ in +etail what happens in this process will in&ol&e using most of theelectromagnetism we ha&e learne+ thus far8 from GaussAs law to 6mpereAs law plus thereasona7le assumption that electromagnetic information propagates at spee+ c in a&acuum.

>ow +o we sha@e an electric fiel+ line8 an+ what +o we gra7 on toK %hat we +o is sha@ethe electric charges that the fiel+ lines are attache+ to. 6fter all8 it is these charges that

pro+uce the electric fiel+8 an+ in a &er real sense the electric fiel+ is roote+ in theelectric charges that pro+uce them. Nnowing this8 an+ assuming that in a &acuum8electromagnetic signals propagate at the spee+ of light8 we can prett much pu==le outhow to ma@e a plane electromagnetic wa&e 7 sha@ing charges. LetAs first figure out howto ma@e a kink in an electric fiel+ line8 an+ then weAll go on to ma@e sinusoi+al wa&es.

*uppose we ha&e an infinite sheet of charge locate+ in thesurface charge +ensit σ 8 as shown in 4igure 13. . .

yz - plane8 initiall at rest8 with

"igure 13.9.9 !lectric fiel+ +ue to an infinite sheet with charge +ensit σ .

4rom GaussAs law +iscusse+ in Chapter #8 we @now that this surface charge will gi&e rise.

to a static electric fiel+ E ) /

. ⎧0+9σ E ) = ⎨

0⎩−9σ 2ε )2ε )

:

i8:

i8

x > )

x < )

913. .3:


41/83

ow8 at t = ) 8 we gra7 the sheet of charge an+ start pulling it downward with constant .&elocit v = −v + . LetAs e amine how things will then appear at a later time t = . 0n

particular8 7efore the sheet starts mo&ing8 letAs loo@ at the fiel+ line that goes through y = ) for t < ) 8 as shown in 4igure 13. . 9a:.


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"igure 13.9.; !lectric fiel+ lines 9a: through y = ) at t < ) 8 an+ 97: at t = .

The ;foot< of this electric fiel+ line8 that is8 where it is anchore+8 is roote+ in the electriccharge that generates it8 an+ that ;foot< must mo&e +ownwar+ with the sheet of charge8 at

the same spee+ as the charges mo&e +ownwar+. Thus the ;foot< of our electric fiel+ line8which was initiall ata is at time t = .

y = ) at t = ) 8 will ha&e mo&e+ a +istance y = −v

+own the y-

%e ha&e assume+ that the information that this fiel+ line is 7eing +ragge+ +ownwar+ will propagate outwar+ from x = ) at the spee+ of light c . Thus the portion of our fiel+ linelocate+ a +istance x >

c along the x-a is from the origin +oesnAt @now the charges are

mo&ing8 an+ thus has not et 7egun to mo&e +ownwar+. (ur fiel+ line therefore mustappear at time t = as shown in 4igure 13. . 97:. othing has happene+ outsi+e of J x J> c O the foot of the fiel+ lineat

x = ) is a +istance y = −v

+own the y-a is8 an+ we

ha&e guesse+ a7out what the fiel+ line must loo@ li@e for ) < J x J< c 7 simplconnecting the two positions on the fiel+ line that we @now a7out at time 9 x = ) an+J x J= c : 7 a straight line. This is e actl the guess we woul+ ma@e if we were +ealingwith a string instea+ of an electric fiel+. This is a reasona7le thing to +o8 an+ it turns outto 7e the right guess.

%hat we ha&e +one 7 pulling +own on the charge+ sheet is to generate a pertur7ation in. . .

the electric fiel+8 E 1 in a++ition to the static fiel+ E ) . Thus8 the total fiel+ E for )


43/83

tan θ = E 1 =

v = v

913. .5:

E ) c c

where.

E 1 =JE 1 Jan+

.

E ) =JE ) J are the magnitu+es of the fiel+s8 an+ θ is the angle with

the x-a is. sing !". 913. .5:8 the pertur7ation fiel+ can 7e written as

.= ⎛ v

E ⎞ D += ⎛ vσ ⎞ D +

913. .,:

1 ⎜ ) ⎟ ⎜ ⎟⎝ c ⎠ ⎝ 2ε ) c⎠

where we ha&e use+ E ) = σ

2ε ) . %e ha&e generate+ an electric fiel+ pertur7ation8 an+.

this e pression tells us how large the pertur7ation fiel+ E 1 is for a gi&en spee+ of thesheet of charge8 v .

This e plains wh the electric fiel+ line has a tension associate+ with it8 Pust as a string.

+oes. The +irection of E 1 is such that the forces it e erts on the charges in the sheetresist the motion of the sheet. That is8 there is an upward electric force on the sheet whenwe tr to mo&e it downward . 4or an infinitesimal area dA of the sheet containing charge

d' = σ dA 8 the upwar+ ;tension< associate+ with the electric fiel+ is. . 2

d " = d' E ⎛ vσ = 9σ dA: ⎞ D += ⎛

vσ dA⎞

D +

913. .':

e 1 ⎜ ⎟ ⎜⎝ 2ε ) c ⎠ ⎝

⎟2ε ) c ⎠

Therefore8 to o&ercome the tension8 the e ternal agent must appl an e"ual 7ut opposite9downward : force

. . 2d " = −d " ⎛ vσ dA⎞

D + 913. . :e t e

= −⎜⎝

⎟

2ε )c ⎠. .

*ince the amount of wor@ +one is d( e t = " e t ⋅ d s 8 the wor@ +one per unit time per unitarea 7 the e ternal agent is

. .2 " s ⎛σ 2 ⎞ 2σ 2

E


44/83

d ( e t = d e t ⋅

d = ⎜ − v

+⎟ ⋅ (−v +) = v 913. . :dAdt dA

dt ⎝ 2ε ) c ⎠ 2ε ) c

%hat else has happene+ in this process of mo&ing the charge+ sheet +ownK %ell8 oncethe charge+ sheet is in motion8 we ha&e create+ a sheet of current with surface current.+ensit 9current per unit length: < = −σ v + . 4rom 6mpereAs law8 we @now that a

.magnetic field has 7een create+8 in a++ition tomagnetic fiel+ 9see ! ample .#:

E 1 ) The current sheet will pro+uce a

. ⎧+9 µ σ v 2: * 8 x > )= 0 ) 913. .1):1 ⎨−9 µ σ v 2:* 8 x < )⎩0 )


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This magnetic fiel+ changes +irection as we mo&e from negati&e to positi&e &alues of x 89across the current sheet:. The configuration of the fiel+ +ue to a +ownwar+ current isshown in 4igure 13. .1) for J x J< c . 6gain8 the information that the charge+ sheet hasstarte+ mo&ing8 pro+ucing a current sheet an+ associate+ magnetic fiel+8 can onl

propagate outwar+ from x = ) at the spee+ of light c . Therefore the magnetic fiel+ is still. .

=ero8 = =

for J x J> c . ote that

E 1 = vσ ? 2ε ) c = 1 = c

913. .11:

B1 µ ) σ v ?2 c µ ) ε )

"igure 13.9.1= Magnetic fiel+ at t = .

The magnetic fiel+. .

1 generate+ 7 the current sheet is perpen+icular to E 1

with a

magnitu+e B1 = E 1 ?c 8 as e pecte+ for a trans&erse electromagnetic wa&e.

ow8 let’s +iscuss the energ carrie+ awa 7 these pertur7ation fiel+s. The energ flu.

associate+ with an electromagnetic fiel+ is gi&en 7 the $o nting &ector / . 4or x > ) 8the energ flowing to the rig&t is

. 1 . . 1 ⎛ vσ D⎞ ⎛ µ σ v

D⎞ ⎛ v2σ 2⎞ D

/ = E × = + × )

* = i 913. .12:1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ µ ) µ ) ⎝ 2ε ) c ⎠ ⎝ 2 ⎠ ⎝ #ε ) c⎠

This is onl half of the wor@ we +o per unit time per unit area to pull the sheet +own8 asgi&en 7 !". 913. . :. *ince the fiel+s on the left carr e actl the same amount of

.energ flu to t&e left 8 9the magnetic fiel+ 1 changes +irection across the plane. x = )

whereas the electric fiel+ E 1 +oes not8 so the $o nting flu also changes across x = ) :.


46/83

*o the total energ flu carrie+ off 7 the pertur7ation electric an+ magnetic fiel+s weha&e generate+ is exactly e'ual to the rate of wor@ per unit area to pull the charge+ sheet+own against the tension in the electric fiel+. Thus we ha&e generate+ pertur7ationelectromagnetic fiel+s that carr off energ at e actl the rate that it ta@es to create them.


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%here +oes the energ carrie+ off 7 the electromagnetic wa&e come fromK The e ternalagent who originall ;shoo@< the charge to pro+uce the wa&e ha+ to +o wor@ against the

pertur7ation electric fiel+ the sha@ing pro+uces8 an+ that agent is the ultimate source of the energ carrie+ 7 the wa&e. 6n e actl analogous situation e ists when one as@swhere the energ carrie+ 7 a wa&e on a string comes from. The agent who originallshoo@ the string to pro+uce the wa&e ha+ to +o wor@ to sha@e it against the restoringtension in the string8 an+ that agent is the ultimate source of energ carrie+ 7 a wa&e ona string.

13.9.# /inusoidal Electromagnetic Wave

>ow a7out generating a sinusoi+al wa&e with angular fre"uenc ω K To +o this8 instea+of pulling the charge sheet +own at constant spee+8 we Pust sha@e it up an+ +own with a

.&elocit v9t : = −v) cosω t + . The oscillating sheet of charge will generate fiel+s which aregi&en 7 /

. .E =

c µ ) σ v) cos ω ⎛ t − x

⎞ +8 =

µ )σ v) cos ω

⎛ t − x

⎞

* 913. .13:

1 2 ⎜ c ⎟ 1 2 ⎜ c⎟⎝ ⎠ ⎝ ⎠

for x > ) an+8 for x < ) 8

. .E = c µ ) σ v) cos ω ⎛ t + x ⎞ +8 = − µ ) σ v) cos ω ⎛ t + x ⎞ *

913. .1#:

1 2 ⎜ c ⎟ 1 2 ⎜ c⎟⎝ ⎠ ⎝ ⎠

0n !"s. 913. .13: an+ 913. .1#: we ha&e chosen the amplitudes of these terms to 7e theamplitu+es of the @in@ generate+ a7o&e for constant spee+ of the sheet8 with E 1 ? B1 = c 8

7ut now allowing for the fact that the spee+ is &ar ing sinusoi+all in time withfre"uenc ω . Hut wh ha&e we put the 9 t − x ?c:cosine function in !"s. 913. .13: an+ 913. .1#:K

an+ 9t + x ? c: in the arguments for the

Consi+er first x > ) . 0f we are sitting at some x > ) at time t 8 an+ are measuring anelectric fiel+ there8 the fiel+ we are o7ser&ing shoul+ not +epen+ on what the currentsheet is +oing at t&at o7ser&ation time t . 0nformation a7out what the current sheet is+oing ta@es a time x ? c to propagate out to the o7ser&er at x > ) . Thus what the o7ser&er at x > ) sees at time t +epen+s on what the current sheet was +oing at an earlier time 8namel t − x ?c . The electric fiel+ as a function of time shoul+ reflect that time +ela +ueto the finite spee+ of propagation from the origin to some x > ) 8 an+ this is the reason the9t − x ? c: appears in !". 913. .13:8 an+ not t itself. 4or x < ) 8 the argument is e actl the


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same8 e cept if x < ) 8 t + x ?c

is the e pression for the earlier time8 an+ not t − x ? c . This

is e actl the time-+ela effect one gets when one measures wa&es on a string. 0f we aremeasuring wa&e amplitu+es on a string some +istance awa from the agent who issha@ing the string to generate the wa&es8 what we measure at time t +epen+s on what the


49/83

agent was +oing at an earlier time8 allowing for the wa&e to propagate from the agent tothe o7ser&er.

0f we note that cos ω 9t − x ? c: = cos (ω t −

kx)

where k = ω

c

is the wa&e num7er8 we see

that !"s. 913. .13: an+ 913. .1#: are precisel the @in+s of plane electromagnetic wa&es.

we ha&e stu+ie+. ote that we can also easil arrange to get ri+ of our static fiel+ E ) 7simpl putting a stationar charge+ sheet with charge per unit area −σ at x = ) . Thatcharge+ sheet will cancel out the static fiel+ +ue to the positi&e sheet of charge8 7ut willnot affect the pertur7ation fiel+ we ha&e calculate+8 since the negati&el -charge+ sheet isnot mo&ing. 0n realit 8 that is how electromagnetic wa&es are generate+--with an o&erallneutral me+ium where charges of one sign 9usuall the electrons: are accelerate+ whilean e"ual num7er of charges of the opposite sign essentiall remain at rest. Thus an

o7ser&er onl sees the wa&e fiel+s8 an+ not the static fiel+s. 0n the following8 we will.assume that we ha&e set E ) to =ero in this wa .

"igure 13.;.( !lectric fiel+ generate+ 7 the oscillation of a current sheet.

The electric fiel+ generate+ 7 the oscillation of the current sheet is shown in 4igure13. .118 for the instant when the sheet is mo&ing down an+ the pertur7ation electric fiel+is up. The magnetic fiel+s8 which point into or out of the page8 are also shown.

%hat we ha&e accomplishe+ in the construction here8 which reall onl assumes that thefeet of the electric fiel+ lines mo&e with the charges8 an+ that information propagates at c

is to show we can generate such a wa&e 7 sha@ing a plane of charge sinusoi+all . Thewa&e we generate has electric an+ magnetic fiel+s perpen+icular to one another8 an+trans&erse to the +irection of propagation8 with the ratio of the electric fiel+ magnitu+e tothe magnetic fiel+ magnitu+e e"ual to the spee+ of light. Moreo&er8 we see +irectl. . .where the energ flu / = E × ? µ ) carrie+ off 7 the wa&e comes from. The agent whosha@es the charges8 an+ there7 generates the electromagnetic wa&e puts the energ in. 0f we go to more complicate+ geometries8 these statements 7ecome much more complicate+in +etail8 7ut the o&erall picture remains as we ha&e presente+ it.


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Let us rewrite slightl the e pressions gi&en in !"s. 913. .13: an+ 913. .1#: for the fiel+sgenerate+ 7 our oscillating charge+ sheet8 in terms of the current per unit length in the. . .sheet8 < 9t : = σ v9t : + . *ince v9t : = −v cos ω t + >it follows that < 9t : = −σ v

cosω t +) Thus8

. . .

E 9 x8t :c µ ) < 9t − x ? c:8 9 x8t : = i × E 1

9 x8t : 913. .15:

1 2 1 c

for x > ) 8 an+

. . .

E 9 x8t :c µ ) < 9t + x ? c:8 9 x8t : = − i ×

E 1 9 x8t : 913. .1,:

1 2 1 c

for .

x < ) . ote that 1 9 x8t : re&erses +irection across the current sheet8 with a Pump of . µ ) < 9t : at the sheet8 as it must from 6mpereAs law. Any oscillating sheet of current must generate the plane electromagnetic wa&es +escri7e+ 7 these e"uations8 Pust as anystationar electric charge must generate a Coulom7 electric fiel+.

13.; /ummar

• The !mpere-Maxwell law rea+s

. .⋅ d s = µ I + µ ε

d Φ E = µ I + I

° ∫ ) ) )dt

) 9 d :

) )

=−

=−

$ote* To a&oi+ possi7le future confusion8 we point out that in a more a+&ance+

electromagnetism course8 ou will stu+ the ra+iation fiel+s generate+ 7 a singleoscillating charge8 an+ fin+ that the are proportional to the acceleration of the charge.This is &er +ifferent from the case here8 where the ra+iation fiel+s of our oscillatingsheet of charge are proportional to the velocity of the charges. >owe&er8 there is nocontra+iction8 7ecause when ou a++ up the ra+iation fiel+s +ue to all the single chargesma@ing up our sheet8 ou reco&er the same result we gi&e in !"s. 913. .15: an+ 913. .1,:


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where I d = ε ) d Φ E dt

is calle+ the displacement current . The e"uation +escri7es how changing electricflu can in+uce a magnetic fiel+.


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0 0

• Gauss’s law for magnetism is.

Φ B = º ∫∫ ⋅ d ! = )S

The law states that the magnetic flu through a close+ surface must 7e =ero8 an+implies the a7sence of magnetic monopoles.

• !lectromagnetic phenomena are +escri7e+ 7 the Maxwell’s equations /. . . .Q

º ∫∫ E ⋅ d !=d Φ

° ∫ E ⋅ d s = − BS ε )

. .dt

. . d Φ E º ∫∫ ⋅ d != )

S ° ∫

⋅ d s = µ ) I + µ ) ε )dt

• 0n free space8 the electric an+ magnetic components of the electromagnetic wa&eo7e a wa&e e"uation/

⎛ ∂2 ∂2 ⎞ ⎧ E y 9 x8t :⎫⎜ ∂ x2 − µ ) ε ) ∂t 2

⎟ ⎨ B 9 x8t :

⎬ = )

⎝ ⎠ ⎩ z ⎭

• The magnitu+es an+ the amplitu+es of the electric an+ magnetic fiel+s in anelectromagnetic wa&e are relate+ 7

E = E ) = ω = c = 1 ≈ 3.)) ×1) m?s

B B) k

• 6 standing electromagnetic wave +oes not propagate8 7ut instea+ the electrican+ magnetic fiel+s e ecute simple harmonic motion perpen+icular to the woul+-

7e +irection of propagation. 6n e ample of a stan+ing wa&e is

E y 9 x8t : = 2 E ) sin kx sin ω t 8 B z 9 x8t : = 2 B) cos kx cos ω t

• The energ flow rate of an electromagnetic wa&e through a close+ surface isgi&en 7

where

d! . .

dt= −º ∫∫ / ⋅ d !


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. . .

/ = 1

µ )E×


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is the )o nting vector 8 an+ / points in the +irection the wa&e propagates.

• The intensit of an electromagnetic wa&e is relate+ to the a&erage energ +ensit 7

I = S = c u

• The momentum transferre+ is relate+ to the energ a7sor7e+ 7

⎧ ∆! 9complete a7sorption:

∆ p = 0 c

02∆! 9complete reflection:

⎩0 c

• The a&erage radiation pressure on a surface 7 a normall inci+entelectromagnetic wa&e is

⎧ I

# = 0c

0 2 I 0⎩ c

9complete a7sorption:

9complete reflection:

13.1=!ppendix 8e&lection o& Electromagnetic Waves at Conducting /ur&aces

>ow +oes a &er goo+ con+uctor reflect an electromagnetic wa&e falling on itK 0n wor+s8what happens is the following. The time-&ar ing electric fiel+ of the incoming wa&e+ri&es an oscillating current on the surface of the con+uctor8 following (hmAs law. Thatoscillating current sheet8 of necessit 8 must generate wa&es propagating in 7oth +irectionsfrom the sheet. (ne of these wa&es is the reflecte+ wa&e. The other wa&e cancels out theincoming wa&e insi+e the con+uctor. Let us ma@e this "ualitati&e +escription "uantitati&e.

⎨

⎨


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"igure 13.1=.1 eflection of electromagnetic wa&es at con+ucting surface


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*uppose we ha&e an infinite plane wa&e propagating in the B x-+irection8 with

. .

E ) = E ) cos (ω t − kx) +8 ) = B) cos (ω t − kx)*

913.1).1:

as shown in the top portion of 4igure 13.1).1. %e put at the origin 9 x = ) : a con+uctingsheet with wi+th , 8 which is much smaller than the wa&elength of the incoming wa&e.

This con+ucting sheet will reflect our incoming wa&e. >owK The electric fiel+ of the. .

incoming wa&e will cause a current ' = E ρ to flow in the sheet8 where ρ is theresisti&it 9not to 7e confuse+ with charge per unit &olume:8 an+ is e"ual to the reciprocalof con+ucti&it σ 9not to 7e confuse+ with charge per unit area:. Moreo&er8 the +irection.of ' will e in t&e same direction as t&e electric field of t&e incoming wave 8 as shown inthe s@etch. Thus our incoming wa&e sets up an oscillating sheet of current with current

. . per unit length . < = ' , . 6s in our +iscussion of the generation of plane electromagneticwa&es a7o&e8 this current sheet will also generate electromagnetic waves 8 mo&ing 7oth tothe right an+ to the left 9see lower portion of 4igure 13.1).1: awa from the oscillatingsheet of charge. sing !". 913. .15: for x > ) the wa&e generate+ 7 the current will 7e

.

E 1 9 x8t :

.

c µ ) /, cos (ω t − kx) +

2

913.1).2:

where / =J ' J . 4or x < ) 8 we will ha&e a similar e pression8 e cept that the argument.will 7e 9 ω t + kx: 9see 4igure 13.1).1:. ote the sign of this electric fiel+ E 1 at. x = ) O itis down 9 − + : when the sheet of current is up 9an+ E is up8 + + :8 an+ &ice-&ersa8 Pust as

.we saw 7efore. Thus8 for x > ) 8 the generate+ electric fiel+ E 1 will alwa s 7e opposite the+irection of the electric fiel+ of the incoming wa&e8 an+ it will tend to cancel out t&eincoming wave for x > ) . 4or a &er goo+ con+uctor8 we ha&e 9see ne t section:

. 2 E

0 =J < J= /, =)

c µ )

913.1).3:

so that for x > ) we will ha&e .E 1 9 x8t : = − E ) cos (ω t − kx) +

913.1).#:

=−

)


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That is8 for a &er goo+ con+uctor8 the electric fiel+ of the wa&e generate+ 7 the currentwill exactly cancel the electric fiel+ of the incoming wa&e for x > ) Q 6n+ thatAs what a&er goo+ con+uctor +oes. 0t supports e actl the amount of current per unit length 0 = 2 E ) ? c µ ) nee+e+ to cancel out the incoming wa&e for x > ) . 4or x < ) 8 t&is samecurrent generates a ;reflecte+< wa&e propagating 7ac@ in the +irection from which the


58/83

original incoming wa&e came8 wit& t&e same amplitude as t&e original incoming wave .This is how a &er goo+ con+uctor totally reflects electromagnetic wa&es. Helow weshall show that 0 will in fact approach the &alue nee+e+ to accomplish this in the limitthe resisti&it ρ approaches =ero.

0n the process of reflection8 there is a force per unit area e erte+ on the con+uctor. This is. . . Pust the v × force +ue to the current ' flowing in the presence of the magnetic fiel+ of . .the incoming wa&e8 or a force per unit &olume of ' × ) . 0f we calculate the total force.d " acting on a c lin+rical &olume with area dA an+ length , of the con+uctor8 we fin+that it is in the + x - +irection8 with magnitu+e

. .d% = , J ' × JdA = ,/B dA =

2 E ) B)dA

913.1).5:

c µ )

so that the force per unit area8 d% = 2 E ) B) = 2S 913.1).,:

dA c µ ) c

or ra+iation pressure8 is Pust twice the $o nting flu +i&i+e+ 7 the spee+ of light c .

%e shall show that a perfect con+uctor will perfectl reflect an inci+ent wa&e. Toapproach the limit of a perfect con+uctor8 we first consi+er the finite resisti&it case8 an+then let the resisti&it go to =ero.

4or simplicit 8 we assume that the sheet is thin compare+ to a wa&elength8 so that theentire sheet sees essentiall the same electric fiel+. This implies that the current +ensit.' will 7e uniform across the thic@ness of the sheet8 an+ outsi+e of the sheet we will see

. .fiel+s appropriate to R an e"ui&alent surface current < 9t : = , ' 9t : . This current sheet willgenerate a++itional electromagnetic wa&es8 mo&ing 7oth to the right an+ to the left8 awa

.from the oscillating sheet of charge. The total electric fiel+8 E 9 x8t : 8 will 7e the sum of the inci+ent electric fiel+ an+ the electric fiel+ generate+ 7 the current sheet. sing !"s.913. .15: an+ 913. .1,: a7o&e8 we o7tain the following e pressions for the total electricfiel+/

. .

. .

.

⎧9 x8t : − c

µ ) < 9t − x

c:80 ) 2 x > )

) )

E


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E 9 x8t : = E ) 9 x8t : + E 1 9 x8t := ⎨ .

c µ .913.1).':

0

⎩0 )

9 x8t : − ) < 9t + x c:82

x < )

.%e also ha&e a relation 7etween the current +ensit ' .

an+ E

from the microscopic form

. .of (hmAs law/ ' 9t : = E 9)8t :

. ρ 8 where E 9)8 t : is the total electric fiel+ at the position of

E


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1 c 0 2

c 0 2

the con+ucting sheet. ote that it is appropriate to use the total electric fiel+ in (hmAslaw -- the currents arise from the total electric fiel+8 irrespecti&e of the origin of that fiel+.Thus8 we ha&e

. . , E 9)8t :< 9t : = , ' 9t : =

ρ

913.1). :

6t x = ) 8 either e pression in !". 913.1).': gi&es

. . . . .

E 9)8 t : = E

.= E

) 9)8 t : + E 1 9)8 t : =

E ) .

9)8 t : − ,c µ ) E 9)8t :

9)8 t : − c µ

) < 9t :2 913.1). :

)2 ρ

.

where we ha&e use+ !". 913.1). : for the last step. *ol&ing for E 9)8 t : 8 we o7tain

.E9)8 t : = E ) 9)8

t :

913.1).1):

sing the e pression a7o&e8 the surface current +ensit in !". 913.1). : can 7e rewrittenas

. .< 9t : = , ' 9t : = , E ) 9)8

t :

913.1).11:

. .

0n the limit where ρ )

9no resistance8 a perfect con+uctor:8 E 9)8 t : = = 8 as can 7e seen

from !". 913.1). :8 an+ the surface current 7ecomes

. .

< 9t : = 2E ) 9)8 t : =

2 E ) cos ω t += 2 B) cos ω t

+913.1).12:


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c µ ) c µ ) µ )

0n this same limit8 the total electric fiel+s can 7e written as

.

. ⎧0( E ) − E ) ) cos (ω t − kx) +==8 x > )

E 9 x8t : = ⎨0 E Ecos9ω t − kx: − cos9ω t + kx:F += 2 E sin ω t sin kx +8 x < )913.1).13:

⎩ ) )

*imilarl 8 the total magnetic fiel+s in this limit are gi&en 7


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. . ..

9 x8t : = 9 x8t : + 9 x8t : = B cos (ω t − kx) * + i ×⎛ E 1 9 x8t : ⎞

) 1 ) ⎜ ⎟⎝ ⎠ 913.1).1#:.

= B) cos (ω t − kx) * − B) cos (ω t − kx) * = =

for x > ) 8an+

.

9 x8t : = B Ecos9ω t − kx: + cos9ω t + kx:F* = 2 B

cos ω t cos kx*

913.1).15:

for x < ) . Thus8 from !"s. 913.1).13: - 913.1).15: we see that we get no electromagneticwa&e for x > ) 8 an+ stan+ing electromagnetic wa&es for x < ) . ote that at x = ) 8 thetotal electric fiel+ &anishes. The current per unit length at x = ) 8

.< 9t : = ) cos ω t +

µ )

913.1).1,:

is Pust the current per length we nee+ to 7ring the magnetic fiel+ +own from its &alue at x < ) to =ero for x > ) .

Sou ma 7e pertur7e+ 7 the fact that in the limit of a perfect con+uctor8 the electric fiel+

&anishes at x = ) 8 since it is the electric fiel+ at x = ) that is +ri&ing the current thereQ 0nthe limit of &er small resistance8 the electric fiel+ re"uire+ to +ri&e an finite current is&er small. 0n the limit where ρ = ) 8 the electric fiel+ is =ero8 7ut as we approach that. .limit8 we can still ha&e a perfectl finite an+ well +etermine+ &alue of ' = Efoun+ 7 ta@ing this limit in !"s. 913.1). : an+ 913.1).12: a7o&e.

13.11)ro?lem-/olving /trateg Traveling Electromagnetic Waves

ρ 8 as we

This chapter e plores &arious properties of the electromagnetic wa&es. The electric an+the magnetic fiel+s of the wa&e o7e the wa&e e"uation. (nce the functional form of either one of the fiel+s is gi&en8 the other can 7e +etermine+ from Ma well’s e"uations.6s an e ample8 let’s consi+er a sinusoi+al electromagnetic wa&e with

.

E 9 z 8t : = E ) sin9kz − ω t : i

The e"uation a7o&e contains the complete information a7out the electromagnetic wa&e/

c

) )

2


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1. Direction of wa&e propagation/ The argument of the sine form in the electric fiel+can 7e rewritten as 9 kz − ω t : = k 9 z − vt :

propagating in the B z -+irection.8 which in+icates that the wa&e is

2. %a&elength/ The wa&elength λ is relate+ to the wa&e num7er k 7 λ = 2π ? k .


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E0 B00

3. 4re"uenc / The fre"uenc of the wa&e8 f 8 is relate+ to the angular fre"uenc ω 7 f = ω ? 2π .

#. *pee+ of propagation/ The spee+ of the wa&e is gi&en 7

v = λ f = 2π ⋅ ω = ω k 2π k

0n &acuum8 the spee+ of the electromagnetic wa&e is e"ual to the spee+ of light8 c .

. . .

5. Magnetic fiel+ / The magnetic fiel+ is perpen+icular to 7oth E which points inthe B x-+irection8 an+ +* 8 the unit &ector along the B z -a is8 which is the +irection of

propagation8 as we ha&e foun+. 0n a++ition8 since the wa&e propagates in the same. . .+irection as the cross pro+uct+irection 9since i × += * :.

E × 8 we conclu+e that must point in the B y-

. .*ince is alwa s in phase with E 8 the two fiel+s ha&e the same functional form.Thus8 we ma write the magnetic fiel+ as

.

9 z 8t : = B) sin9kz − ω t : +

where B) is the amplitu+e. sing Ma well’s e"uations one ma show that

B) = E ) 9k ?ω : = E ) ? c in &acuum.

,. The $o tning &ector/ sing !". 913.,.5:8 the $o nting &ector can 7e o7taine+ as

. 1 . . 1 E B sin 2 9kz − ω t :/ = E × = ( E sin9kz − ω t : i ) × (

Bsin9kz − ω t : +) = ) ) *

µ µ * ) + * ) + µ ) ) )

'. 0ntensit / The intensit of the wa&e is e"ual to the a&erage of S /

2 2

I = S 5 sin 2 9kz − ω t :

= E ) B) =2 µ ) E )2c µ )

= cB)2 µ )

. a+iation pressure/ 0f the electromagnetic wa&e is normall inci+ent on a surfacean+ the ra+iation is completel reflected 8 the ra+iation pressure is

2 I E B E 2 B2

# = = ) ) = ) = )


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c c µ c2 µ µ )) )


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13.1#/olved )ro?lems

13.1#.1)lane Electromagnetic Wave

*uppose the electric fiel+ of a plane electromagnetic wa&e is gi&en 7

.

E 9 z 8 t : = E ) cos (kz − ω t ) i913.12.1:

4in+ the following "uantities/

9a: The +irection of wa&e propagation..

97: The correspon+ing magnetic fiel+ .

/olutions

9a: H writing the argument of the cosine function as kz − ω t = k 9 z − ct :we see that the wa&e is tra&eling in the B z +irection.

where ω = ck 8

97: The +irection of propagation of the electromagnetic wa&es coinci+es with the. . .

+irection of the $o nting &ector which is gi&en 7 / = E × ? µ ) . 0n a++ition8 E an+. .D

.are perpen+icular to each other. Therefore8 if E = E 9 z 8t : i an+ / =

S * 8 then

. . . .= B9 z 8t : + . That is8 points in the B y-+irection. *ince E an+ are in phase with each

other8 one ma write.

9 z 8t : = B cos9kz − ω t : +

.

913.12.2:

To fin+ the magnitu+e of 8 we ma@e use of 4ara+a ’s law/

. . d ΦE ⋅ d s = − B

dt 913.12.3:

which implies ∂ E x∂ z

∂ B y=− ∂t 913.12.#:

4rom the a7o&e e"uations8 we o7tain

)


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− E ) k sin9kz − ω t : = − B) ω sin9kz −ω t :

or E ) =

ω = c

913.12.5:

913.12.,:

B) k


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Thus8 the magnetic fiel+ is gi&en 7

.

9 z 8t : = 9 E )

?c: cos9kz − ω t : +

913.12.':

13.1#.#,ne-Dimensional Wave Equation

erif that8 for ω = kc 8 E 9 x8t : = E ) cos (kx − ω t )

B9 x8t : = B)

cos(

kx − ω t )

913.12. :

satisf the one-+imensional wa&e e"uation/

⎛ ∂2 1⎜ − ∂

2 ⎞ ⎧ E 9 x8t :⎫⎟ ⎨ ⎬ = )

913.12. :

⎝ ∂ x2 c 2 ∂t 2 ⎠ ⎩ B9 x8t :⎭

/olution

Differentiating

E = E ) cos (kx − ω t )

with respect to x gi&es

∂ E = −kE sin (kx − ω t ) 8∂ E = −k 2

E cos (kx − ω t )

913.12.1):

∂ x ) ∂ x2 )

*imilarl 8 +ifferentiating E with respect to t iel+s

∂ E = ω E sin (kx − ω t ) 8∂ E = −ω 2

E cos(kx − ω t )

913.12.11:

2

2


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∂t ) ∂t 2 )

Thus8 2 2 2∂ E − 1

∂ E = ⎛ −k 2 + ω

⎞ E cos (kx − ω t ) =

)913.12.12:

∂ x2 c2 ∂t 2 ⎜ c2⎟ )⎝ ⎠

where we ha&e ma+e use+ of the relation ω = kc . (ne ma follow a similar proce+ure to&erif the magnetic fiel+.


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"igure 13.1#.#


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4rom the c lin+rical s mmetr of the s stem8 we see that the magnetic fiel+ will 7e.

circular8 centere+ on the z -a is8 i.e.8 5 B6 9see 4igure 13.12.2.:

Consi+er a circular path of ra+ius r " 7etween the plates. sing the a7o&e formula8 weo7tain

B (2π r ) = ) + µ ε d ⎛ Qπ r

2⎞ = µ ) r dQ

913.12.1#:

) )dt⎜

π " 2ε ⎟ 2 dt

or

. µ ) r dQ 6913.12.15:

2π " 2 dt .

The $o nting / &ector can then 7e written as

. . . Q/ =

1E × =

1 ⎛

Q* ⎞

×⎛ µ ) r

d 6 D⎞

µ µ ⎜

π " 2ε ⎟ ⎜

2π " 2 dt ⎟

) )⎝ ) ⎠ ⎝ ⎠ 913.12.1,:⎛ Qr ⎞ ⎛ dQ⎞ D= −⎜ ⎟ ⎜ ⎟ r

⎝ 2π

2

"#

ε ⎝ dt ⎠. ote that for dQ ? dt >

)/ points in the −r +irection8 or ra+iall inwar+ towar+ the

center of the capacitor.

97: The energ per unit &olume carrie+ 7 the electric fiel+ is u E = ε ) E ?2 . The total energstore+ in the electric fiel+ then 7ecomes

! = u 1 = ε

) E 2 (π " 2&) = 1ε ⎛

Q ⎞

2

π "2

&= Q & 913.12.1':

⎝

2

) "⎠

=

)

⎠

2

⎝ )⎠

2


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E E 2

)


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) ⎜ π " 2ε

⎟2π " 2ε

Differentiating the a7o&e e pression with respect to t 8 we o7tain the rate at which thisenerg is 7eing store+/

2d! E = d⎛ Q & ⎞ =Q&

⎛ dQ⎞⎜ 2 ⎟ 2 ⎜ ⎟ 913.12.1 :

dt dt ⎝ 2π " ε ) ⎠ π " ε ) ⎝ dt ⎠

(n the other han+8 the rate at which energ flows into the capacitor through the c lin+erat r U " can 7e o7taine+ 7 integrating / o&er the surface area/

. . ⎛/ ⋅ d ! = SA = Qr dQ⎞

π "& =Q& ⎛ dQ ⎞

913.12.1 :

° ∫ " ⎜ 2 # ⎟ (2 ) 2 ⎜ ⎟

⎝

2π ε o " dt⎠

ε ) π "

⎝ dt ⎠

which is e"ual to the rate at which energ store+ in the electric fiel+ is changing.

13.1#.()o nting 2ector o& a Conductor

6 c lin+rical con+uctor of ra+ius a an+ con+ucti&it σ carries a stea+ current I which is+istri7ute+ uniforml o&er its cross-section8 as shown in 4igure 13.12.3.

"igure 13.1#.3

.9a: Compute the electric fiel+ E insi+e the con+uctor.

.97: Compute the magnetic fiel+ Pust outsi+e the con+uctor.

9c: Compute the $o nting &ector / at the surface of the con+uctor. 0n which +irection.

+oes / pointK


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9+: H integrating / o&er the surface area of the con+uctor8 show that the rate at whichelectromagnetic energ enters the surface of the con+uctor is e"ual to the rate at whichenerg is +issipate+.

/olutions

9a: Let the +irection of the current 7e along the z -a is. The electric fiel+ is gi&en 7

. 'E5 = I

*

913.12.2):

σ σπ a 2

where " is the resistance an+ l is the length of the con+uctor.

97: The magnetic fiel+ can 7e compute+ using 6mpere’s law/. .

° ∫ ⋅ d s = µ ) I enc913.12.21:

Choosing the 6mperian loop to 7e a circle of ra+ius r 8 we ha&e B92π r : = µ ) I 8 or

. µ I = ) 62π r

913.12.22:

9c: The $o nting &ector on the surface of the wire 9 r U a: is. .

E × 1 ⎛

I D⎞ ⎛ µ I ⎞ ⎛ I

2⎞

/ = = ⎜

*⎟ ×⎜

) 6⎟ =

−⎜ ⎟ rD 913.12.23:

2 2 3

µ ) µ ) ⎝ σπ a ⎠ ⎝ 2π a ⎠ ⎝ 2π σ a ⎠.

otice that / points ra+iall inwar+ towar+ the center of the con+uctor.

9+: The rate at which electromagnetic energ flows into the con+uctor is gi&en 7

d! . . ⎛ I 2 ⎞ I 2l # = = º ∫∫ / ⋅ d ! = ⎜ ⎟ 2π al= 913.12.2#:

.


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dt S ⎝ 2σπ 2a

3

⎠ σπ a 2

>owe&er8 since the con+ucti&it σ is relate+ to the resistance " 7

The a7o&e e pression 7ecomes

σ = 1

= ρ l = l A" π a 2 "

913.12.25:

# = I 2 " 913.12.2,:

which is e"ual to the rate of energ +issipation in a resistor with resistance " .

13.13 Conceptual :uestions

1. 0n the 6mpere-Ma well’s e"uation8 is it possi7le that 7oth a con+uction current an+ a+isplacement are non-&anishingK

2. %hat causes electromagnetic ra+iationK

3. %hen ou touch the in+oor antenna on a T 8 the reception usuall impro&es. %h K


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#. ! plain wh the reception for cellular phones often 7ecomes poor when use+ insi+e asteel-frame+ 7uil+ing.

5. Compare soun+ wa&es with electromagnetic wa&es.

,. Can parallel electric an+ magnetic fiel+s ma@e up an electromagnetic wa&e in &acuumK

'. %hat happens to the intensit of an electromagnetic wa&e if the amplitu+e of the electricfiel+ is hal&e+K Dou7le+K

13.1( !dditional )ro?lems

13.1(.1/olar /ailing

0t has 7een propose+ that a spaceship might 7e propelle+ in the solar s stem 7 ra+iation pressure8 using a large sail ma+e of foil. >ow large must the sail 7e if the ra+iation forceis to 7e e"ual in magnitu+e to the *unAs gra&itational attractionK 6ssume that the mass of the ship an+ sail is 1,5) @g8 that the sail is perfectl reflecting8 an+ that the sail isoriente+ at right angles to the *un’s ra s. Does our answer +epen+ on where in the solar s stem the spaceship is locate+K

13.1(.# 8e&lections o& True %ove

9a: 6 light 7ul7 puts out 1)) % of electromagnetic ra+iation. %hat is the time-a&erageintensit of ra+iation from this light 7ul7 at a +istance of one meter from the 7ul7K %hatare the ma imum &alues of electric an+ magnetic fiel+s8from the 7ul7K 6ssume a plane wa&e.

E ) an+ B) 8 at this same +istance

97: The face of our true lo&e is one meter from this 1)) % 7ul7. %hat ma imum surfacecurrent must flow on our true lo&eAs face in or+er to reflect the light from the 7ul7 into

our a+oring e esK 6ssume that our true lo&eAs face is 9what elseK: perfect--perfectlsmooth an+ perfectl reflecting--an+ that the inci+ent light an+ reflecte+ light are normalto the surface.

13.1(.3 Coaxial Ca?le and )ower "low

6 coa ial ca7le consists of two concentric long hollow c lin+ers of =ero resistanceO theinner has ra+ius a 8 the outer has ra+ius 8 an+ the length of 7oth is l 8 with l >> . Theca7le transmits DC power from a 7atter to a loa+. The 7atter pro&i+es an electromoti&eforce ε 7etween the two con+uctors at one en+ of the ca7le8 an+ the loa+ is aresistance " connecte+ 7etween the two con+uctors at the other en+ of the ca7le. 6


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current I flows +own the inner con+uctor an+ 7ac@ up the outer one. The 7atter chargesthe inner con+uctor to a charge −Q an+ the outer con+uctor to a charge +Q .


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"igure 13.1(.1.

9a: 4in+ the +irection an+ magnitu+e of the electric fiel+ E e&er where..

97: 4in+ the +irection an+ magnitu+e of the magnetic fiel+ e&er where.

9c: Calculate the $o nting &ector / in the ca7le..

9+: H integrating / o&er appropriate surface8 fin+ the power that flows into the coa ialca7le.

9e: >ow +oes our result in 9+: compare to the power +issipate+ in the resistorK

13.1(.( /uperposition o& Electromagnetic Waves

!lectromagnetic wa&e are emitte+ from two +ifferent sources with

. .E 9 x8t : = E cos9kx − ω t : +8 E 9 x8t : = E cos9kx − ω t + φ : +

1 1) 2 2)

9a: 4in+ the $o nting &ector associate+ with the resultant electromagnetic wa&e.

97: 4in+ the intensit of the resultant electromagnetic wa&e

9c: epeat the calculations a7o&e if the +irection of propagation of the secon+electromagnetic wa&e is re&erse+ so that

. .

E 9 x8t : = E cos9kx − ω t : +8 E 9 x8t : = E cos9kx + ω t + φ : +1 1) 2 2)

13.1(. /inusoidal Electromagnetic Wave

The electric fiel+ of an electromagnetic wa&e is gi&en 7.

E 9 z 8t : = E ) cos9kz − ω t : 9 i + +:


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9a: %hat is the ma imum amplitu+e of the electric fiel+K.

97: Compute the correspon+ing magnetic fiel+ ..

9c: 4in+ the $on ting &ector / .

9+: %hat is the ra+iation pressure if the wa&e is inci+ent normall on a surface an+ is perfectl reflecte+K

13.1(.08adiation )ressure o& Electromagnetic Wave

6 plane electromagnetic wa&e is +escri7e+ 7. .E = E sin9kx −ω t : +8 = B sin9kx −ω t :*

) )

where E ) = cB) .

9a: *how that for an point in this wa&e8 the +ensit of the energ store+ in the electric fiel+e"uals the +ensit of the energ store+ in the magnetic fiel+. %hat is the time-a&erage+ total 9electric plus magnetic: energ +ensit in this wa&e8 in terms of E ) K 0nterms of B) K

97: This wa&e falls on an+ is totall a sor ed 7 an o7Pect. 6ssuming total a7sorption8 show

that the ra+iation pressure on the o7Pect is Pust gi&en 7 the time-a&erage+ total energ+ensit in the wa&e. ote that the +imensions of energ +ensit are the same as the+imensions of pressure.

9c: *unlight stri@es the !arth8 Pust outsi+e its atmosphere8 with an a&erage intensit of 135)%?m 2. %hat is the time a&erage+ total energ +ensit of this sunlightK 6n o7Pect in or7ita7out the !arth totall a7sor7s sunlight. %hat ra+iation pressure +oes it feelK

13.1(.7 Energ o& Electromagnetic Waves

9a:0f the electric fiel+ of an electromagnetic wa&e has an rms 9root-mean-s"uare: strength of 3.) ×1) −2 ?m 8 how much energ is transporte+ across a 1.))-cm 2 area in one hourK

97: The intensit of the solar ra+iation inci+ent on the upper atmosphere of the !arth isappro imatel 135) %?m 2. sing this information8 estimate the energ containe+ in a1.))-m 3 &olume near the !arth’s surface +ue to ra+iation from the *un.


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13.1(.9 Wave Equation

Consi+er a plane electromagnetic wa&e with the electric an+ magnetic fiel+s gi&en 7. .E 9 x8t : = E 9 x8t :* 8 9 x8t : = B 9 x8t : +

z y

6ppl ing arguments similar to that presente+ in 13.#8 show that the fiel+s satisf thefollowing relationships/

∂ E z = ∂ B y ∂ B y =

∂ E z 8

∂ x ∂t µ ε

∂ x ) ) ∂t

13.1(.; Electromagnetic )lane Wave

6n electromagnetic plane wa&e is propagating in &acuum has a magnetic fiel+ gi&en 7

. = B f 9ax + t : + f 9u: = ⎧1 ) < u < 1

) ⎨else

The wa&e encounters an infinite8 +ielectric sheet at x U ) of such a thic@ness that half ofthe energ of the wa&e is reflecte+ an+ the other half is transmitte+ an+ emerges on the

far si+e of the sheet. The

+* +irection is out of the paper.

9a: %hat con+ition 7etween a an+ must 7e met in or+er for this wa&e to satisf Ma well’se"uationsK

.97: %hat is the magnitu+e an+ +irection of the E fiel+ of the incoming wa&eK

9c: %hat is the magnitu+e an+ +irection of the energ flu 9power per unit area: carrie+ 7the incoming wa&e8 in terms of B) an+ uni&ersal "uantitiesK

9+: %hat is the pressure 9force per unit area: that this wa&e e erts on the sheet while it isimpinging on itK

13.1(.1=/inusoidal Electromagnetic Wave

6n electromagnetic plane wa&e has an electric fiel+ gi&en 7

.⎛ 2π , ⎞E = 93)) ?m: cos x − 2 ×1) t *

)⎩

π


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⎜ 3

⎟⎝ ⎠

where x an+ t are in *0 units an+

* is the unit &ector in the B z -+irection. The wa&e is

propagating through ferrite8 a ferromagnetic insulator8 which has a relati&e magnetic

permea7ilit κ m = 1)))

an+ +ielectric constant κ = 1) .


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9a: %hat +irection +oes this wa&e tra&elK

97: %hat is the wa&elength of the wa&e 9in meters:K

9c: %hat is the fre"uenc of the wa&e 9in >=:K

9+: %hat is the spee+ of the wa&e 9in m?s:

generac i on de ondas em

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