Lecture Physics A2: Electromagnetic Field and Wave

Electromagnetic Field and Wave

Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology

Maxwell’s Equation

Maxwell discovered that the basic principles of electromagnetism can be expressed in terms of the four equations that now we call Maxwell’s equations:

(1) Gauss’s law for electric ﬁelds; (2) Gauss’s law for magnetic ﬁelds, showing no existence of magnetic

monopole. (3) Faraday’s law; (4) Ampere’s law, including displacement current;

Maxwell’s Equations

Diﬀerential form:

~E =

Integral form: Gauss’ Law ~E

d~S = r · ⇢ "0 · Qinside "0 I

Gauss’ Law for Magnetism

~B d~S = 0 ~B = 0 · r ·

I Faraday’s Law

~E d~l = ~E = – dB dt · @ ~B @t r ⇥

I Ampere’s Law

~B d~l = µ0Ienclosed + µ0"0 ~B = µ0 ~J + µ0"0 dE dt · @ ~E @t r ⇥ I

Macroscopic Scale

Microscopic Scale

Gauss’s Law for Electric Field

The ﬂux of the electric ﬁeld (the area integral of the electric ﬁeld) over any closed surface (S) is equal to the net charge inside the surface (S) divided by the permittivity ε0.

d~S =

Qinside "0

~E dxdyˆn =

d~S = ˆndS = ˆndxdy

~E dxdycos✓ =

· Qinside "0 Qinside "0

dx

Edxdy =

dy

Qinside "0

→ dS →

ES = E(4⇡r2) =

Qinside "0

Coulomb’s Law

E =

Qinside (4⇡r2)"0

Gauss’s Law of Magnetism

Gauss’s law of magnetism states that the net magnetic ﬂux through any closed surface is zero

d~S = 0

The number of magnetic ﬁeld lines that exit equal to the number for magnetic ﬁeld lines that enter the closed surface

→ E

d~S =

Qinside "0

Faraday’s Law

The electric ﬁeld around a closed loop is equal to the negative of the rate of change of the magnetic ﬂux through the area by the loop.

d~l =

~E d~l

dB dt

~Ed~l = Edlcos✓ · = Edl

(θ = 0)

E(2⇡R) =

dB dt

e.m.f =

dB dt

e.m.f =

dB dt

= Ed

W = F d = Eqd W d V = Ed = e.m.f

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic ﬁeld over a closed path is equal to the total current going through any surface bounded by the closed path

~B d~l = µ0Ienclosed + µ0"0

dE dt ·

1. Time-changing electric fields induces magnetic fields

2. Displacement current

cause Magnetic field

Conduction currents ( motion of charged particles)

Time changing electric fields

also cause Magnetic field

=> Time changing electric fields is equivalent to a current. We call it dispalcement current.

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic ﬁeld over a closed path is equal to the total current going through any surface bounded by the closed path

~B d~l = µ0Ienclosed + µ0"0

dE dt ·

(when θ = 0)

~B d~l = Bdlcos✓ = Bdl

~B d~l = µ0Ienclosed +

dE Bdl = B(2⇡R) dt ·

I B(2⇡R) = µ0I

B =

µ0I 2⇡R

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic ﬁeld over a closed path is equal to the total current going through any surface bounded by the closed path

~B d~l = µ0"0 + µ0Ienclosed

dE dt ·

For S1:

~B d~l = µ0Ienclosed

I For S2:

~B d~l = 0

Two diﬀerent situations in even one case!

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic ﬁeld over a closed path is equal to the total current going through any surface bounded by the closed path

~B d~l = µ0"0 + µ0Ienclosed

dE dt ·

Displacement current

Electric ﬂux is deﬁned as

~E d~S = E =

· Qinside µ0

For S2:

~B d~l = µ0Ienclosed + µ0"0 dQ µ0 = µ0"0 dt d dt dQ dt · Q "0 ◆ ✓ I

Recall: Maxwell’s Equations

Diﬀerential form:

~E =

Integral form: Gauss’ Law ~E

d~S = r · ⇢ "0 · Qinside "0 I

Gauss’ Law for Magnetism

~B d~S = 0 ~B = 0 · r ·

I Faraday’s Law

~E d~l = ~E = – dB dt · @ ~B @t r ⇥

I Ampere’s Law

~B d~l = µ0Ienclosed + µ0"0 ~B = µ0 ~J + µ0"0 dE dt · @ ~E @t r ⇥ I

Macroscopic Scale

Microscopic Scale

Convert Intergral form to Diﬀerential form

~E d~l = ~E d~S = dB dt · · Qinside "0 I I

~E d~S = dV ~Bd~S

· ⇢ "0 d~l = ~E · d d d~l = ~Bd~S dt dt ·

I ~E I

Stokes’ theorem: the circulation of a ﬁeld E around the loop l that bounds a surface S is equal to the ﬂux of curl E over S

Z Z ZV Divergence theorem: the ﬂux penetrating a closed surface S that bounds a volume V is equal to the divergence of the ﬁeld E inside the volume

~E)dS ~E)dS ~E d~S = ~E)dV ( d~l = ( ~E · d~l = ( r ⇥ · r ⇥ · r · ~E I I

~E +

dS = 0

( )dV = 0

r ⇥

r · Z Z ZV ⇢ ~E "0 Z Z ZV

@ ~B @t ! @ ~B @t

~E = ~E = – r · ⇢ "0 r ⇥

Divergence Operator

Divergence at a point (x,y,z) is the measure of the vector ﬂow out of a surface surrounding that point.

= ˆx

+ ˆy

+ ˆz

Fig. 4: Zero divergence

~E = ˆxEx + ˆyEy + ˆzEz @ @z

@ @x

@ @y

Fig. 3: No variation → zero divergence

~E =

@Ex @x

@Ey @y

@Ez @z

r ·

@Ez

~E = + +

≡ (rate of change of E in x direction) + (rate of change of E in y direction) + (rate of change of E in z direction)

Fig. 5

Fig. 1: Positive divergence at P

Fig. 2: Negative divergence at P

@Ex 1 @x @Ey @y @z r ·

Curl Operator in a Spherical Coordinate

Curl is a measure of the rotation of a vector ﬁeld.

~E = ˆxEx + ˆyEy + ˆzEz

= ˆx

+ ˆy

+ ˆz

@ @x

@ @y

@ @z

Fig. 1: Non-zero curl

~E =

Fig. 2: How much is the curl?

r ⇥

ˆx ˆy ˆz @ @ @ @x @y @z Ex Ey Ez

= ˆx

+ ˆy

+ ˆz

5 @Ey @z

@Ez @x

@Ex @y

@Ez @y

@Ex @z

@Ey @x

✓

◆

✓

◆

✓

◆

≡ (how much does an object in y-z plane rotate) + (how much does an object in x-z plane rotate) + (how much does an object in x-y plane rotate)

Curl =

Circulation Area

Curl Operator in a Cylindrical Coordinate

Curl is a measure of the rotation of a vector ﬁeld.

~E = ˆ⇢E⇢ + ˆE + ˆzEz

= + +

1 ⇢ @ @⇢ 1 ⇢ @ @ @ @z r

⇢

r ⇥

ˆ 1 1 ⇢ ˆ⇢ ⇢ ˆz @ @ @ 3 @⇢ @ @z E⇢ E Ez

@(⇢E)

~E =

@(⇢E) @z

ˆ⇢ ⇢

@E⇢ @z

ˆz ⇢

@E⇢) @z

r ⇥

@Ez @

@Ez @⇢

✓

◆

✓

◆

✓

◆

~E = 2 6 4 7 5

Diﬀerential form: Gauss’ Law for Magnetism

⟺

~B d~S = 0 ~B = 0 ·

1 r

@rBr @r

1 r

@B @

@Bz @z

r ·

Bin I ~B = = 0 ~B = r · Bout

= 0

1 a

@ @r

µ0I 2a

✓

◆

volume r ·

B = BA

B =

µ0I 2⇡a

Diﬀerential form: Faraday’s Law

⟺

~E d~l = ~E = – dB dt · @ ~B @t r ⇥ I

curl of ~E =

⇢

@ ~B @t r ⇥

Circulation Area @ ~B @t

~E = E =

ˆ 1 1 ⇢ ˆz ⇢ ˆ⇢ @ @ @ 3 @z @⇢ @ E⇢ E Ez 0 0 7 5 @ ~B ⇢ 2 @t ˆz ⇢ ~E = 2 6 4 ˆz ⇢ r ⇥

@ ~B @t

~E = ˆz ⇢ ~E = @ ~B @t 1 2 r ⇥ ~E(2⇡⇢) ⇡⇢2 = @ ~B @t ⇢ 2

Diﬀerential form: Ampere’s Law with Maxwell’s Corr

⟺

~B d~l = µ0Ienclosed + ~B = µ0 ~J + µ0"0 dE dt · @ ~E @t r ⇥ I

J: Current density J =

I A I ⇡R2

~B = = µ0 ~J

Circulation Area r ⇥

B(2⇡R) ⇡R2 = µ0 ~J

B(2⇡R) ⇡R2 = µ0 Ienclosed ⇡R2

B = µ0

Ienclosed 2⇡R

Diﬀerential form: Ampere’s Law with Maxwell’s Corr

⟺

~B d~l = µ0Ienclosed + ~B = µ0 ~J + µ0"0 dE dt · @ ~E @t r ⇥ I

Start from:

~B = µ0 ~J

r ⇥ Take divergence:

( ~J

r · r ⇥

~J = ~E) = "0

r · ("0r · @ ~E @t ! r ·

~B) = µ0r · @ @⇢ = @t @t ⇢ ( :charge density)

We need to add displacement current term

Diﬀerential form: Ampere’s Law with Maxwell’s Corr

~B = µ0 ~J

r ⇥

1 ⇢ ˆ⇢ @ @⇢ 0

1 ⇢ ˆz @ 3 @z 0

1 ⇢ ˆ⇢ @ @⇢ B⇢

= 3 r ⇥ r ⇥

ˆ 1 ⇢ ˆz @ @ @ @z ⇢B Bz ˆ @ @ ⇢B

@⇢B

~B =

⇢

7 5 7 5

@⇢B @z

ˆz ⇢

@0 @

r ⇥

@0 @⇢

@⇢

✓

◆

✓

◆

✓

◆

~B = 2 6 4 ˆ⇢ ⇢ ~B = 2 6 4 @0 @z

~B = =

ˆz ⇢ @⇢B @⇢ µ0I 2⇡⇢2 ˆz r ⇥

Varying current causes a circulating magnetic ﬁeld

How Electromagnetism is a Wave in Vacuum?

~E = –

~B = µ0"0

@ ~E @t

r ⇥

@ ~B @t

r ⇥

~E)

~B)

~B = µ0"0

~E = –

r ⇥ @t

r ⇥ r ⇥

r ⇥ @t

r ⇥ r ⇥

~E)

(

2 ~E =

µ0"0 µ0"0

~B)

(

2 ~B = µ0"0

r ·

@ @t

@ ~B @ ~E @t ! @t !

@ @t

@ ~B @t !

2 ~E

2 ~B

µ0"0 µ0"0

@2 ~B @t2 = 0 r r

Wave equation is in form

@2 ~E @t2 = 0 1 3 108m/s

where c =

1 v2

d2y dx2

µ0 = 4⇡

⇥

d2y dt2 = 0 (cid:15482) Both Electric and

" = 8.85

⇥

7 Wb A.m 12 C2 N.m2

⇥ pµ0"0 ⇡

Magnetism are waves

!t)

E = E0cos(kx

B = B0cos(kx

Magnetism vs Electric

Let’s work on one dimension

~E = –

r ⇥

kx)]ˆj ~E = [E0sin(!t = kx)]ˆi [B0sin(!t r ⇥ r ⇥ @ @t

@ ~B @t @ ~B @t @ ~B @t

~E = kx)]ˆi k[E0sin(!t kx)]ˆi = [!B0sin(!t r ⇥

(cid:15483)

kE0 = !B0 = = = f

! k 2⇡f 2⇡/ E0 B0

= c

E0 B0

Energy of an Electromagnetic Wave

Start from the magnetic density:

c = 3 108m/s B2 = ⌘B = ⇥ pµ0"0 ⇡ E2 c2 1 2µ0 ◆

(Electric energy density)

⌘B = µ0"0E2 =

1 2µ0 ✓ "0 2 1 2µ0

For an electromagnetic wave in free space, half of the energy is in the electric ﬁeld and another half is the magnetic ﬁeld

⌘ = ⌘E + ⌘B

⌘ = E2 E2 +

Total energy:

"0 "0 2 2 ⌘ = "0E2

Power density

Since waves are spread out in space and time, energy density is often a more useful concept than energy. The power of a wave should probably replaced with the more useful concept of its power density. The energy content of a wave ﬁlls a volume of space deﬁnes energy density as energy per volume u V

The power density is the power per area

⌘ =

S =

P A W m2



Poynting Vector

Power density is deﬁned as

S = = =

P A 1 A u t 1 A ⌘V t

S = = = = ⌘c

⌘V At ⌘(Al) At ⌘(Act) At

S = ⌘c = B2

E B 1 µ0 ✓ ◆

S = EB S = ~E ~B

Poynting Vector

The magnitude of the Poynting vector represents the rate at which energy passes through a unit surface area perpendicular to the direction of wave propagation. The direction of the Poynting vector is along the direction of wave propagation.

⇥ 1 µ0 1 µ0