Basic Theoretical Physics: A Concise Overview P18 pptx

A practical example of such a time-dependent electric dipole so-called Hertz dipole is a radio or mobile-phone antenna, driven by an alternating current of frequency ω.. Due to the ω4-de

170 20 Maxwell’s Equations II: Electromagnetic Waves

20.5 Hertz’s Oscillating Dipole (Electric Dipole

Radiation, Mobile Phones)

It can be readily shown that the electromagnetic field generated by a time-dependent dipolep(t) at r  = 0, viz

B(r, t) = μ0

4πcurl

˙

p t − r c

r

E(r, t) = 1

4πε0

curl curlp t − r

c

for t ∈ (−∞, ∞), solves everywhere all four Maxwell equations (If B(r, t)

obeys Maxwell’s equations, then so doesE(r, t) and vice versa, in accordance

with (20.14).)

A practical example of such a time-dependent electric dipole (so-called Hertz dipole) is a radio or mobile-phone antenna, driven by an alternating

current of frequency ω Since the current is given explicitly by

I(t) ≡ | ˙p(t)| ,

the retarded vector potential has the following form, which is explicitly used

in equation (20.14):

A(r, t) = μ0

4πr p˙t − r

c



.

In particular, the asymptotic behavior of the fields in both the near-field and far-field range can be derived without difficulty, as follows:

a) In the near-field range (for r  λ, where λ is the wavelength of light

in a vacuum, corresponding to the frequency ω, i.e., λ = ω

2πc) one can approximate

p t − r c

p(t)

r .

After some elementary transformation, using the identity

curl curlv ≡ (grad div − ∇2)v

and the spherical wave equation given previously, one then obtains:

E(r, t) ∼=3(p(t) · r)r − r2p(t)

which is the quasi-static result

E(r, t) ∝ r −3 for r → 0

with which we are already familiar, while simultaneously

B(r, t) ∼= μ0

4πr3p(t) × r , i.e , ∝ r˙ −2 for r → 0 ,

and is thus less strongly divergent (i.e., asymptotically negligible w.r.t

E(r, t)) for r → 0 Here the position-dependence of the denominator

dominates

b) In the far-field range (for r

ator dominates, e.g.,

∇ × pt − r

c



∼

=−1 c

ˆ

r × p˙

t − r c

r

/

, with r = r/r ˆ

Here, for r

E(r, t) ∼= 1

4πε0c2r



ˆ

r ׈r × ¨pt − r

c



and

B(r, t) ∼=− μ0

4πcr



ˆ

r × ¨pt − r

c



i.e.,

S = E × H ∼= ˆr (sin θ)

2

16π2ε0c3r2



¨pt − r

c

2

where θ is the angle between ¨ p and ˆr.

As for planar waves, the propagation vector

ˆ

r := r/r

and the vectors E and cB form a right-handed rectangular trihedron, where

in addition the vector E in the far-field range lies asymptotically in the plane

defined by ¨ p and r.

In the far-field range the amplitudes of bothE and B are ∝ ω2r −1 sin θ;

the Poynting vector is therefore ∝ ω4r −2 (sin θ)2ˆr The power integrated

across the surface of a sphere is thus∝ ω4, independent of the radius! Due to

the ω4-dependence of the power, electromagnetic radiation beyond a limiting

frequency range is biologically dangerous (e.g X-rays), whereas low-frequency radiation is biologically harmless The limiting frequency range (hopefully)

seems to be beyond the frequency range of present-day mobile phones, which transmit in the region of 109Hz

20.6 Magnetic Dipole Radiation; Synchrotron Radiation

For vanishing charges and currents Maxwell’s equations possess a symmetry

which is analogous to that of the canonical equations of classical mechanics

( ˙q = − ∂ H

∂p; ˙p = ∂ H

∂q , see Part I), called symplectic invariance: the set of

equations does not change, ifE is transformed into cB, and B into

−c −1 E

 (E, cB) →



0, +1

−1, 0

 

E

c B



.

To be more precise, if in equation (20.14)B is transformed into −D and

E is transformed into H, while simultaneously ε and μ are interchanged

172 20 Maxwell’s Equations II: Electromagnetic Waves

and the electric dipole moment is replaced by a magnetic one (corresponding

to B = μ0H + J, D = ε0E + P ), one obtains the electromagnetic field

produced by a time-dependent magnetic dipole For the Poynting vector in

the far-field range one thus obtains instead of (20.17):

S = E × H ∼= ˆr (sin θ m)2

16π2μ0c3r2



 ¨mt − r

c

2

where θ mis again the angle between ¨m and ˆr.

We shall now show that magnetic dipole radiation for particles with

non-relativistic velocities is much smaller – by a factor ∼ v2

c2 – than electric dipole radiation Consider an electron moving with constant angular velocity ω in the xy −plane in a circular orbit of radius R The related electric dipole

mo-ment is

p(t) = eR · (cos(ωt) sin(ωt), 0)

On average the amplitude of this electric dipole moment is p0 = eR The

corresponding magnetic dipole moment is (on average)

m0= μ0πR2eω

2π ,

where we have again used – as with the calculation of the gyromagnetic ratio – the relation

I = eω

2π .

With

ω = v R

we thus obtain

m0=1

2evR Therefore, as long as p(t) and m(t) oscillate in their respective amplitudes

p0 and m0 with identical frequency ω as in cos ωt, we would get on average

the following ratio of the amplitudes of the respective Poynting vectors:4

|S m |

|Se| =

m2ε0

p2μ0

= v

2

Charged relativistic particles in a circular orbit are sources of intense,

po-larized radiation over a vast frequency range of the electromagnetic spectrum,

e.g., from the infrared region up to soft X-rays In a synchrotron, electrons

travelling at almost the speed of light are forced by magnets to move in

4 The factor 4 (= 22) in the denominator of this equation results essentially from the fact that the above formulap(T ) = eR ·(cos(ωt), sin(ωt)) can be interpreted

as follows: there are in effect two electric dipoles (but only one magnetic dipole)

involved

a circular orbit The continual acceleration of these charged particles in their circular orbit causes high energy radiation to be emitted tangentially to the path To enhance the effectivity the electrons usually travel through special

structures embedded in the orbit such as wigglers or undulators.

Synchrotron radiation is utilized for all kinds of physical and biophysical

research at various dedicated sites throughout the world

20.7 General Multipole Radiation

The results of this section follow directly from (20.14) for electric dipole ra-diation (Hertz dipole) and the corresponding equations for magnetic dipole

radiation (see Sect 20.6) We recall that a quadrupole is obtained by a limit-ing procedure involvlimit-ing the difference between two exactly opposite dipoles, one of which is shifted with respect to the other by a vectorb(= b2) ; an oc-tupole is obtained by a similar shift (withb(= b3)) from two exactly opposite quadrupoles, etc

As a consequence, the electromagnetic field of electric octupole radiation, for example, is obtained by application of the differential operator

(b3· ∇)(b2· ∇)

on the electromagnetic field of a Hertz dipole:

a) In the near-field range, i.e., for r  λ, one thus obtains the following quasi-static result for the E-field of an electric 2 l-pole:

E ∼=−gradφ(r, t) ,

with

φ( r, t) ∼= 1

4πε0r l+1

l



m= −l

c l,m (t)Y l,m (θ, φ) , (20.20)

where Y l,m are spherical harmonics.

The coefficients of this expansion depend on the vectors b1, b2, ,b l, and on time

b) In the far-field range, i.e., for r

(b i · ∇)

*

f t − r c

r

+

by5 1

r(b l · ˆr)



−∂

c∂t



ft − r c



and one obtains for the electric 2l-pole radiation asymptotically:

5 It should be noted that the vector f (t − r

c) is always perpendicular to ˆr, cf.

(20.16)

174 20 Maxwell’s Equations II: Electromagnetic Waves

1)

E ∝ 1

r ·

l 0

ν=2

(b ν · ˆr) ·



−∂

c∂t

l

ˆ

r × pt − r

c



2)

c B ∼= ˆr × E and

3)

S ≡ E × H , i.e., for p(t) = p0cos ωt

on average w.r.t time:

S(r) ∝ ω 2(l+1)

r2

l+1 0

ν=2

(b ν · ˆr)2

c2 |ˆr × p0|2

ˆ

Here we have l = 1, 2, 3 for dipole, quadrupole and octupole radiation.

The dependence on the distance r is thus universal, i.e., S ∝ r −2 for all l;

the dependence on the frequency is simple ( S ∝ ω 2(l+1) ); only the angular dependence of electric multipole radiation is complicated However, in this

case too, the three vectors ˆ r, E and cB form a right-handed rectangular

trihedron, similar to e x , e y , e z Details can be found in (20.21).6 For

magnetic multipole radiation the results are similar.

20.8 Relativistic Invariance of Electrodynamics

We have already seen in Part I that classical mechanics had to be amended as

a result of Einstein’s theory of special relativity In contrast, Maxwell’s theory

is already relativistically invariant per se and requires no modification (see

below)

It does no harm to repeat here (see also Section 9.1) that prior to Ein-stein’s theory of relativity (1905) it was believed that a special inertial frame

existed, the so-called aether or world aether, in which Maxwell’s equations had their usual form, and, in particular, where the velocity of light in vacuo

had the value

c ≡ √ 1

ε0μ0

,

whereas in other inertial frames, according to the Newtonian (or Galilean)

additive behavior of velocities, the value would be different (e.g., c → c+v) In

their well known experiments, Michelson and Morley attempted to measure the motion of the earth relative to the aether and thus tried to verify this

behavior Instead, they found (with great precision): c → c.

6 Gravitational waves obey the same theory with l = 2 Specifically, in the

distri-bution of gravitational charges there are no dipoles, but only quadrupoles etc

In fact, Hendryk A Lorentz from Leiden had already established before Einstein that Maxwell’s equations, which are not invariant under a Galilean

transformation, are invariant w.r.t a Lorentz transformation – as it was later

called – in which space and time coordinates are “mixed” (see Section 9.1) Furthermore, the result of Michelson and Morley’s experiments follows nat-urally from the Lorentz transformations However, Lorentz interpreted his results only as a strange mathematical property of Maxwell’s equations and (in contrast to Einstein) not as a scientific revolution with respect to our

basic assumptions about spacetime underlying all physical events.

The relativistic invariance of Maxwell’s equations can be demonstrated most clearly in terms of the Minkowski four-vectors introduced in Part I a) The essential point is that in addition to

˜

x := (x, y, z, ict) ,

the following two quadruplets,

1)

˜

A :=



A, i Φ

c



and ,

2)

˜

j := (

are Minkowski four-vectors, whereas other quantities, such as the d’Alem-bert operator, are Minkowski scalars (which are invariant), and the fields themselves,E plus B (six components) correspond to a skew symmetric

tensor

F μ,ν(=−F ν,μ) generated from ˜A, viz

F μ,ν :=∂A ν

∂x μ − ∂A μ

∂x ν

(e.g., F 1,2 =−F 2,1 = B3), with x1:= x, x2:= y, x3:= z, and x4:= ict.

b) One defines the “Minkowski nabla”

˜

∇ :=



∂

∂x ,

∂

∂y ,

∂

∂z ,

∂

∂ict



=



∇, ∂

∂ict



.

Similarly to Euclidian spaceR3, where the Laplace operator∇2 (≡ Δ)

is invariant (i.e., does not change its form) under rotations, in Minkowski

spaceM4 the d’Alembert operator ˜∇2 (≡ ) is invariant under pseudo-rotations.

In addition, similar to the fact that the divergence of a vector field has

a coordinate-invariant meaning with respect to rotations inR3, analogous results also apply for the Minkowski divergence, i.e one has an invariant meaning of ˜∇ · ˜v with respect to pseudo-rotations in M

176 20 Maxwell’s Equations II: Electromagnetic Waves

c) For example, the continuity equation,

divj +

∂t = 0 ,

has a simple invariant relativistic form (which we shall use below):

˜

∇ · ˜j :=

4



ν=1

∂j ν

d) Analogously, gauge transformations of the kind

A → A + gradg(r, t), Φ → Φ − ∂g( r, t)

∂t

can be combined to

˜

A(˜ x) → ˜ A(˜ x) + ˜ ∇g(˜x)

We are now prepared for the explicit Minkowski formulation of Maxwell’s equations As mentioned, the homogeneous equations II and III,

divB = 0 ; curlE = − ∂ B

∂t

are automatically satisfied by introducing the above skew symmetric field

tensor

F μν := ∂ μ A ν − ∂ ν A ν , with μ, ν = 1, , 4 ;

which is analogous to the representation of E and B by a scalar potential

plus a vector potential

The remaining inhomogeneous Maxwell equations I and IV,

div 0 and curlB ≡ μ0j ,

simply yield the following result, with Einstein’s summation convention7:

∂ μ F νμ = ∂ μ ∂ ν A μ − ∂ μ ∂ μ A ν ≡ μ0j ν

With the Lorentz gauge,

divA + 1

c2

∂φ

∂t = ∂ μ A μ = 0 ,

8

the first term on the l.h.s vanishes, i.e., we again obtain the d’Alembert-Poisson equation

− ˜ ∇2A˜≡ μ0˜j

7

Using the Einstein convention one avoids clumsy summation symbols: If an index appears twice, it is summed over

8

Here we again use the permutability of partial derivatives

From the “simple” Lorentz transformations for the x- and t-components

of the Minkowski four-potential ˜A, for a transition between different inertial

frames the following “more complicated” Lorentz transformations for

elec-tromagnetic fields result: The longitudinal components E x and B x remain unchanged, whereas one obtains for the transverse components:

B ⊥(r, t) = B  ⊥(r  , t ) + v

c2 × E (r  , t )



1− v2

c2

E ⊥(r, t) = E  ⊥(r  , t )− v × B (r  , t )



1− v2

c2

These results can be used to obtain the E- and B-fields of a moving point

charge from the Coulomb E  -field in the co-moving frame.9

9 See the exercises at http://www.physik.uni-regensburg.de/forschung/krey, sum-mer 2002, file 9

21 Applications of Electrodynamics

in the Field of Optics

21.1 Introduction: Wave Equations;

Group and Phase Velocity

Firstly we shall remind ourselves of the relationship between the frequency

ν = 2π

ω and wavelength λ =

2π

k (k = wavenumber)

of an electromagnetic wave in vacuo:

ω = 2πν = c · k = c · 2π

λ , or λ · ν = c

Secondly, electromagnetic waves cover an extremely wide spectral range For example, radio waves have wavelengths from 1 km or more (long-wave) via 300 m (medium-wave) to about 50 m (short-wave) This range is followed

by VHF (very high frequency), then the range of television and mobile-phone frequencies from≈ 100 MHz to 10 GHz; then we have radar, light waves,

X-rays and, at very short wavelengths or high frequencies, γ-X-rays.

It is useful to remember that the wavelengths of visible light range from

λ ≈ 8000 ˚ A (or 800 nm, red ) down to ≈ 4000 ˚ A (400 nm, violet )1On the lower

frequency side of the visible range come infrared and far-infrared, and on the high-frequency side ultraviolet and soft-X-ray radiation.

Thirdly, in connection with X-rays and γ-radiation, it is useful to

re-member that these phenomena arise from quantum transitions, see Part III (Fermi’s “golden rules”), according to the formula

ΔE ≡ E i − E f ≡ hν ,

i.e by transitions from a higher initial energy E i to a lower final energy E f The radiation may have a continuous distribution of frequencies (so-called

bremsstrahlung, or braking radiation), or it may contain a discrete set of

spectral “lines” The quantity h is Planck’s constant:

h = 6.625 · 10 −34Ws2≡ 4.136 · 10 −15 eVs

1

Some readers may prefer the characteristic atomic length 1 ˚A, whereas others use units such as 1 nm (≡ 10 ˚A) Which one is more appropriate, depends on the problem, on the method used, and on personal preferences

X-rays have typical energies of ΔE ≈ 10 keVto ≈ 1 MeV, characteristic

for the electron shell of atoms, whereas for γ-radiation one is dealing with excitations of nuclei, i.e., ΔE ≈ 1 MeVup to 1 GeV.

Fourthly, Planck’s formula for black-body radiation: The total energy of

the electromagnetic field contained in a volume V at Kelvin temperature T

is given by

U (T ) ≡ V

∞



0

dνu(ν, T ) ,

with the spectral energy density

u(ν, T ) = 8πν

2

c3

hν

For the surface temperature of the sun, i.e., for T ≈ 6000 K, the function u(ν, T ) has a pronounced maximum in the green range, i.e., for

ν = c

λ with λ ≈ 6000 ˚ A(= 600 nm)

See Fig 21.1:

The vacuum velocity of electromagnetic waves (e.g., light) is

c0:= (ε0μ0)−1

.

In polarizable matter, the (stationary) velocity of electromagnetic waves is smaller:

c m=c0

n ,

Fig 21.1 Planck’s black-body radiation

formula For the reduced frequency f (≡ hν/(k B T ) in the text) Planck’s function

P (f ) := f3/(exp f − 1) is shown as a

double-logarithmic plot It has a pronounced

maxi-mum around f ≈ 2

transformation, are invariant w.r.t a Lorentz transformation – as it was later

called – in which space and time coordinates are “mixed”...

In fact, Hendryk A Lorentz from Leiden had already established before Einstein that Maxwell’s equations,...

8

Here we again use the permutability of partial derivatives

From