[ Pobierz całość w formacie PDF ]
the only directly observable quantities are the fields themselves (and quantities
derived from them) and not the potentials. On the other hand, the treatment
becomes significantly simpler if we use the potentials in our calculations and
then, at the final stage, use Equation (3.7) and Equation (3.11) above to calcu-
late the fields or physical quantities expressed in the fields.
Inserting (3.11) and (3.7) into Maxwell s equations (1.43) on page 14 we
obtain, after some simple algebra and the use of Equation (1.9) on page 5, the
general inhomogeneous wave equations
1 "2 Á(t,x) " 1 "
Æ - "2Æ = + " · A + Æ (3.12a)
c2 "t2 µ0 "t c2 "t
1 "2 1 "
A - "2A = µ0j(t,x) - " " · A + Æ (3.12b)
c2 "t2 c2 "t
Draft version released 13th November 2000 at 22:01. Downloaded fromhttp://www.plasma.uu.se/CED/Book
main
2000/1
page 3
36 ELECTROMAGNETIC POTENTIALS
These two second order, coupled, partial differential equations, representing in
all four scalar equations (one for Æ and one each for the three components A1,
A2, and A3 of A) are completely equivalent to the formulation of electrodynam-
ics in terms of Maxwell s equations, which represent eight scalar first-order,
coupled, partial differential equations.
3.3.1 Electromagnetic gauges
Lorentz equations for the electromagnetic potentials
As they stand, Equations (3.12) look complicated and may seem to be of lim-
ited use. However, if we write Equation (3.7) on the preceding page in the
form " × A(t,x) = B(t,x) we can consider this as a specification of " × A. But
we know from Helmholtz theorem that in order to determine the (spatial beha-
viour) of A completely, we must also specify " · A. Since this divergence does
not enter the derivation above, we are free to choose " · A in whatever way we
like and still obtain the same physical results! This is somewhat analogous to
the freedom of adding an arbitrary scalar constant (whose grad vanishes) to the
potential energy in classical mechanics and still get the same force.
With a judicious choice of " · A, the calculations can be simplified consid-
erably. Lorentz introduced
1 "
" · A + Æ = 0 (3.13)
c2 "t
which is called the Lorentz gauge condition, because this choice simplifies the
system of coupled equations (3.12) on the previous page into the following set
of uncoupled partial differential equations which we call the Lorentz inhomo-
geneous wave equations:
1 "2 Á(t,x)
def
2
Æ a" Æ - "2Æ = (3.14a)
c2 "t2 µ0
1 "2
def
2
A a" A - "2A = µ0j(t,x) (3.14b)
c2 "t2
2
where is the d Alembert operator discussed in Example M.5 on page 171.
We shall call (3.14) the Lorentz equations for the electromagnetic potentials.
Gauge transformations
We saw in Section 3.1 on page 33 and in Section 3.2 on page 34 that in electro-
statics and magnetostatics we have a certain mathematical degree of freedom,
Downloaded fromhttp://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01.
main
2000/1
page 3
3.3 THE ELECTROMAGNETIC SCALAR AND VECTOR POTENTIALS 37
up to terms of vanishing gradients and curls, to pick suitable forms for the
potentials and still get the same physical result. In fact, the way the electro-
magnetic scalar potential Æ(t,x) and the vector potential A(t,x) are related to
the physically observables gives leeway for similar manipulation of them
also in electrodynamics. If we transform Æ(t,x) and A(t,x) simultaneously into
new ones Æ (t,x) and A (t,x) according to the mapping scheme
"
Æ(t,x) ’! Æ (t,x) = Æ(t,x) + “(t,x) (3.15a)
"t
A(t,x) ’! A (t,x) = A(t,x) - "“(t,x) (3.15b)
where “(t,x) is an arbitrary, differentiable scalar function called the gauge
function, and insert the transformed potentials into Equation (3.11) on page 35
for the electric field and into Equation (3.7) on page 35 for the magnetic field,
we obtain the transformed fields
" " " " "
E = -"Æ - A = -"Æ - "“ - A + "“ = -"Æ - A (3.16a)
"t "t "t "t "t
B = " × A = " × A - " × "“ = " × A (3.16b)
where, once again Equation (M.78) on page 175 was used. Comparing these
expressions with (3.11) and (3.7) we see that the fields are unaffected by the
gauge transformation (3.15). A transformation of the potentials Æ and A which
leaves the fields, and hence Maxwell s equations, invariant is called a gauge
transformation. A physical law which does not change under a gauge trans-
formation is said to be gauge invariant. By definition, the fields themselves
are, of course, gauge invariant.
The potentials Æ(t,x) and A(t,x) calculated from (3.12) on page 35, with
an arbitrary choice of " · A, can be further gauge transformed according to
(3.15) above. If, in particular, we choose " · A according to the Lorentz con-
dition, Equation (3.13) on the facing page, and apply the gauge transformation
(3.15) on the resulting Lorentz equations (3.14) on the preceding page, these
equations will be transformed into
1 "2 " 1 "2 Á(t,x)
Æ - "2Æ + “ - "2“ = (3.17a)
c2 "t2 "t c2 "t2 µ0
1 "2 1 "2
A - "2A - " “ - "2“ = µ0j(t,x) (3.17b)
c2 "t2 c2 "t2 [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl karpacz24.htw.pl
the only directly observable quantities are the fields themselves (and quantities
derived from them) and not the potentials. On the other hand, the treatment
becomes significantly simpler if we use the potentials in our calculations and
then, at the final stage, use Equation (3.7) and Equation (3.11) above to calcu-
late the fields or physical quantities expressed in the fields.
Inserting (3.11) and (3.7) into Maxwell s equations (1.43) on page 14 we
obtain, after some simple algebra and the use of Equation (1.9) on page 5, the
general inhomogeneous wave equations
1 "2 Á(t,x) " 1 "
Æ - "2Æ = + " · A + Æ (3.12a)
c2 "t2 µ0 "t c2 "t
1 "2 1 "
A - "2A = µ0j(t,x) - " " · A + Æ (3.12b)
c2 "t2 c2 "t
Draft version released 13th November 2000 at 22:01. Downloaded fromhttp://www.plasma.uu.se/CED/Book
main
2000/1
page 3
36 ELECTROMAGNETIC POTENTIALS
These two second order, coupled, partial differential equations, representing in
all four scalar equations (one for Æ and one each for the three components A1,
A2, and A3 of A) are completely equivalent to the formulation of electrodynam-
ics in terms of Maxwell s equations, which represent eight scalar first-order,
coupled, partial differential equations.
3.3.1 Electromagnetic gauges
Lorentz equations for the electromagnetic potentials
As they stand, Equations (3.12) look complicated and may seem to be of lim-
ited use. However, if we write Equation (3.7) on the preceding page in the
form " × A(t,x) = B(t,x) we can consider this as a specification of " × A. But
we know from Helmholtz theorem that in order to determine the (spatial beha-
viour) of A completely, we must also specify " · A. Since this divergence does
not enter the derivation above, we are free to choose " · A in whatever way we
like and still obtain the same physical results! This is somewhat analogous to
the freedom of adding an arbitrary scalar constant (whose grad vanishes) to the
potential energy in classical mechanics and still get the same force.
With a judicious choice of " · A, the calculations can be simplified consid-
erably. Lorentz introduced
1 "
" · A + Æ = 0 (3.13)
c2 "t
which is called the Lorentz gauge condition, because this choice simplifies the
system of coupled equations (3.12) on the previous page into the following set
of uncoupled partial differential equations which we call the Lorentz inhomo-
geneous wave equations:
1 "2 Á(t,x)
def
2
Æ a" Æ - "2Æ = (3.14a)
c2 "t2 µ0
1 "2
def
2
A a" A - "2A = µ0j(t,x) (3.14b)
c2 "t2
2
where is the d Alembert operator discussed in Example M.5 on page 171.
We shall call (3.14) the Lorentz equations for the electromagnetic potentials.
Gauge transformations
We saw in Section 3.1 on page 33 and in Section 3.2 on page 34 that in electro-
statics and magnetostatics we have a certain mathematical degree of freedom,
Downloaded fromhttp://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01.
main
2000/1
page 3
3.3 THE ELECTROMAGNETIC SCALAR AND VECTOR POTENTIALS 37
up to terms of vanishing gradients and curls, to pick suitable forms for the
potentials and still get the same physical result. In fact, the way the electro-
magnetic scalar potential Æ(t,x) and the vector potential A(t,x) are related to
the physically observables gives leeway for similar manipulation of them
also in electrodynamics. If we transform Æ(t,x) and A(t,x) simultaneously into
new ones Æ (t,x) and A (t,x) according to the mapping scheme
"
Æ(t,x) ’! Æ (t,x) = Æ(t,x) + “(t,x) (3.15a)
"t
A(t,x) ’! A (t,x) = A(t,x) - "“(t,x) (3.15b)
where “(t,x) is an arbitrary, differentiable scalar function called the gauge
function, and insert the transformed potentials into Equation (3.11) on page 35
for the electric field and into Equation (3.7) on page 35 for the magnetic field,
we obtain the transformed fields
" " " " "
E = -"Æ - A = -"Æ - "“ - A + "“ = -"Æ - A (3.16a)
"t "t "t "t "t
B = " × A = " × A - " × "“ = " × A (3.16b)
where, once again Equation (M.78) on page 175 was used. Comparing these
expressions with (3.11) and (3.7) we see that the fields are unaffected by the
gauge transformation (3.15). A transformation of the potentials Æ and A which
leaves the fields, and hence Maxwell s equations, invariant is called a gauge
transformation. A physical law which does not change under a gauge trans-
formation is said to be gauge invariant. By definition, the fields themselves
are, of course, gauge invariant.
The potentials Æ(t,x) and A(t,x) calculated from (3.12) on page 35, with
an arbitrary choice of " · A, can be further gauge transformed according to
(3.15) above. If, in particular, we choose " · A according to the Lorentz con-
dition, Equation (3.13) on the facing page, and apply the gauge transformation
(3.15) on the resulting Lorentz equations (3.14) on the preceding page, these
equations will be transformed into
1 "2 " 1 "2 Á(t,x)
Æ - "2Æ + “ - "2“ = (3.17a)
c2 "t2 "t c2 "t2 µ0
1 "2 1 "2
A - "2A - " “ - "2“ = µ0j(t,x) (3.17b)
c2 "t2 c2 "t2 [ Pobierz całość w formacie PDF ]