Slowly Varying Fields in Vacuum
3.4 The Potentials
When the fields are time dependent,E can no longer be found simply as the gradient of a scalar potential since ∇ × E = 0. In all cases∇ · B ≡0, so that we can still expressB as B =∇ × A. From Faraday’s law we have
∇ × E =−∂ B
∂t =− ∂
∂t(∇ × A)
=−∇ × ∂ A
∂t
(3–28) or
∇ ×
E +∂ A
∂t
= 0 (3–29)
We conclude thatnot E, but instead E +∂ A/∂t, is expressible as the gradient of a scalar potential. We have thenE+∂A/∂t=−∇V , or
E =−∇V −∂ A
∂t (3–30)
Taking the divergence of (3–30), we obtain the equation
∇ · E =∇ ·
−∇V −∂ A
∂t
or, equating the right hand side toρ/ε0,
− ∇2V − ∂
∂t
∇ · A
= ρ
ε0 (3–31)
Similarly, putting B = (∇ × A) into Amp` ere’s law we obtain the analogous equation for the vector potentialA.
∇ × ∇ × A
=∇∇ · A
− ∇2A =µ0J−µ0ε0 ∂
∂t
∇V +∂ A
∂t
or
∇2A−µ0ε0∂2A
∂t2
−∇
∇ · A+µ0ε0∂V
∂t
=−µ0J (3–32) 3.4.1 The Lorentz Force and Canonical Momentum
In classical mechanics it is frequently useful to write conservative forces as gradients of potential energies. Although the forces associated with B are not conservative, we can nonetheless find a canonical momentum whose time derivative is the gradient of a quasipotential. We begin by writing the Lorentz force as the rate of change of momentum of a charge in an electromagnetic field,
dp
dt =q(E +v×B) (3–33)
=q
−∇V −∂ A
∂t +v×(∇ × A)
(3–34) The last term of (3–34) may be expanded using (9)
∇(v·A) = ( v·∇)A+ (A·∇)v+A×(∇ × v) +v×(∇ × A)
Ifv, although a function of time, is not explicitly a function of position, we may eliminate spatial derivatives ofvfrom the expansion of∇(v ·A) to obtain v×(∇ × A) = ∇(v ·A) −(v·∇) A. Replacing the last term of (3–34) by this equality we find
dp dt =q
−∇V −∂ A
∂t +∇(v ·A) −(v·∇) A
(3–35) The pair of terms
∂ A
∂t + (v·∇) A ≡ d A
dt (3–36)
is called the convective derivative7 ofA. Substituting d A/dtfor the pair of terms
7As a charge moves through space, the change inA it experiences arises not only from the temporal change inA but also from the fact that it samplesAin different locations.
d A = A [r(t+dt), t+dt]−A[r(t), t]
= '
A[r(t) +vdt, t+dt]−A[r(t) +vdt, t](
+'
A[r(t) +vdt, t]−A[r(t), t](
= ∂ A
∂tdt+ (v·∇)Adt
(3–36), equation (3–35) may be written dp
dt =q
−∇ (V −v·A) −d A dt
(3–37) Grouping like terms gives the desired result:
d dt
p+q A
=−∇
qV −qv·A
(3–38) The argument of ∇ on right hand side of (3–38) is the potential that enters La- grange’s equation as the potential energy of a charged particle in an electromagnetic field and the termmvi+qAi on the left hand side is the momentum conjugate to the coordinatexi (see Exercise 3.9).
3.4.2 Gauge Transformations
As we have mentioned previously,A is not unique since we can add any vector field whose curl vanishes without changing the physics. We see now from (3-30) that, concomitant with any change inA, we also require a compensating change inV in order to keep E (and hence the physics) unchanged.
Recall that a curl free field must be the gradient of a scalar field; hence we write as before,A=A+∇Λ. Let us denote the correspondingly changed potential as V. The magnetic induction field is invariant under this change and we can use (3–30) to express the electric field in terms of the changed () potentials,
E =−∇V −∂ A
∂t (3–39)
=−∇V − ∂
∂t
A+∇Λ
=−∇
V+∂Λ
∂t
−∂ A
∂t (3–40)
or in terms of the unchanged potentials E =−∇V −∂ A/∂t. Comparison of the terms givesV =V −∂Λ/∂t.
The pair of coupled transformations
A=A+∇Λ V =V −∂Λ
∂t
(3–41)
given by (3–41) is called a gauge transformation and the invariance of the fields under such a transformation is called gauge invariance. The transformations are useful for recasting the somewhat awkward equations (3–31) and (3–32) into a more elegant form. Although many different choices of gauge can be made, theCoulomb gauge and theLorenz gauge are of particular use.
In statics it is usually best to choose Λ so that∇ · A vanishes, a choice known as the Coulomb gauge. With this choice (3–31) and (3–32) reduce to Poisson’s equation,
∇2V =− ρ
ε0 (3–42)
and ∇2A =−µ0J (3–43)
If the fields are not static it is frequently still useful to adopt the Coulomb gauge.
The “wave” equation (3–32) for the vector potential now takes the form
∇2A−µ0ε0∂2A
∂t2 =−µ0J+µ0ε0∇ ∂V
∂t
(3–44) Any vector can be written as a sum of curl free component (calledlongitudinal) and a divergence free component (calledsolenoidal ortransverse). As we expect∇V to be curl free, such a decomposition may be useful. Labelling the components of such a resolution ofJby subscriptsl ands we write
J=Jl+Js (3–45)
Substituting this into the general vector identity (13),∇× (∇× J) =∇(∇· J)−∇2J, we obtain separate equations for JlandJs:
∇2Js=−∇ × (∇ × J) (3–46) and
∇2Jl=∇( ∇ · J) (3–47)
Although (3–46) and (3–47) can be solved systematically, the reader is invited to consider (1–19) where V is shown to be the solution of∇2V =−ρ/ε0. We know the solution to this equation to be (1–17). Inserting our form of the inhomogeneity instead of−ρ/ε0we obtain the solutions
Js(r) = 1 4π
∇×[∇×J( r)]d3r
|r−r| (3–48)
Jl(r) =− 1 4π
∇[∇·J( r)]d3r
|r−r| (3–49)
Focussing on (3–49), we integrate “by parts”, noting from (5) thatA∇B=∇(AB)−
B ∇Aso that we write ∇[∇·J( r)]d3r
|r−r| =
∇
∇·J(r)
|r−r|
d3r−
(∇·J)∇ 1
|r−r|
d3r (3–50) The first of the integrals in (3–50) may be integrated using (19) to get
∇
∇·J(r)
|r−r|
d3r=
∇·J(r)
|r−r|
dS (3–51)
The volume integral (3–49) was to include all current meaning that zero current crosses the boundary allowing us to set the integral (3–51) to zero. Focussing now on the remaining integral on the right hand side of (3–50), we use∇|r−1r|=−∇ |r−1r|
to get
Jl=−1 4π∇
∇·J(r)d3r
|r−r| (3–52)
Finally with the aid of the continuity equation (1–24) we replace∇ · J by−∂ρ/∂t to obtain
Jl= 1 4π∇
(∂ρ/∂t)d3r
|r−r| =∇ ∂
∂t
ρ(r)d3r
|r−r| =ε0∇ ∂V
∂t (3–53)
Returning now to (3–44). we find that the last term,µ0ε0∇(∂V /∂t) precisely cancels the longitudinal component of−µ0J. Equation (3–4) may therefore be recast as
∇2A−µ0ε0∂2A
∂t2 =−µ0Js (3–54)
while the electric potential obeys
∇2V =−ρ
0 (3–55)
It is interesting to note that in the Coulomb gauge, V obeys the static equation giving instantaneous solutions (with no time lapse to account for the propagation time of changes in the charge density). The vector potential, on the other hand, obeys a wave equation which builds in the finite speed of propagation of disturbances in Js. We will meet the solenoidal current again in Chapter 10 when we deal with multipole radiation.
In electrodynamics, it is also frequently useful to choose ∇ · A =−µ0ε0∂V /∂t, a choice known as the Lorenz gauge. With this choice, equations (3–31) and (3–32) take the form of a wave equation:
∇2V −µ0ε0∂2V
∂t2 =−ρ
ε0 (3–56)
∇2A−µ0ε0∂2A
∂t2 =−µ0J (3–57)
It is clear that in the Lorenz gaugeV andA obey manifestly similar equations that fit naturally into a relativistic framework.
The invariance of electromagnetism under gauge transformations has profound consequences in quantum electrodynamics. In particular, gauge symmetry permits the existence of a zero mass carrier of the electromagnetic field.