Scalar and Vector Potential-02

(1)

LECTURE NOTES 11

POTENTIALS & FIELDS

The Potential Formulation: Scalar and Vector PotentialsV r t

( )

, and A r t

( )

,

How do the instantaneous sources ρ_tot

( )

r t, and J_tot

( )

r t, (total electric charge density and

total electric current density) generate the electric and magnetic fields E r t

( )

, and B r t

( )

, ? We seek general solutions to the full Maxwell Equations:

(1) Gauss’ Law:

( )

, 1 _tot

( )

,

o

E r t ρ r t

ε

∇i =

(2) No Magnetic Charges: ∇iB r t

( )

, =0

(3) Faraday’s Law: E r t

( )

, B r t

( )

,

t

∂

∇× = −

∂

(4) Ampere’s Law: B r t

( )

, _{o tot}J

( )

r t, _{o o} E r t

( )

,

t

μ μ ε ∂

∇× = +

∂

If ρ_tot

( )

r t, and J_tot

( )

r t, are given (i.e. specified), then what are the corresponding fields

( )

, and

( )

,

E r t B r t ?

⇒ For the static case {i.e. ρ, , ,J E B≠ fcns t

( )

}: Coulomb’s Law

( )

1

( )

₂ ˆ

4

tot v o

r

E r ρ dτ

πε ′

′ ′

=

_∫

_r

r

and the Biot-Savart Law

( )

₂ ˆ

4

tot o

v

J r

B r μ dτ

π ′

′ × ′

=

∫

r

r provide the answers {n.b. r ≡ −r r′ }. ⇒ However, we’re interested in the generalization of Coulomb’s Law and the Biot-Savart Law

for time-dependent situations…

This is not an easy problem!! The approach we will adopt here is to represent the

( )

, and

( )

,

E r t B r t fields in terms of their corresponding potentials V r t

( )

, and A r t

( )

, .

In electrostatics, ∇×E r t

( )

, =0 enabled us to write: E r t

( )

, = −∇V r t

( )

, .

In electrodynamics, we cannot do this because: ∇×E r t

( )

, ≠0. { E r t

( )

, B r t

( )

,

t

∂

∇× = −

∂ }

However, in electrodynamics ∇iB r t

( )

, =0 (still). ⇒ B r t

( )

, = ∇×A r t

( )

,

(2)

Insert B r t

( )

, = ∇×A r t

( )

, into Faraday’s Law: E r t

( )

, B r t

( )

, A r t

( )

,

t t

⎛ ⎞

∂ ∂

∇× = − = −∇×⎜_⎜ ⎟_⎟

∂ _⎝ ∂ _⎠

Then we see that: E r t

( )

, A r t

( )

, 0

t

⎡ ∂ ⎤

∇×_⎢ + _⎥=

∂

⎢ ⎥

⎣ ⎦ i.e. the curl of the quantity

( )

,

, A r t

E r t

t

⎡ ∂ ⎤

+

⎢ _∂ ⎥

⎢ ⎥

⎣ ⎦

does vanish for any/all

( )

r t, , so ∴ we can define: E r t

( )

, A r t

( )

, V r t

( )

,

t

⎡ ∂ ⎤

+ ≡ −∇

⎢ _∂ ⎥

⎢ ⎥

⎣ ⎦

or: E r t

( )

, V r t

( )

, A r t

( )

,

t

∂

= −∇ −

∂ ⇐n.b. reduces to E r

( )

= −∇V r

( )

for the electrostatics case.

Since we used ∇iB=0 and ∇× = −∂ ∂E B t in deriving the above relation, Maxwell’s equations (2) ∇iB=0 and (3) Faraday’s Law: ∇× = −∂ ∂E B t are automatically satisfied. What about equation (1) Gauss’ Law: ∇iE= −ρ ε_tot _o and (4) Ampere’s Law: ∇× =B μ_{o tot}J +ε μ_o _o∂ ∂E t?? (1) Gauss’ Law:

( )

, 1 _tot

( )

, o

E r t ρ r t

ε

∇i = with: E r t

( )

, V r t

( )

, A r t

( )

,

t

∂

= −∇ −

∂

=

( )

_,

( )

, 2

( )

_,

(

( )

_,

)

1

( )

_,

tot o

A r t

V r t V r t A r t r t

t t ε ρ

⎡ ∂ ⎤ ∂

∇ −∇_⎢ − _⎥= −∇ − ∇ =

∂ ∂

⎢ ⎥

⎣ ⎦

i i

Electrodynamics version of the Poisson equation: 2

( )

,

(

( )

,

)

1 tot

( )

, o

V r t A r t r t

t ε ρ

∂

∇ + ∇ = −

∂ i

(4) Ampere’s Law:

B r t

( )

, _{o tot}J

( )

r t, _{o o} E r t

( )

,

t

μ μ ε ∂

∇× = +

∂

with: E r t

( )

, V r t

( )

, A r t

( )

,

t

∂

= −∇ −

∂

and: B r t

( )

, = ∇×A r t

( )

,

( )

(

)

( )

2

( )

2

, ,

, _{o tot} , _{o o} V r t _{o o} A r t

A r t J r t

t t

μ μ ε ⎛∂ ⎞ μ ε ∂

∇× ∇× = − ∇_⎜ _⎟−

∂ ∂

⎝ ⎠

But: _{∇× ∇×}

(

_A

) ( )

_{≡ ∇ ∇}_i_A _{− ∇}2_A

( )

2

( )

2

, ,

, _{o o} A r t , _{o o} V r t _{o tot} ,

A r t A r t J r t

t t

μ ε μ ε μ

⎛ ∂ ⎞ ⎛ ∂ ⎞

∇ − − ∇ ∇ + = −

⎜ ⎟ ⎜ ⎟

⎜ _∂ ⎟ _⎝ _∂ _⎠

⎝ ⎠ i ⇐

n.b. These two equations contain all of the information

that is contained in Maxwell’s 4 equations !!!

n.b. this relation is

actually 3 separate eqns. for x, y and z!

(3)

These two equations may seem “ugly” at this point in time, but watch what we can do with them:

a.) Add:

( )

2 2

, ,

0 _o _o V r t _o _o V r t

t t

ε μ ∂ ε μ ∂

= − +

∂ ∂ to the first equation, and group terms as follows:

1st eqn:

( )

2 2

2

, , 1

, _o _o , _o _o _tot ,

o

V r t V r t

V r t A r t r t

t t t

ε μ ε μ ρ

ε

⎛_∇ ₋ ∂ ⎞₊ ∂ ⎛_∇ ₊ ∂ ⎞_{= −}

⎜ _∂ ⎟ _∂ ⎜ _∂ ⎟

⎝ ⎠

⎝ ⎠ i

2nd eqn:

( )

2 2

2

, ,

, _o _o A r t , _o _o V r t _{o tot} ,

A r t A r t J r t

t t

ε μ ε μ μ

⎛ ∂ ⎞ ⎛ ∂ ⎞

∇ − − ∇ ∇ + = −

⎜ ⎟ ⎜ ⎟

⎜ _∂ ⎟ _⎝ _∂ _⎠

⎝ ⎠ i

b.) Define: L r t

( )

, A r t

( )

, _o _o V r t

( )

,

t

ε μ ∂

⎛ ⎞

≡ ∇_⎜ + _⎟

∂

⎝ i ⎠.

c.) Define the D’Alembertian (aka “Box2”) operator:

2 2

2 2 2

1

_o _o

t c t

ε μ ∂ ∂

≡ ∇ − = ∇ −

∂ ∂

Then:

1st eqn: 2

( )

_,

( )

, 1

( )

_,

tot o

L r t

V r t r t

t ε ρ

∂

+ = −

∂

2nd eqn: 2

( )

_,

( )

_,

( )

_,

o tot

A r t − ∇L r t = −μ J r t

d.) Divide 1st eqn. by c, then use ₁ 2

o o c

ε μ = relation:

1st eqn: 21

( )

_, 1

( )

, o

( )

_,

( )

_,

tot o tot

o o

L r t

V r t r t c r t

c c t cε μμ ρ μ ρ

∂

+ = − = −

∂

2nd eqn: 2

( )

_,

( )

_,

( )

_,

o tot

A r t − ∇L r t = −μ J r t

Thus:

1st eqn: 21V r t

( )

, 1 L r t

( )

, _oc _tot

( )

r t,

c c t μ ρ

∂

+ = −

∂

“time-like”

2nd eqn: 2A r t

( )

, − ∇L r t

( )

, = −μo Jtot

( )

r t, “space-like”

We shall see later on in the semester (when we get to relativistic electrodynamics), that

relativisticfourvectors exist {valid in anyinertialreferenceframe} for the relativistic 4-vector

potential A r tμ

( )

, ≡

(

V r t c A r t

( )

, ,

( )

,

)

{where μ=0,1, 2,3; μ =0is the temporal component

of the relativistic 4-vector, and μ =1, 2,3 are e.g. the x, y, z spatial components of the 4-vector},

the 4-current density J_totμ

( )

r t, ≡

(

cρ_tot

( )

r t J, , _tot

( )

r t,

)

.

The covariant relativistic 4-gradient operator: 1 ,

x c t

μ μ

∂ ⎛ ∂ ⎞

∂ ≡ ≡ +_⎜ ∇_⎟

∂ ⎝ ∂ ⎠, whereas

the contravariant relativistic 4-gradient operator: 1 ,

x c t

μ ∂ ⎛ ∂ ⎞

∂ ≡ ≡ −_⎜ ∇_⎟

∂ ⎝ ∂ ⎠.

Do you see/canyou see any interesting parallels between these two equations???

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The relativistic D’Alembertian / “Box2” operator can thus be written in 4-vector notation as:

2

2 2

1

x x c t

μ

μ μ

μ

∂ ∂ ∂

∂ ∂ = = = ∇ −

∂ ∂ ∂

We can write: L r t

( )

, A r t

( )

, 1₂ V r t

( )

, A r t

( )

, A r t

( )

,

c t x

μ

μ μ μ

∂ ∂

⎛ ⎞

≡ ∇_⎜ + _⎟= = ∂

∂ ∂

⎝ i ⎠

We can also write:

( )

, 1

( )

, 1

( )

,

( )

,

( )

, v

( )

,

v

L r t L r t

L r t L r t L r t A r t

c t c t x

μ μ

μ

∂ ⎛ ∂ ⎞ ∂

−∇ + = − −_⎜ + ∇_⎟ = − = −∂ = −∂ ∂

∂ ⎝ ∂ ⎠ ∂

Thus, the two equations:

1st eqn: 21

( )

_, 1

( )

,

( )

_,

o tot

L r t

V r t c r t

c c t μ ρ

∂

+ = −

∂

2nd eqn: 2

( )

, , _o _tot ,

A r t − ∇L r t = −μ J r t

can be written as a single equation, very compactly (& elegantly!) in relativistic electrodynamics as:

( )

,

( )

,

( )

,

v v

v A r t vA r t oJ r t

μ μ _μ μ

∂ ∂ − ∂ ∂ = −

Very shortly, we will learn that as a consequence of the gauge-invariant nature of the

electromagnetic interaction, for the choice of working in the so-called Lorenzgauge, namely that:

( )

2

( )

, ,

1

, , , 0

v

v v v

V r t A r t

L r t A r t A r t

c t x

∂ ∂

⎛ ⎞

≡ ∇_⎜ + _⎟= = ∂ =

∂ ∂

⎝ i ⎠

If L r t

( )

, =0 ∀

( )

r t, , then ∂L r t

( )

, ∂ =t 0 ∀

( )

r t, . Hence, in the Lorenzgauge:

( )

,

( )

,

v

v A r t oJ r t

μ _μ μ

∂ ∂ = − or equivalently: 2A r tμ

( )

, = −μ_oJμ

( )

r t,

This single potential equation (actually 4 separate equations, since μ =0,1, 2,3) contain(s) all of the information that is contained in Maxwell’s 4 field equations !!!

(1) Gauss’ Law:

( )

, 1 _tot

( )

, o

E r t ρ r t

ε

∇i = _{(3) Faraday’s Law:} E r t

( )

, B r t

( )

,

t

∂

∇× = −

∂

(2) No Magnetic Charges: ∇iB r t

( )

, =0_{(4) Ampere’s Law:} B r t

( )

, _{o tot}J

( )

r t, _{o o} E r t

( )

,

t

μ μ ε ∂

∇× = +

(5)

Griffiths Example 10.1

Find the electric charge and current distributions ρ_tot

( )

r t, and J_tot

( )

r t, that produce:

( )

, 0

V r t =

( )

(

)

2

ˆ, for 4

,

0, for

ok _ct _x _z _x _ct

c A r t

x ct

μ

⎧ ₋ _<

⎪ = ⎨

⎪ _≥

⎩

Note that the parabola:

(

ct− x

)

2 =

( )

ct 2−2ct x + x2 =0 at ct= x .

Since V r t

( )

, =0 ∀ ,

( )

r t , we know that ρ_tot

( )

r t, =0 ∀ ,

( )

r t .

However, A r t

( )

, zˆ tells us that there must be some kind of current present, and in the ˆz-direction.

In the process of solving this problem, we will determine what kind of current is present…

Note that A A z= _zˆ i.e. note that A is an explicit fcn(x) only, and points (only) in the ˆz-direction.

First, we determine E r t

( )

, and B r t

( )

, from A r t

( )

, :

( )

,

( )

,

(

)

ˆ

2 o

A r t k

E r t ct x z

t

μ

∂

= − = − −

∂ for x <ct, and: E r t

( )

, =0 for x ≥ct.

Thus:

0

z

A A

y =

∂ ∇× =

∂

0

y

A z =

∂ −

∂

0

ˆ Ax

x z =

⎛ ⎞

∂

⎜ _{⎟ +}

⎜ ⎟ _∂

⎜ ⎟

⎝ ⎠

0

ˆ y

z A

A y

x x

=

⎛ ⎞ _∂

∂

⎜ ₋ ⎟ ₊

⎜ _∂ ⎟ _∂

⎜ ⎟

⎝ ⎠

0

x

A y =

∂ +

∂ ˆ z ˆ

A

z y

x

⎛ ⎞

∂

⎜ _{⎟ = −}

⎜ ⎟ _∂

⎜ ⎟

⎝ ⎠

since A_x = A_y =0

Hence:

For x > 0

( )

(

)

2

(

)

ˆ ˆ

, ,

4 2

ok ok

B r t A r t ct x y ct x y

c x c

μ ∂ μ

= ∇× = − − = ± −

∂ for x <ct, and B=0 for x ≥ct.

For x < 0

n.b. For x ≥ct, E r t

( )

, and B r t

( )

, =0 n.b. B x t_y

( )

, has a discontinuity for t > 0 at x = 0 !!

( )

,

z

E x t

x ct ct

−

@ t =t

@ t =0

1 2μokct −

( )

, y

B x t

x ct ct

−

@ t =t

@ t =0

1 2μokt −

@ t =t 1 2μokt +

@ t =0 where k = constant

and c=1 ε μ_o _o

( )

, z

A x t

x

@t t=

@t=0

ct

− ct

1 4μokt

(6)

It can be shown that indeed ∇iE r t

( )

, =0, ∇iB r t

( )

, =0 {and ∇iA r t

( )

, =0} {n.b. please explicitly check/work these out yourselves!}, and that indeed:

( )

, ˆ

2 ok

E r t μ y

∇× =∓

( )

, ˆ

2 ok

B r t z

c μ

∇× = −

( )

, ˆ

2 o

B r t k

y t

μ

∂

= ±

∂

( )

,

ˆ 2 o

E r t kc

z t

μ

∂

= − ∂

⇒Maxwell’s eqn’s all satisfied, with ρ_tot

( )

r t, =0 and J_tot

( )

r t, =0.

• However, we have seen before {déjà vu!} that the discontinuity in B r t

( )

, @ x = 0 for t>0

heralds/signals the presence of a surfacecurrent K_free

( )

r t, .

• In our case {here}, the surface current K_free

( )

r t, lies in the y-z plane {n.b. A K_free zˆ!} The boundary condition associated with a free surface current (see P435 Lect. Notes 24, p. 13-17 and/or Griffiths equation 7.63 (iv), p. 333) is:

1 2 1 2

1 2

1 1

ˆ free

H H B B K n

μ μ

− = − = × with: μ₁ =μ₂ =μ_o {here}

⇒ B₁ −B₂ =μ_oK_free×nˆ at x=0

i.e.:

(

0,

)

0

(

0,

)

0 ˆ 0

y x y x o free x

B x< t ₌ −B x> t ₌ =μ K ×n ₌ n.b. nˆ points from medium 2 → 1 (nˆ= −xˆ) See Griffiths p. 331-2, figures 7.46 and 7.47 ˆz − =x nˆ ˆ

medium 1 (x < 0)

K_free

ˆy ⇐ n.b. K_free

( )

r t, lies in the y-z plane K_free

xˆ= −nˆ medium 2 (x > 0) ∴ ˆ ˆ ˆ

2 2

o o

o free

kt kt

y y K x

μ μ _μ

− − = − ×

But: ˆ ˆx y z× = ˆ, ˆy z x× =ˆ ˆ, z x yˆ× =ˆ ˆ ⇒ ˆkty K= _free×xˆ ⇒ K_free

( )

t =ktzˆ at x = 0 in y-z plane. Physically, this corresponds to a uniform, but time-dependent free surface current K_free

( )

t =ktzˆ which flows in the ˆ+z direction at x = 0 in the y-z plane. It starts up from zero at time t = 0, its strength K_free

( )

t =kt increases linearly with time t.

n.b. ⇒ “News” travels outward at speed c, note that E and B are still zero for points x ≥ct!!! Region 1: x < 0 Region 2: x > 0

(7)

Gauge Transformations

Using v

( )

, v

( )

,

( )

,

v A r t vA r t oJ r t

μ μ _μ μ

∂ ∂ − ∂ ∂ = − enables us to reduce the problem of finding the six components associated with the two vectors E E x E y E z= _xˆ+ _yˆ+ _zˆ and B B x B y B z= _xˆ+ _yˆ+ _zˆ

down to four components – the scalar potential V and the vector potential A A x A y A z= _xˆ+ _yˆ+ _zˆ.

As we saw last semester in P435, B r t

( )

, = ∇×A r t

( )

, and E r t

( )

, = −∇V r t

( )

, − ∂A r t

( )

, ∂t

do not enable us to uniquely define / specify / determine the scalar and vector potentials V r t

( )

,

and A r t

( )

, ; only potential differencesV2 – V1 and A2−A1 are physically meaningful…

We are free to impose extra conditions on V r t

( )

, and A r t

( )

, aslongas E r t

( )

, and B r t

( )

,

remain unchanged by the imposition of these extra conditions…

This freedom to impose extra conditions on V r t

( )

, and A r t

( )

, without changing

( )

, and

( )

,

E r t B r t is known as gaugefreedom or (more formally) as gaugeinvariance.

Suppose we have e.g. two sets of potentials

{

V r t A r t

( ) ( )

, , ,

}

and

{

V r t A r t′

( ) ( )

, , ′ ,

}

that

correspond to the same physical fields E r t

( )

, and B r t

( )

, . These two sets of potentials must

be related to each other by:

( )

,

( )

,

( )

,

A r t′ =A r t +α r t and: V r t′

( )

, =V r t

( )

, +β

( )

r t,

Because: B r t

( )

, = ∇×A r t

( )

, = ∇×A r t′

( )

, = ∇×

(

A r t

( )

, +α

( )

r t,

)

⇒ ∇×α

( )

r t, ≡0 But if:

(

∇×α

( )

r t,

)

=0, then since

(

∇×∇f r t

( )

,

)

≡0 (always!)

⇒ we can always write α

( )

r t, as the gradient of a scalar function λ

( )

r t, , e.g. α

( )

r t, ≡ ∇λ

( )

r t, .

And if:

( )

,

, ,

, , ,

A r t

E r t V r t

t

A r t A r t r t

V r t V r t r t

t t t

α β

∂

= −∇ −

∂ ′

∂ ∂ ∂

′

= −∇ − = −∇ − ∇ − −

∂ ∂ ∂

Then we see that the scalar function β

( )

r t, must be related to α

( )

r t, by:

( )

r t,

( )

r t, 0

t α

β ∂

∇ + =

∂

But: α

( )

r t, ≡ ∇λ

( )

r t, ⇒

( )

r t,

(

( )

r t,

)

0

t λ

β ∂ ∇

∇ + =

∂ ⇒

( )

,

, r t 0

r t

t

λ

β ∂

⎛ ⎞

∇_⎜ + _⎟=

∂

⎝ ⎠

which must hold for arbitrary/any/all space-time points

( )

r t, ⇒

( )

r t,

( )

r t, 0

t λ

β +∂ =

(8)

Note that

( )

r t,

( )

r t, 0

t

λ

β ∂

⎛ ⎞

∇_⎜ + _⎟=

∂

⎝ ⎠ can also be satisfied if

( )

,

_{( )}

, r t

r t t

t

λ

β ∂ κ

⎛ ⎞

+ =

⎜ _∂ ⎟

⎝ ⎠ ,

i.e. the scalar function κ

( )

t depends only on time, t.

Thus we see that:

( )

r t,

( )

r t,

( )

t t

λ

β = −∂ +κ

∂ .

But we can always “absorb” κ

( )

t into λ

( )

r t, by adding

( )

0

t t

t κ t dt

′=

′= ′

∫

to λ

( )

r t, , i.e.

( )

,

( )

, t t₀

( )

t

r t r t t dt

λ′ =λ +

∫

_′=′=κ ′ .

We can then redefine the “new” λ′

( )

r t, . ⇒ “old”λ

( )

r t, .

Note also that since the scalar function κ

( )

t depends only on time t, this will not affect the

gradient of ∇λ

( )

r t, in any way, and hence α

( )

r t, = ∇λ

( )

r t, is completelyunaffected by this!

Thus: A r t′

( )

, = A r t

( )

, +α

( )

r t, = A r t

( )

, + ∇λ

( )

r t, or: ∇λ

( )

r t, =A r t′

( )

, −A r t

( )

, ≡ ΔA r t

( )

, And: V r t

( )

, V r t

( )

,

( )

r t,

t λ

∂

′ = −

∂ or:

( )

,

_{( )}

, , ,

r t

V r t V r t V r t

t λ

∂

′

− = − ≡ Δ

∂

Thus, for any scalar function λ

( )

r t, , we can always add ∇λ

( )

r t, to A r t

( )

, provided that

we simultaneouslysubtract ∂λ

( )

r t, ∂t from V r t

( )

, .

⇒ This will leave the E r t

( )

, and B r t

( )

, -fields unchanged / invariant under this so-called

gaugetransformation!

• Gauge transformations can be exploited to adjust ∇iA r t

( )

, .

• In magnetostatics, we chose ∇iA r t

( )

, =0 ( = the Coulomb gauge).

• In electrodynamics, the situation is not always so clear cut!! The most convenient “gauge”

choice depends on the detailed nature of the problem! ∃ Many gauges to work with…. Coulomb Gauge: ∇iA r t

( )

, =0 (Most useful in electro/magnetostatics)

Lorenz Gauge: A r t

( )

, _{o o} V r t

( )

,

t

μ ε ∂

∇ = −

∂

i (Most useful in electrodynamics)

Ludwig Valentin Lorenz, Danish physicist – a contemporary of J.C. Maxwell, ca. 1867 – not to be confused with Hendrick A. Lorentz, Dutch physicist & contemporary of Albert Einstein… {see/read J.D. Jackson & L.B. Okun’s “Historical Roots of Gauge Invariance” Feb. 22, 2001; On the web at: http://arxiv.org/vc/hep-ph/papers/0012/0012061v2.pdf}.

The two most popular gauges for use in E&M

(9)

• Gauge Transformation(s) are intimately connected to the choice of an {inertial} reference

frame. “Hints” of this connection, e.g. in the context of special relativity – length contraction and time-dialation between two {inertial} reference frames are the ∇λ

( )

r t, and

( )

r t, t λ′

∂ ∂ {where

( )

0

, , t t

t

r t r t t dt

λ λ ′=κ

′=

′ = +

∫

′} terms that can be added to

{subtracted from} the vector potential A r t

( )

, {scalar potential V r t

( )

, }, respectively.

The Coulomb Gauge: ∇iA r t

( )

, =0

Here, the scalar potential V r t

( )

, is “easy” to calculate, but in comparison, the vector potential

( )

,

A r t is “difficult” to calculate. The inhomogeneous differential equation for V r t

( )

, is:

( )

2_{V r t}_, _{A r t}_,

t

∂

∇ + ∇

∂

(

i

)

( )

1

, tot o

r t ρ ε

= − ⇒ 2

( )

_, 1

( )

_,

tot o

V r t ρ r t

ε

∇ = −

Then for: 2

( )

_, 1

( )

_,

TOT o

V r t ρ r t

ε

∇ = − if we set: V r

(

= ∞,t

)

=0

The solution to Poisson’s equation is:

( )

, 1

( )

, 4

tot v o

r t

V r t ρ dτ

πε ′

′ ′

=

∫

r

But V r t

( )

, alone does not determine E r t

( )

, in electrodynamics – we mustalso know A r t

( )

,

as well! In the Coulomb gauge, the differential equation for the vector potential A r t

( )

, is:

( )

2 2

2

,

, o o ,

A r t

A r t A r t

t

μ ε ∂

∇ − − ∇ ∇

∂ i

( )

,

_{( )}

,

o o o tot

V r t

J r t

t

μ ε ∂ μ

⎛ ⎞

+ = −

⎜ _∂ ⎟

⎝ ⎠

becomes:

( )

2 2

2

, ,

, _{o o} A r t _{o tot} , _{o o} V r t

A r t J r t

t t

μ ε ∂ μ μ ε ⎛∂ ⎞

∇ − = − + _∇⎜ _⎟

∂ _⎝ ∂ _⎠ ⇐

In the Coulomb Gauge, the scalar potential at time t, V r t

( )

, is determined by the distribution of electric charge at the “rightnow” time t – which is acausal, because EM signals/information cannot

propagate faster than the speed of light c! However, changes in the electric charge density distribution

( )

tot r

ρ ′ at the source point r′ take a finite time to be observed at the observation point r !!!

( )

, 1

( )

,

4

tot v o

r t

V r t ρ dτ

πε ′

′ ′

=

_∫

r

Thus in the Coulomb Gauge, the scalar potential V r t

( )

, instantaneously reflects all changes

inρtot

( )

r t′, . However, recall that V r t

( )

, byitself is not a physically measurable quantity – only potential differences such as ΔV t₂₁

( )

≡V r t

( )

₂, −V r t

( )

₁, or ΔV r

(

₁,Δ ≡t

)

V r t

(

₁, ₂

)

−V r t

(

_{1 1},

)

are!

The “usual” Poisson’s equation

with: _r ≡ −r r′ and:

2 2

r r′

= = −

r r

n.b. Also depends

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An astronaut standing on the surface of the moon can only e.g. directly measure E r t

( )

, , which manifestly involves both V r t

( )

, and A r t

( )

, : E r t

( )

, = −∇V r t

( )

, − ∂A r t

( )

, ∂t, thus in

the Coulomb gauge, while the scalar potential V r t

( )

, instantaneously reflects all changes in

( )

, tot r t

ρ ′ , the combination −∇V r t

( )

, − ∂A r t

( )

, ∂t does not have this instantaneous,

“right-now” behavior in the Coulomb gauge, i.e. the combination −∇V r t

( )

, − ∂A r t

( )

, ∂t is causal in the Coulomb Gauge, and thus E r t

( )

, changes in a causally-connected manner only after EM

“news” / information arrives at the appropriate / causally-related time interval Δt, as a direct consequence of the propagation speed (= c in free space/vacuum) of this EM “news” /

information. Thus, in the Coulomb gauge, the causal behavior is carried by / encrypted into the

vector potential A r t

( )

, , in satisfying the above differential equation for A r t

( )

, !

The Lorenz Gauge:

( )

, o o

( )

,

V r t A r t

t

μ ε ∂

∇ = −

∂

i ⇒

( )

,

( )

, o o

( )

, 0

V r t

L r t A r t

t

μ ε ∂

≡ ∇ + =

∂ i

Here we obtain:

(a) 2_{V r t}

( )

_, L r t

( )

, t

∂ +

∂

1 tot

( )

, o

r t

ρ ε

= − ⇒ 2

( )

, 1 _tot

( )

,

o

V r t ρ r t

ε

= −

and:

(b) 2_{A r t}

( )

_, _{− ∇}_{L r t}

( )

_,

( )

_,

o totJ r t μ

= − ⇒ 2A r t

( )

, = −μ_{o tot}J

( )

r t, where:

2

2 2

1

c t

∂ ≡ ∇ −

∂

.

Thus, we see from 2

( )

_,

( )

_,

tot o

V r t = −ρ r t ε

and 2

( )

_,

( )

_,

o tot

A r t = −μ J r t

that the

Lorenz gauge puts the {time-like} scalar potential V r t

( )

, and the {space-like} vector potential

( )

,

A r t on an equal footing! (i.e. a “democratic” treatment of V r t

( )

, and A r t

( )

, ).

In relativity, V r t c

( )

, and A r t

( )

, respectively are the temporal and spatial components of the relativistic 4-vector potentialA r tμ

( )

, and cρ_tot

( )

r t, and J_tot

( )

r t, respectively are the temporal and spatial components the relativistic 4-vector current Jμ

( )

r t, , where the index μ =0 : 3 {0,1, 2,3}= . In tensor notation: 2_{A r t}μ

( )

_, ₌_Jμ

( )

_{r t}_, _⇐_{4-D {space-time} Poisson Equation in the Lorenz gauge!}

where: Aμ ≡

(

V c A A A, _x, _y, _z

)

and Jμ ≡

(

c J J Jρ, ,_x _y, _z

)

and

2

2 2

1 v

v

x x c t

∂ ∂ ∂

≡ = ∇ −

∂ ∂ ∂

n.b. Convention: repeated indices (here = v) are summedover (i.e. summed over v = 0:3).

n.b. Griffiths claims that he will use the Lorenz gauge for the remainder of his book {we will also!}. The whole of electrodynamics thus reduces to solving the inhomogeneous 4-D Poisson’s

(11)

Continuous Electric Charge and Current Distributions Advanced and Retarded Potentials

We derived {see above}the following relations for the potentials in the Lorenz gauge:

L r t

( )

, A r t

( )

, _{o o} V r t

( )

, 0

t

μ ε ∂

≡ ∇ + =

∂

i 2_{A r t}μ

( )

_, ₌_Jμ

( )

_{r t}_, _⇐

4-D Poisson Equation with: A r tμ

( )

, ≡

(

V r t c A r t

( )

, ,

( )

,

)

=

(

V r t c A r t A r t A r t

( )

, , _x

( )

, , _y

( ) ( )

, , _z ,

)

and: Jμ

( )

r t, ≡

(

cρ

( ) ( )

r t J r t, , ,

)

=

(

cρ

( ) ( ) ( ) ( )

r t J r t J r t J r t, , _x , , _y , , _z ,

)

Or: 2

( )

_, 1

( )

_,

tot o

V r t ρ r t

ε

= −

where:

2 2

2 2 2

1

o o

x_γ xγ c t μ ε t

∂ ∂ ∂ ∂

≡ = ∇ − = ∇ −

∂ ∂ ∂ ∂

( )

2 _, _,

o tot

A r t = −μ J r t

Or:

( )

2

2 _, , 1 _,

o o tot

o

V r t

V r t r t

t

μ ε ρ

ε

∂

∇ − = −

∂

( )

2

2 _, , _,

o o o tot

V r t

A r t J r t

t

μ ε ∂ μ

∇ − = −

∂

For static situations (no time dependence) these reduce to the familiar “4-D” Poisson Equation:

( )

2 1

tot o

V r ρ r

ε

∇ = − ⇒ solution is:

( )

1

( )

4

tot v o

r

V r ρ dτ

πε ′

′ ′

=

∫

r

( )

2

o tot

A r μ J r

∇ = − ⇒ solution is:

( )

4

tot o

v

J r

A r μ dτ

π ′

′ ′

=

∫

r

(12)

( )

(

)

(

)

(

)

2 2

2

2 2

r

2 2 2

r

c t

c t t

r r c t t

+ Δ

= + −

′

= − + −

r r

(

r

)

c t

c t t

Δ

= −

r

Field/Observation Point P r t

( )

,

Source Point S r t

(

′, _r

)

ct

r ct

The Light-Cone in Space-Time:

c time⋅

space

( )

0,0

r

Now consider what happens if the sources ρ_tot

( )

r t′, and J_tot

( )

r t′, in the volume v’ are

time-dependent:

An observer at the field point P r t

( )

, detects changes in the potentials and/or the EM fields

at the time t. However, those changes observed at the field point P r t

( )

, result from changes in

(

, r

)

tot r t

ρ ′ and / or J_tot

(

r t′, _r

)

that occurred at earlier time(s), because it takes a finite amount of

time for EM “news” (i.e. changes) that occur at a source point S r t

(

′, _r

)

to propagate to the field / observation point P r t

( )

, .

In terms of a space-time light-cone diagram (for propagation of EM news in free space = vacuum):

(

r

)

c t c t tΔ = − =_r

or: Δ = −t

(

t tr

)

= r c

or: t t= +r r c

or: tr = −t r c = retardedtime

with: _r = r t

( )

−r t′

( )

_r

and: _r =r t

( )

−r t′

( )

_r

(13)

Due to causality, it takes a finite time Δ = − =t t t_r r c= −r r c′ for a changee.g. in the

electric charge density ρ_tot

(

r t′, _r

)

that occurs at the source point S r t

(

′, _r

)

at the earlier, retarded

time trto propagate to the field/observation point P r t

( )

, at the later time, t > tr: t t= +r r c.

In free space (the vacuum) this EM “news” / information propagates with speed c = 3 × 108_m_/_sec_.

We also point out {here, for completeness’ sake} that both the source point S r t

(

′, _r

)

and field/

observation point P r t

( )

, are at rest (e.g. in the labframe – an inertial {i.e. non-accelerating} referenceframe). The electrodynamics of this situation will be different e.g. if the observer is movingrelative to the source, or if both the source and the observer are moving with respect to

a chosen reference frame (e.g. the lab frame). Special relativity deals with these situations… Thus, for non-static source volume charge density/current density distributions ρ_tot

(

r t′, _r

)

and J_tot

(

r t′, _r

)

, the scalar and vector potentials V r t

( )

, and A r t

( )

, at the observation/field point

( )

,

P r t at the later, causal time t t= +_r r c (t > tr) are related to the sources ρ_tot

(

r t′, _r

)

and

(

, r

)

tot

J r t′ at the source point S r t

(

′, _r

)

at the earlier, so-called retarded time, t_r = −t r c

by the following relations:

( )

(

r

)

r

, 1

, 4

tot v o

r t

V r t ρ dτ

πε ′

′

=

∫

r with tr = −t r c

( )

(

r

)

r

, ,

4

tot o

v

J r t

A r t μ dτ

π ′

′

=

∫

r and r = r t

( )

−r t′

( )

r

These expressions for the potentials are known as retardedpotentials because changes in the

source volume charge density and/or current density distributions ρ_tot

(

r t′, _r

)

and J_tot

(

r t′, _r

)

at source point(s) S r

( )

′ occurring at the (earlier) “retardedtime”, tr < t, take a time interval

r

t t t c r r c′

Δ = − =_r = − to propagate from the source point S r t

(

′, _r

)

to the observation/field point P r t

( )

, arriving there at the later, causal time t t= +_r _r c, where _r = r t

( )

−r t′

( )

_r .

• This is exactly analogous to the situation where an observer is looking out into the night sky. Light (= EM radiation) from a star a distance _r_star = r_obs

( )

t −r_star′

( )

t_r away, arriving on Earth at time t had to have left the surface of that star at an earlier time: t_r = −t r_star c .

The transit/propagation time of the light from the star to the Earth is:Δ = − =t t t_r _r_star c.

• From our own star (the sun), this time interval is:

11 8

1.496 10

3 10 /

Earth Sun m

t

c c m s

⊕− − ×

Δ = = =

×

r r ₅₀₀_seconds_{= 8.3}_minutes

Thus, we see that causality over astronomical distances is significant, but is also important even for laboratory/everyday distance scales.

Retarded Scalar and

Vector Potentials:

(14)

In the previous semester’s course (P435) we saw that, for static sources:

( )

1

( )

4

tot v o

r

E r V r ρ dτ

πε ′

′ ′

= −∇ = −∇

∫

r where: r = −r r′ and ∇ = fcn r

( )

only

( )

1

( )

4

tot v

o

r

E r ρ dτ

πε ′

′

⎛ ⎞

′

= _−∇⎜ _⎟

⎝ ⎠

∫

_r where: ρ_tot

( )

r′ = fcn r

( )

′ only

( )

2

1

ˆ 4

tot v o

r

E r ρ dτ

πε ′

′ ′

= +

_∫

r r

( )

4

tot o

v

J r

B r A r μ dτ

π ′

′ ′

= ∇× = ∇×

∫

r where: Jtot

( )

r′ = fcn r

( )

′ only

( )

4

tot o

v

J r

B r μ dτ

π ′

⎛ ′ ⎞

′

= ∇×⎜_⎜ ⎟_⎟

⎝ ⎠

∫

_r

( )

2

ˆ 4

tot o

v

J r

B r μ dτ

π ′

′ × ′

=

∫

r

Now we cannot simply “generalize” these to the time-dependent case merely by adding t and tr

arguments to the

{

E and B

}

and

{

ρ_tot and J_tot

}

(field and source) variables respectively!!

i.e.

( )

(

r

)

r 2

, 1

ˆ ,

4

tot v o

r t

E r t ρ dτ

πε ′

′

≠

_∫

r r Nyet !!!

( )

(

r

)

r 2

ˆ , ,

4

tot o

v

J r t

B r t μ dτ

π ′

′ ×

′

≠

_∫

r

r Nyet

2_!!!

The reason why these expressions are not correct is simple: The causal connection between

t and tr has not been properly taken into account in the above two formulae: t_r = −t r c with

( )

r

r t r t′

= −

r .

Properly taking into account this causal connection we need to realize that:

(

, r

)

(

,

)

tot r t tot r t c

ρ ′ =ρ ′ −_r

(

, r

)

(

,

)

tot tot

J r t′ =J r t′ −_r c

Thus, in order to correctly / properly calculate E r t

( )

, and B r t

( )

, we need to back up and calculate these relations much more carefully!!!

i.e. ρ_tot and J_tot are now also implicit functions of r = −r r′

via the causal relation t_r = −t _r c and hence are implicit

(15)

In calculating e.g. the Laplacian of the retarded scalar potential V r t

( )

, , it is critical

to realize that the integrand in

( )

(

r

)

r , 1 , 4 tot v o r t

V r t ρ dτ

πε ′

′

=

∫

r depends on r in two places: Explicitly, in the denominator of the integrand, because: r = r t

( )

−r t′

( )

_r , and also Implicitly, in the numerator of the integrand, because: t_r = −t r c t r r c= − − ′ .

Thus:

( )

(

r

)

(

)

r , , 1 1 , 4 4 tot tot v v o o

r t r r c r t

V r t d d

r r ρ

ρ

τ τ

πε ′ πε ′

′ − − ′ ′ ′ ′ = = ′ −

∫

_r

∫

Then since ∇ = fcn r

( )

only, then:

r

t = −t _r c t r r c= − − ′

( )

(

)

(

)

(

)

r r r , 1 , 4 , , 1 1 4 4 tot v o tot tot v v o o r t

V r t d

r t r r c r t d d r r ρ τ πε ρ ρ τ τ πε πε ′ ′ ′ ′ ⎛ ⎞ ′ ∇ _{= ∇⎜} _⎟ ⎝ ⎠ ⎛ ′ − − ′ ⎞ ′ ⎛ ⎞ ′ ′ = ∇⎜ ⎟ = ∇⎜_⎜ ₋ _′ ⎟_⎟ ⎝ ⎠ _⎝ _⎠

∫

r r

But ρ_tot

(

r t′, _r

)

is an explicit fcn r

( )

′ and also an implicit fcn r

( )

because t_r = −t r c t r r c= − − ′ .

By the chain rule:

(

r

)

(

)

(

)

r r

,

1 1 1 1

, , 4 4 tot tot tot v v o o r t

d r t r t d

ρ

τ ρ ρ τ

πε ′ πε ′

′ ⎛ ⎞ _′ ⎡ _′ _′ ⎛ ⎞⎤ _′ ∇_⎜ _⎟ = _⎢ ∇ + ∇_{⎜ ⎟}_⎥ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠

∫

_r

∫

_r _r

But:

(

)

(

r

)

(

)

r r r

r

, ,

, , , tot tot r

tot tot tot

r r r t r t

r t r t r t t t

c c t t

ρ ρ

ρ ′ ρ ⎛ ′ ⎞ ρ ⎛ ′ − ⎞ ∂′ ′ ∂ ′

∇ = ∇ _⎜ − _⎟= ∇ _⎜ − _⎟= ∇ = ∇

∂ ∂

⎝ ⎠ _⎝ _⎠

r

n.b. In the last step, note that we have used the fact that since t_r = −t r c with r = −r r′ ≠ fcn t

( )

(because {here} the source and the observer are notmovingrelative to each other – i.e. they are at fixed/stationary points, e.g. in the lab reference frame), therefore ∂ = ∂t_r t and thus: ∂ ∂ = ∂ ∂t_r t.

Since∇ = fcn r

( )

only: tr t t

c ⎛ ⎞ ∇ = ∇ −_⎜ _⎟= ∇ ⎝ ⎠ r 0

1 1 1 _ˆ

c c c

≡

− ∇ = − ∇ = −_r _r _r

where: ∇ = ∇ − =_r r r′ _rˆ and: 2

ˆ 1 1 r r ⎛ ⎞ ∇_{⎜ ⎟}= ∇ = − ′ − ⎝ ⎠ r r r

Thus:

(

)

(

r

)

(

r

)

(

)

(

)

r r r r

, , ˆ 1

ˆ

, tot tot , ,

tot tot tot

r t r t

r t t r t r t

t t c c c

ρ ρ

ρ ′ ∂ ′ ∂ ′ ⎛−∇ ⎞ ρ ′ ⎛ ⎞ ρ ′

∇ = ∇ = _⎜ _⎟= _⎜− _⎟= −

∂ ∂ _⎝ _⎠ ⎝ ⎠

r r _r

where:

(

)

(

r

)

r , , tot tot r t r t t ρ

ρ ′ ≡∂ ′

∂

See Appendices A & B

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Thus: r

( )

(

r

)

(

r

)

1 1 1

, , ,

4 v tot tot

o

V r t ρ r t ρ r t dτ

πε ′

⎡ _′ _′ ⎛ ⎞⎤ _′

∇ = _⎢ ∇ + _{∇⎜ ⎟}_⎥

⎝ ⎠

⎣ ⎦

∫

_r _r

But:

(

r

)

(

r

)

1

ˆ

, ,

tot r t tot r t

c

ρ ′ ρ ′

∇ = − _r and ∇⎛ ⎞_{⎜ ⎟}1 = − ˆ₂

⎝ ⎠ r

r r {from above}

Therefore: r

( )

(

r

)

(

r

)

2

, ˆ 1 ˆ , , 4 tot tot v o r t

V r t r t d

c ρ ρ τ πε ′ ′ − ⎡ _⎛ _⎞⎤ ′ ′ ∇ = _⎢ − _⎜ _⎟_⎥ ⎝ ⎠ ⎣ ⎦

∫

r r

If we now take the divergence of ∇V r tr

( )

, , i.e. the Laplacian of V r tr

( )

, :

( )

(

( )

)

(

)

(

)

(

)

(

)

r 2

r r r 2

r r , ˆ 1 ˆ , , , 4 ˆ ˆ 1 1 , , 4 tot tot v o tot tot v o r t

V r t V r t r t d

c

r t r t

c ρ ρ τ πε ρ ρ πε ′ ′ ′ ⎡ _′ ⎛ ⎞⎤ _′ ∇ = ∇ ∇ = ∇ −_⎢ − _⎜ _⎟_⎥ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎧ ⎛ ⎞ _′ _′ ⎛ ⎞ = _⎨− _⎢_{⎜ ⎟}∇ + ∇ _{⎜ ⎟}_⎥ ⎩ ⎣⎝ ⎠ ⎝ ⎠⎦

∫

i i i i r r r r r r r r

(

r

)

(

)

2 2

ˆ ˆ

−⎡_⎢_⎜⎛ _⎟⎞∇ρ_tot r t′, +ρ_tot r t′, _r ∇_⎜⎛ ⎞_⎟_⎥⎤_⎬⎫dτ′

⎝ ⎠ ⎝ ⎠

⎣ i i ⎦⎭

r r

Since: _tot

(

r t, _r

)

1 _tot

(

r t, _r

)

ˆ

c

ρ ′ ρ ′

∇ = − _r {from above}

Then:

(

r

)

(

r

)

(

r

)

1 1 _ˆ

, , , , ,

tot tot tot tot tot

r r

r t r t r t r t r t

c c c c

ρ ′ ρ ⎛ ′ ⎞ ρ ⎛ ′ − ⎞′ ρ ′ ρ ′

∇ = ∇ _⎜ − _⎟= ∇ _⎜ − _⎟= − ∇ = −

⎝ ⎠ _⎝ _⎠

r _r _r

where:

(

)

(

)

(

)

2

r r

r 2

, ,

, tot tot

tot

r t r t

r t

t t

ρ ρ

ρ ′ ≡∂ ′ =∂ ′

∂ ∂ and: ∇ =r rˆ

and: ∇ _{⎜ ⎟}⎛ ⎞ˆ = ∇⎛_⎜r rˆ ˆ− ′⎞_⎟= 1₂

⎝ ⎠ ⎝ ⎠

i r

r r r and:

( )

3 2 ˆ 4πδ ⎛ ⎞ ∇ _⎜ _⎟= ⎝ ⎠

i r _r

r

( )

3

δ _r = 3-D delta fcn for _r = −r r′

Thus:

( )

(

)

(

) ( )

(

)

( )

(

) ( )

( ) r r 2 3

r 2 r

2 r 3 2 2 , , , 1 1

, 4 ,

4

,

1 1 1

, 4 tot tot tot v o tot tot r v v o o r t V r t

r t

V r t r t d

c

r t

d r t d

c t

ρ ρ

πρ δ τ

πε

ρ

τ ρ δ τ

πε ε ′ ′ ′ = ⎡ ⎛ ′ ⎞ ⎤ ′ ′ ∇ = _⎢+ _⎜ _⎟− _⎥ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ ′ ⎡ ⎤ ∂ _′ _′ _′ = _⎢ _⎥− ∂ ⎣ ⎦

∫

r r r r

Or:

( )

2 r

2 2

r r 2 2

,

1 1

, , _tot ,

o

V r t

V r t V r t r t

c t ε ρ

∂

= ∇ − = −

∂

Thus, we see that the retarded scalar potential V r t_r

( )

, does indeed satisfy the inhomogeneous wave equation / the 4-D Poisson’s equation!

Using the chain rule: See Appendix D See Appendix C See Appendix A

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The same methodology can be carried out for the retarded vector potential A r t_r

( )

, with the same results {please work through this yourselves !!!}, such that:

( )

2 r

( )

2 2

r r 2 2

,

1 1

, , _tot ,

o

V r t

V r t V r t r t

c t ε ρ

∂

= ∇ − = −

∂

( )

2 r

( )

2 2

r r 2 2

, 1

, , A r t _{o tot} ,

A r t A r t J r t

c t μ

∂

= ∇ − = −

∂

where: r

( )

(

r

)

, 1

, 4

tot v o

r t

V r t ρ dτ

πε ′

′

=

_∫

r

( )

(

r

)

r

, ,

4

tot o

v

J r t

A r t μ dτ

π ′

′

=

_∫

r

Note that because the D’Alembertian operator 2 2 2

2 2

1

c t

∂ ≡ ∇ −

∂

explicitly involves the second

derivative with respect to time, _{∂ ∂}2 _t2₍_i_._e_{. note that it is}

quadratic {not linearly } dependent in

the time variable t ), therefore the D’Alembertian operator does not distinguish past from future!

Thus, there exist equally mathematically-acceptable, but physically unacceptable , acausal

solutions (i.e. ones which violatecausality), known as the so-called advancedpotentials

(n.b. the above proof(s) are also valid for the advancedpotential solutions) where: AdvancedTime: t_a = +t r c with ta > t and thus: t t= −a r c.

and: a

( )

(

a

)

, 1

, 4

tot v o

r t

V r t ρ dτ

πε ′

′

=

∫

r

( )

(

a

)

a

, ,

4

tot o

v

J r t

A r t μ dτ

π ′

′

=

∫

r

with:

( )

2 a

2 2

a a 2 2

,

1 1

, , _tot ,

o

V r t

V r t V r t r t

c t ε ρ

∂

= ∇ − = −

∂

( )

2 a

( )

2 2

a a 2 2

, 1

, , A r t _{o tot} ,

A r t A r t J r t

c t μ

∂

= ∇ − = −

∂

The advanced potentials are entirely consistent with Maxwell’s Equations, but violate

causality – because they predict potentials now (at time t) that depend on the charge and current

distributions at a future time t_a = +t r c ⇐ We do not observe such things in our universe!

{n.b. this has not stopped physicists from seriously looking for such things as tachyons, etc.} • In our universe, direct/empirical observation tells us that electromagnetic influences /

changes / disturbances propagate with time going forward, not going backward in time

– i.e. the universe that we live in behaves causally.

Retarded potentials associated with

the retarded time: t_r = −t r c.

Advanced potentials are associated

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• The macroscopic theory of electrodynamics must be manifestlytime-reversalinvariant

(i.e. even under the operation of time reversal, t→−t) because at the microscopic/ elementaryparticlephysics level, the electromagnetic interaction manifestly obeys

time-reversal invariance. This is not a trivial point, because e.g. the microscopic weak interaction violates time-reversal invariance in certain situations!! MacroscopicT-violating effects can

also manifest themselves here too (e.g. weak interaction induced regeneration of 0

S

K in a 0

L

K

beam passing through matter; the so-called Bvs. A phase of superfluid LHe3

{where ∃ a macroscopic B-field vs. a macroscopic E-field, respectively})….

Griffiths Example 10.2:

An infinitely long straight wire carries a time-dependent current I(t) = 0 for t ≤ 0, I(t) = Io for t > 0.

I(t)

Io n.b. t = 0 is the time atthewire

⇒ thus: t = 0 is tr = 0 !!!

0 t

Find / determine the resulting E and B fields at an observation point P r t

( )

, a radial distance ρ

from the axis of the wire. We choose the current-carrying long wire to lie along the ˆz-axis as shown in the figure below:

n.b. We assume that the ∞-long line current is {always} electrically neutral, hence the retarded

scalar potential V_r

( )

ρ,t =0 everywhere ∀

( )

ρ,t .

The retarded vector potential A_r

( )

ρ,t is: _r

( )

,

( )

r ˆ

4 z o

z

I t

A ρ t μ dz z

π =+∞ =−∞

=

∫

r where

2 _z2

ρ

= +

r

and t t= +_r _r c where tr = 0 is the time that the current is switched on.

ˆ

x

ˆ

z

ˆ

y

Field/Observation Point P r t

( )

, Source

Point

(

, r

)

S r t′

( )

r t

r =ρ

( )

r r t

r z

′ ′ =

2 2

r r′ ρ z

= = − = +

r r

( )

r

r t r t′

= −

r