1
Monday, October 10: Potentials
I’ve already remarked on the similarity between the electrostatic force be
tween two point charges,
F
=
q
1
q
2
r
2
,
(1)
and the gravitational force between two point masses,
F
=

G
m
1
m
2
r
2
.
(2)
The similarities between the two suggest that we can profitably describe
electrostatic phenomena using some techniques stolen (excuse me, I mean
“borrowed”) from Newtonian gravitational theory.
For instance, in dealing with gravity, it is useful to use the scalar field
known as the gravitational potential. The gravitational potential Φ
grav
(
~
r
) is
defined such that the force
~
F
grav
acting on a unit mass is
~
F
grav
=

~
∇
Φ
grav
.
(3)
For an arbitrary mass density
ρ
(
~
r
), the gravitational potential is
Φ
grav
(
~
r
) =

G
Z
ρ
(
~
r
0
)

~
r
0

~
r

d
3
r
0
.
(4)
Thus, you find the gravitational potential by smoothing the mass density
field with a very broad (1
/r
) smoothing function. Minima in the potential
correspond to dense regions in the universe.
The gravitational potential itself cannot be directly measured. We can
detect its gradient at a given point by determining the force on a unit mass.
We can detect the difference in potential between two points by determining
the energy needed to lift a particle from the point of lower potential to the
point of higher potential. However, you can always add an arbitrary constant
to the gravitational potential,
Φ
grav
→
Φ
grav
+
C ,
(5)
without changing any of the physics. For a mass distribution of finite extent,
it is conventional to choose
C
so that Φ
grav
→
0 as
~
r
→ ∞
; however, this is
just a convention. The concept of the “potential” has proved to be very useful
for Newtonian gravity (largely because it’s easier to deal with the scalar field
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Φ
grav
than with the vector field
~
F
). It would be doubly useful if we could
also express electromagnetic phenomena in terms of potentials.
Alas, there are a few complicating factors. It is true that the electrostatic
force between two point charges is directly analogous to the gravitational
force between two point masses. However, if the point charges are moving
relative to each other, there is a magnetic force between them that has no
analogy in Newtonian gravity. Thus, we need an extension of potential theory
that will allow us to deal with magnetism.
There’s another complicating
factor, as well. Newtonian gravity only deals with slowly moving particles
(
v
¿
c
). In studying electromagnetism, we must be prepared to deal with
highly relativistic charged particles.
With these caveats in mind, let’s see how we can define potentials for
the electric field strength
~
E
(
~
r, t
) and the magnetic flux density
~
B
(
~
r, t
). Let’s
start with the simplest of Maxwell’s equations:
~
∇ ·
~
B
= 0
.
(6)
Because of the vector identity
~
∇ ·
(
~
∇ ×
~
f
) = 0
(7)
for an arbitrary vector function
~
f
, we see that the divergenceless magnetic
flux density can be written in the form
~
B
(
~
r, t
) =
~
∇ ×
~
A
(
~
r, t
)
,
(8)
where
~
A
(
~
r, t
) is the
electromagnetic vector potential
.
1
If we write
~
B
as the curl of
~
A
, we can write another of Maxwell’s equa
tions,
~
∇ ×
~
E
=

1
c
∂
~
B
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '05
 RYDEN
 Magnetic Field, Fundamental physics concepts, magnetic ﬂux density

Click to edit the document details