Chapter 10. Potentials and Fields
10.1.1 Scalar and Vector Potentials
• Vector potential
• Electric field for the time-varying case.
T
B A
0
V/m
t t t
t V
V t
B A
E A E
E A
E A
B 0
Due to charge distribution
Due to time-varying current J
10.1 The Potential Formulation
10.1.1 Scalar and Vector Potentials
0
E
2 2
0 0 2 0 0 0
V
t t
A A A J
&
V t
E A B A
2
0
V t
A
0 0 0
t
B J E
2
2 2
0 0 2
: d'Alembertian
t
0 0
L V
t
A
2
0 2
0
V L
t L
A J
(4-dimensional operator, good for special relativity)
: Gauge (to make E & B fields unchanged under transformation)
10.1.3 Coulomb Gauge and Lorentz Gauge
A 0
0 0
L V
t
A
Coulomb Gauge:
2
0
V
2
0
V t
A
2 2
0 0 2 0 0 0
V
t t
A A A J
2 2
0 0 2 0 0 0
V
t t
A A J
Advantage is that the scalar potential is particularly simple to calculate;
Disadvantage is that A is particularly difficult to calculate.
10.1.3 Coulomb Gauge and Lorentz Gauge
0 0
L V
t
A
2
0
V t
A
2 2
0 0 2 0 0 0
V
t t
A A A J
Lorentz Gauge:
0 0 L 0
V
t
A
2 2
0 0 2
0
V V
t
2 2
0 0 2 0
t
A A J
2
0 2
0
V
A J
V and A have the same differential operator of l’Alembertian (4-dim. operator)
Under the Lorentz gauge, the whole of electrodynamics reduces to the problem
of solving the inhomogeneous wave equation for specified sources.
10.2 Continuous Distributions
0 0
1 ',
, '
4
, ', '
4
r V
r V
V r t r t d
R
r t r t d
R
J
A
Time-varying charges and currents generate retarded scalar potential, retarded vector potential.
Potentials at a distance
r
from the source at time tdepend on the values of and J at an earlier time (t -
r
/u) Retarded in time
r t ',
'
d R
' r
tr t R u/ p t R c/ in vacuum r
V t
E A
B A
R r r '
Note, since both the r and t
rhave r dependence, grad(V) to get E is not simple!
t
r t R c /
( r) R c tt
10.2.2 Jefimenko's Equations (Retarded E and B fields)
Jefimenko's Equations (Retarded E and B fields)
0 0
1 ',
, '
4
, ', '
4
r V
r V
V r t r t d
R
r t r t d
R
J
A
V t
E A
B A
R r r '
t
r t R c /
( ', )
r r r r r
r
r t t t t
t t
Similarly,
time-dependent generalization of Coulomb's law
time-dependent generalization of the Biot Savart law
r t ',
'
d R
' r
r
( r) R c tt
r t ',
'
d R
'
r r
(
r) R c t t
Taking the divergence,
The retarded potential also satisfies the inhomogeneous wave equation.
Jefimenko's Equations (Retarded E and B fields)
(Example 10.2)
An infinite straight wire carries the current that is turned on abruptly at t = 0.
Find the resulting electric and magnetic fields.
The retarded vector potential at point P is
For t < s/c, the "news" has not yet reached P, and the potential is zero. For t > s/c, only the segment contributes (outside this range tris negative, so I(tr) = 0); thus
The electric field is
The magnetic field is
Jefimenko's Equations (Retarded E and B fields)
10.3.1 Lienard-Wiechert Potentials Potentials of moving point charges
It might suggest to you that the retarded potential of a point charge is simply
But, if the source is moving,
(Proof: This is a purely geometrical effect)
v
J
Lienard-Wiechert potentials for a moving point charge
v
10.3 (Moving) Point Charges
Lienard-Wiechert potentials for a moving point charge
v
(Example 10.3)
Find the potentials of a point charge moving with constant velocity.
For convenience, let's say the particle passes through the origin at time t = 0, so
Consider a point charge q that is moving on a specified trajectory
The retarded time is determined implicitly by the equation
By squaring:
To fix the sign, consider the limit v = 0: should be (t - r/c);
Lienard-Wiechert potentials for a moving point charge
v
(Example 10.3)
Find the potentials of a point charge moving with constant velocity.
(continued)
Therefore,
10.3.2 The Fields of a Moving Point Charge
v
Let’s begin with the gradient of V:
v
The Fields of a Moving Point Charge
v
The Fields of a Moving Point Charge
v
If the velocity and acceleration are both zero, E reduces to the old electrostatic result:
Now, we can say the Lorentz force exerting on a test charge Q by any configuration of a charge (q):
The entire theory of classical electrodynamics is contained in that equation.
The Fields of a Moving Point Charge
v
(Example 10.4) Calculate the Electric and magnetic fields of a point charge moving with constant velocity.
When v2 « c2, they reduce to
Coulomb's law, Biot-Savart law for a point charge
Because of the sin2 in the denominator, the field of a fast-moving charge is flattened out like a pancake in the direction perpendicular to the motion.
In forward and backward directions E is reduced by a factor (I - v2/c2) relative to the field of a charge at rest;
in the perpendicular direction it is enhanced by a factor
w(t) R