Chapter 10. Potentials and Fields 10.1 the Potential Formulation

Chapter 10. Potentials and Fields 10.1 The Potential Formulation

10.1.1 Scalar and Vector Potentials

• Vector potential BA  T  B 0

• Electric field for the time-varying case.

BA  EAE 0 tt  t A E V t A  E V   V/m t Due to time-varying current J

Due to charge distribution  10.1.1 Scalar and Vector Potentials

A EBA V  &    t     E 2V A  0 t 0 2 E 2 A V BJ   AAJ002  00  0 000t tt

2 22 00 2 : d'Alembertian (4-dimensional operator, good for special relativity) t V L A 00 : Gauge (to make E & B fields unchanged under transformation) t

2 L  V  t 0 2 AJL 0 10.1.3 Coulomb Gauge and Lorentz Gauge

  2V  A t 0

2A V V 2AAJ    002  00 0 L A 00 tt t

Coulomb Gauge:  A 0  2V   0

2 2  A V AJ  002  0 00  tt

 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

  2V  A t 0

2A V V 2AAJ    002  00 0 L A 00 tt t

V Lorentz Gauge: A  L  0 00t 2 2  V  2   V 00 2 V  t 0 0 2 2 2  A AJ   AJ0 00t 2 0

 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

10.2.2 Jefimenko's Equations (Retarded E and B fields)

Time-varying charges and currents generate retarded scalar potential, retarded vector potential.

R  ct() t  Potentials at a distance r from the source at time t r depend on the values of  and J at an earlier time (t - r/u) R d '

 Retarded in time ttRutRcrp/  / in vacuum  rt',  r r' 1  rt', r  Vrt,' d R rr'  A 4 V  R E V  0 t  J rt', r ttRcr  /  Art,' 0 d BA V  4  R 

Note, since both the r and tr have r dependence, grad(V) to get E is not simple! Jefimenko's Equations (Retarded E and B fields)

 rt', 1   r  A R  ct() tr Vrt,' d R rr'  E V  4 V  R  0 t R d '  J rt', r ttRcr  /  Art,' 0 d BA 4 V  R   rt',  r  r'

   rr(',)rt   t r  t rr t ttr

time-dependent generalization of Coulomb's law

Similarly,

time-dependent generalization of the Biot Savart law Jefimenko's Equations (Retarded E and B fields)

R  ct() tr R d ' r' r

 rt',  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 tr is negative, so I(tr) = 0); thus

The electric field is

The magnetic field is 10.3 (Moving) Point Charges

10.3.1 Lienard-Wiechert Potentials  Potentials of moving point charges

v 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)

Lienard-Wiechert potentials for a moving point charge

J  v Lienard-Wiechert potentials for a moving point charge

(Example 10.3) Find the potentials of a point charge moving with constant velocity.

 Consider a point charge q that is moving on a specified trajectory

For convenience, let's say the particle passes through the origin at time t = 0, so

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

(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

Let’s begin with the gradient of V: The Fields of a Moving Point Charge

v The Fields of a Moving Point Charge

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

w(t) R

(Example 10.4) Calculate the Electric and magnetic fields of a point charge moving with constant velocity.

 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

 When v2 « c2, they reduce to

 Coulomb's law, Biot-Savart law for a point charge