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 AAJ002 00 0 000t 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 AJL 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 AJ002 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 00t 2 2 V 2 V 00 2 V t 0 0 2 2 2 A AJ AJ0 00t 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 / Art,' 0 d BA 4 V 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 / Art,' 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 v (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 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: The Fields of a Moving Point Charge 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 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.