# 10 Potentials and Fields 10 1 The Potential

- Slides: 28

10 Potentials and Fields 10. 1 The Potential Formulation 10. 1. 1 Scalar and Vector Potentials. 10. 1. 2. Gauge Transformations. 10. 1. 3. Coulomb Gauge and Lorentz Gauge. 10. 2 Continuous Distributions 10. 2. 1. Retarded Potentials. 10. 2. 2. Jefimenko's Equations. 10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials. 10. 3. 2 The Fields of a Moving Point Charge. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 1

10. 1 The Potential Formulation 10. 1. 1 Scalar and Vector Potentials Definitions We seek the general solution to Maxwell's equations. contain all the information in Maxwell's equations 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 2

10. 1 The Potential Formulation 10. 1. 1 Scalar and Vector Potentials Example Find the charge and current distributions that would give rise to the potentials 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 3

10. 1 The Potential Formulation 10. 1. 1 Scalar and Vector Potentials Example Find the charge and current distributions that would give rise to the potentials Boundary conditions 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 4

10. 1 The Potential Formulation 10. 1. 2 Gauge Transformations Formula By how much can they differ? The term in parentheses is therefore independent of position (it could, however, depend on time); call it k(t): There are many famous gauges in the literature; We'll see the two most popular ones. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 5

10. 1 The Potential Formulation 10. 1. 3 Coulomb Gauge and Lorentz Gauge The Coulomb gauge The Coulomb Gauge: 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 6

10. 1 The Potential Formulation 10. 1. 3 Coulomb Gauge and Lorentz Gauge The Lorentz gauge The Lorentz Gauge: From now on we shall use the Lorentz gauge exclusively, and the whole of electrodynamics reduces to the problem of solving the inhomogeneous wave equation for specified sources. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 7

10. 2 Continuous Distributions 10. 2. 1 Retarded Potentials Retarded potentials The Lorentz Gauge In the static case Note Guess Because the integrands are evaluated at the retarded time, these are called retarded potentials. To prove them, we must show that they 1 - Satisfy the inhomogeneous wave equation. 2 - Meet the Lorentz condition. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 8

10. 2 Continuous Distributions 10. 2. 1 Retarded Potentials V satisfies the inhomogeneous wave equation Show that satisfies the inhomogeneous wave equation This proof applies to the advanced potentials in which the charge and the current densities are evaluated at the advanced time 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields The advanced potentials violate the principle of causality. 9

10. 2 Continuous Distributions 10. 2. 1 Retarded Potentials Example The wire is electrically neutral. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 10

10. 2 Continuous Distributions 10. 2. 2 Jefimenko's Equations Electric field Given the retarded potentials From slide 9, we calculated the gradient of V The time-dependent generalization of Coulomb's law, to which it reduces in the static case. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 11

10. 2 Continuous Distributions 10. 2. 2 Jefimenko's Equations Magnetic field Given the retarded potentials The time-dependent generalization of the Biot-Savart law, to which it reduces in the static case. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 12

10. 2 Continuous Distributions 10. 2. 2 Jefimenko's Equations Jefimenko's equations In practice Jefimenko's equations are of limited utility, since it is typically easier to calculate the retarded potentials and differentiate them, rather than going directly to the fields. Note They provide strong support for the quasistatic approximation. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 13

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials One retarded point at a time Since no charged particle can travel at the speed of light, it follows that only one retarded point contributes to the potentials, at any given moment. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 14

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials Delta of a function 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 15

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials Potential of a point charge The retarded time is determined implicitly by the equation 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 16

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials Potential of a point charge 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 17

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials Vector potential of a point charge 21 Mar 2017 Liénard-Wiechert Potentials Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 18

10. 3 Point Charges 10. 3. 1 Liénard-Wiechert Potentials Example Find the potentials of a point charge moving with constant velocity. Assume the particle passes through the origin at time t = 0, so that squaring The retarded time But, The minus sign is the correct sign. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 19

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Gradient of V Finding the electric and magnetic fields of a point charge in arbitrary motion, using the Liénard-Wiechert potentials 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 20

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Gradient of V 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 21

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Gradient of V From previous slide. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 22

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Electric field A similar calculation yields 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 23

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Magnetic field From the previous slides 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields The magnetic field of a point charge is always perpendicular to the electric field, and to the vector from the retarded point. 24

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Lorentz force The first term The second term falls off as the inverse square of the distance from the particle. If the velocity and acceleration are both zero, It alone survives and reduces to the old electrostatic result It is responsible for electromagnetic radiation. It is sometimes called the generalized Coulomb field. Because it does not depend on the acceleration, it is also known as the velocity field. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 25

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Example Calculate the electric of a point charge moving with constant velocity. Assume the particle passes through the origin at time t = 0, so that From previous example See next slide. 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 26

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Example 21 Mar 2017 Aljalal-phys 306 -162 -Ch 10: Potentials and Fields 27

10. 3 Point Charges 10. 3. 2 The Fields of a Moving Point Charge Example Calculate the magnetic fields of a point charge moving with constant velocity. Lines of B circle around the charge. 21 Mar 2017 Coulomb's law Aljalal-phys 306 -162 -Ch 10: Potentials and Fields Biot-Savart law for a point charge 28

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