8 Conservation Laws 8 1 Charge and Energy

8 Conservation Laws 8. 1 Charge and Energy 8. 1. 1 The Continuity Equation 8. 1. 2 Poynting's Theorem 8. 2 Momentum 8. 2. 1 Newton's Third Law in Electrodynamics 8. 2. 2 Maxwell's Stress Tensor 8. 2. 3 Conservation of Momentum 8. 2. 4 Angular Momentum 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 1

8. 1 Charge and Energy 8. 1. 1 The Continuity Equation Formula Conservation of charge: If the total charge in some volume changes, then exactly that amount of charge must have passed in or out through the surface. The current flowing out through the boundary �� The above is true for any volume, hence The continuity equation 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 2

8. 1 Charge and Energy 8. 1. 1 The Continuity Equation Derivation from Maxwell’s equations Continuity equation can be derived from the Maxwell’s equations Gauss’s law Faraday’s law Ampere’s law with Maxwell's correction 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 3

8. 1 Charge and Energy 8. 1. 1 The Continuity Equation Next step The continuity equation We want to construct the corresponding equations for conservation of energy and conservation of momentum. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 4

Energy stored in electromagnetic fields 8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem The work necessary to assemble a static charge distribution (against the Coulomb repulsion of like charges) The work required to get currents going (against the back emf) The total energy stored in electromagnetic fields is We will derive this equation more general 1 y in the context of the energy conservation law for electrodynamics. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 5

8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem Power per unit volume The work done per unit time, per unit volume (power delivered per unit volume) 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 6

8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem Power in terms of the fields Ampere’s law Faraday’s law Poynting's theorem; The "work-energy theorem" of electrodynamics. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 7

8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem Poynting's theorem The total energy stored in the fields The work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, less the energy that flowed out through the surface. The energy per unit time, per unit area, transported by the fields (the energy flux density) The Poynting's theorem 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 8

8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem Differential version of Poynting's theorem The differential version of Poynting's theorem The continuity equation 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 9

8. 1 Charge and Energy 8. 1. 2 Poynting's Theorem Example We want to find Assuming the electric field is uniform, the electric field is parallel to the wire The magnetic field is “circumferential”; at the surface Since E and B are independent of time. The Poynting vector points radially inward. Its magnitude is The energy per unit time passing in through the surface of the wire is The energy per unit time delivered to the wire is 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 10

8. 2 Momentum 8. 2. 1 Newton's Third Law in Electrodynamics Imagine a point charge q traveling along the x axis at a constant speed v. Because it is moving, its electric field is not given by Coulomb's law. Nevertheless, E still points radially outward from the instantaneous position of the charge. 05 Feb 2017 A moving point charge does not constitute a steady current, its magnetic field is not given by the Biot-Savart law. Nevertheless, it's a fact that B still circles around the axis in a manner suggested by the right-hand rule. Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 11

Forces on two moving charges 8. 2 Momentum 8. 2. 1 Newton's Third Law in Electrodynamics In electrostatics and magnetostatics the third law holds, but in electrodynamics it does not. Conservation of momentum rests on the cancellation of internal forces, which follows from the third law. Momentum conservation is rescued in electrodynamics by the realization that the fields themselves carry momentum. Whatever momentum is lost to the particles is gained by the fields. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 12

8. 2 Momentum Force per unit volume 8. 2. 2 Maxwell's Stress Tensor The force per unit volume Gauss’s law In terms of fields alone 05 Feb 2017 Ampere’s law Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 13

8. 2 Momentum Maxwell stress tensor 8. 2. 2 Maxwell's Stress Tensor The force per unit volume: Maxwell stress tensor: is a vector The force per unit volume: 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 14

8. 2 Momentum Total force 8. 2. 2 Maxwell's Stress Tensor The force per unit volume: In the static case, the second term drops out, and the electromagnetic force on the charge configuration can be expressed entirely in terms of the stress tensor at the boundary. Force direction 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws surface orientation 15

8. 2 Momentum Example – method 1 8. 2. 2 Maxwell's Stress Tensor From symmetry, the net force is in the z-direction 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 16

8. 2 Momentum Example – method 2 8. 2. 2 Maxwell's Stress Tensor For the bowl, The net force is in the z-direction, so it suffices to calculate The force on the "bowl" is 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 17

8. 2 Momentum Example – method 2 8. 2. 2 Maxwell's Stress Tensor The force on the "bowl" is For the disk, The net force is in the z-direction, so it suffices to calculate The force on the “disk" is The net force on the northern hemisphere is 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 18

8. 2 Momentum Example – method 3 8. 2. 2 Maxwell's Stress Tensor For the disk, For the outer portion, Outer portion y Disk The force on the “outer portion" is The net force on the northern hemisphere is 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws x 19

8. 2 Momentum Integral form 8. 2. 3 Conservation of Momentum According to Newton's second law, the force on an object is equal to the rate of change of its momentum Rate of change of the momentum stored in the electromagnetic fields themselves 05 Feb 2017 The momentum per unit time flowing in through the surface Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 20

8. 2 Momentum Differential form 8. 2. 3 Conservation of Momentum The continuity equation 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 21

8. 2 Momentum Example 8. 2. 3 Conservation of Momentum The momentum in the fields The cable is not moving, and the fields are static. If the center of mass of a localized system is at rest, its total momentum must be zero. There is "hidden" mechanical momentum associated with the flow of current, and this exactly cancels the momentum in the fields. The hidden momentum is actually a relativistic effect and It will be discussed in Chapter 12. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 22

8. 2 Momentum Example 8. 2. 3 Conservation of Momentum Suppose we turn up the resistance, so the current decreases. The changing magnetic field will induce an electric field This is precisely the momentum originally stored in the fields. The cable will not recoil because an equal and opposite impulse is delivered by the simultaneous disappearance of the hidden momentum. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws is similar to a magnetic field of a long solenoid. 23

8. 2 Momentum Angular momentum 8. 2. 4 Angular Momentum The density of angular momentum: 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 24

8. 2 Momentum Example 8. 2. 4 Angular Momentum Before the current was switched off There was an electric field in the region between the cylinders There was a magnetic field inside the solenoid The momentum density was constant The angular momentum density was The total angular momentum in the fields volume 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 25

8. 2 Momentum Example 8. 2. 4 Angular Momentum The total angular momentum in the fields When the current is turned off The changing magnetic field induces a circumferential electric field, given by Faraday's law The torque on the outer cylinder is The outer cylinder picks up an angular momentum The torque on the inner cylinder is The inner cylinder picks up an angular momentum The angular momentum lost by the fields is precisely equal to the angular momentum gained by the cylinders. The total angular momentum (fields plus matter) is conserved. 05 Feb 2017 Aljalal-phys 306 -162 -Ch 08 -Conservation Laws 26