Chapter 8 Conservation Laws 8.1 Charge and Energy 8.2 Momentum

8.1 Charge and Energy The continuity equation: conservation of charge Poynting’s theorem and conservation of energy

8.1.1 The continuity equation: conservation of charge Global conservation of charge: the total charge in the universe is constant. Local conservation of charge: the change of the total charge in some volume exactly equals to the amount of charge passing in or out through the surface.

8.1.2 Poynting’s theorem and conservation of energy The work done on charge The rate at which work is done on all charges in a volume Ampere-Maxwell law

8.1.2 (2)

8.1.2 (3) Poynting theorem : U em, the total energy stored in electromagnetic field. Poynting vector energy flux density

The work done on the charges by the EM force is equal to the decrease in energy stored in the field, less the energy flowed out through the surface (4) Poynting theorem in differential form: continuity eq.

8.1.2 (5) answer: Ex 8.1

8.2 Momentum Newton’s third law in electrodynamics Maxwell’s stress tensor Conservation of momentum

8.2.1 Newton’s third law in electrodynamics Newton’s third law in trouble ? The fields themselves carry momentum.

8.2.2 Maxwell’s stress tensor The total electromagnetic force on the charge in volume The force per unit volume

8.2.2 (2)

Maxwell stress tensor, e.g., shear pressure (3)

The total force on the charges in (4)

Ex.8.2 net force on the northern hemisphere of a uniformly charged solid sphere? [Problem 2.43] solution:The net force is on For the bowl (5) R Q

8.2.2 (6)

For the equatorial disk (7) y x Inside sphere

For the r>R and z=0 area, (8) the net force >

8.2.3 Conservation of momentum, the momentum stored in the electromagnetic fields. The density of momentum in the fields conservation of momentum is the electromagnetic stress (force per unit area) is the momentum flux density

Ex. 8.3 What is the electromagnetic momentum stored in the fields? solution: Power transported from the battery to the resistor (2)

The momentum in the fields (There is a hidden mechanic momentum to cancel this EM momentum to maintain the motionless cable and the static fields. This hidden momentum is due to a relativistic effect as discussed in chapter 12,Ex ) (3)