Intermission Page 343, Griffith

Intermission Page 343, Griffith

Chapter 8. Conservation Laws (Page 346, Griffith) Lecture : Electromagnetic Power Flow

Flow of Electromagnetic Power Electromagnetic waves transport throughout space the energy and momentum arising from a set of charges and currents (the sources). If the electromagnetic waves interact with another set of charges and currents in a receiver, information (energy) can be delivered from the sources to another location in space. The energy and momentum exchange between waves and charges and currents is described by the Lorentz force equation. Poynting s theorem concerns the conservation of energy for a given volume in space. Poynting s theorem is a consequence of Maxwell s equations.

Poynting s Theorem is the Work-energy theorem Conservation of Energy 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. dw dt d 1 2 1 2 1 E J dv E B dv E B ds dt 2 2 V V S Work done by the EM field Total energy stored in the EM field Energy flowed out through the surface Poynting s Theorem dw dt du 1 S EB em S dt S ds : Poynting vetor

Proof: Total energy stored in Electromagnetic fields is 1 1 2 2 Uem E B dv 2 2 V Suppose we have some charge and current configuration, produces fields E and B at time t. Work done on a charge q in the interval dt is dw F dl q( E vb) vdt qe vdt E J dw dt Now, q d and v J. ( d) Ev ( Ev) d EJ d V V V is the work done per unit time, per unit volume (or, power/volume) Using the Ampere-Maxwell law, EJ

E J dw dt d 1 2 1 2 1 E J dv E B dv E B ds dt 2 2 V V S Note that 1 S EB

Poynting Vector in Time Domain We define a new vector called the (instantaneous) Poynting vector as S E H ( W/m 2 ) The Poynting vector has the same direction as the direction of propagation. The Poynting vector at a point is equivalent to the power density of the wave at that point.

8.2 Momentum of EM fields (Griffiths) 8.2.1 Newton's Third Law in Electrodynamics Imagine a point charge q traveling In 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. Generalized Coulomb field (velocity field) If the velocity and acceleration are both zero, Radiation field (acceleration field) Since, moreover, 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.

Newton's Third Law in Electrodynamics Now suppose two identical charges, q 1 and q 2, moving with a same speed along the x and y axes. At an instantaneous time t, this may be regarded as an electromagnetostatic case. F F F eq, eq, 2 1 F mq, mq, 2 1 The electromagnetic force of q 1 on q 2 is equal, but not opposite to the force of q 1 and q 2. violation of Newton s third law! In electrostatics and magnetostatics the third law holds, but in electrodynamics it does not. Is it true? The third law must be hold ALL THE TIME! We know that the proof of momentum conservation rest on the cancellation of internal forces. Therefore, let s prove momentum conservation in electrodynamics! The fields themselves carry momentum. Only when the field momentum is added to the mechanical momentum of the charges, momentum conservation (or, the third law) is restored.

8.2.2 Maxwell s stress tensor The fields themselves carry momentum. To find out the field momentum, let s calculate the total EM force on the charges in volume V in terms of Poynting vector S and stress tensor T: The force per unit volume is It can be simplified by introducing the Maxwell stress tensor T

Force density in a symmetry form of E and B. Maxwell s stress tensor: T ij The total EM force on the charges in volume V is * In static case *T ij The force per unit area (or stress) on the surface

Example 8.2 (p. 353) Find the net force on the northern hemisphere exerted by the southern hemisphere. Q For the bowl, In static case The net force on the bowl is obviously in the z-direction, Meanwhile, for the equatorial disk, Combining them, the net force on the northern hemisphere is

8.2.3 Conservation of momentum for EM fields According to Newton's second law, the force on an object is equal to the rate of change of its momentum: When the object has an electric charge distribution in V, The first integral represents momentum stored in the electromagnetic fields: The second integral represents momentum per unit time flowing in through the surface: In differential form, : densities of mechanical momentum : densities of field momentum If V is all of space, then no momentum flows in and out, P mech P em constant Solve Problems. 8.5 and 8.6

Example 8.3 (p. 356) What is the electromagnetic momentum stored in the fields? uniform charge per unit length outer conductor (radius b) inner conductor (radius a) The momentum in the fields is:

8.2.4 Angular Momentum Electromagnetic fields carry energy density and momentum density of The electromagnetic fields also carry density of angular momentum of [Example 8.4] Two long cylindrical shells are coaxial with a solenoid carrying current I. When the current in the solenoid is gradually reduced, the cylinders begin to rotate. Where does the angular momentum comes from? = constant over the volume of Total angular momentum in the fields: current I

[Example 8.4] continued. When the current is turned off, the changing magnetic field induces a circumferential electric field, given by Faraday's law: Torque on the outer cylinder: Angular momentum of the outer cylinder: Torque on the inner cylinder: Angular momentum of the outer cylinder: Total angular momentum of the inner and outer cylinders: Which is the same as the angular momentum of the field: Angular momentum lost by fields is precisely equal to angular momentum gained by the cylinders, the total angular momentum (fields plus matter) is conserved