hw408 - Classical Electrodynamics Physics 6561 Homework 4....

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Classical Electrodynamics Physics 6561 Homework 4. Due: In class on Tuesday, 9/30 1. Angular momentum of EM field 12.19 from Jackson: Source-free EM fields exist in a localized region of space. Consider the various conserva- tion laws that are contained in the integral of α M αβγ = 0 over all of space, where M is as defined in class T αβ x γ - T αγ x β . (a) Show that when β,γ are both spatial indices conservation of the total field angular momentum follows. (b) Show that when β = 0 the conservation law is d ~ X dt = c 2 ~ P EM E EM , where ~ X is the center of mass of the electromagnetic field defined by ~ X Z ud 3 x = Z ~xud 3 x, where u is the electromagnetic energy density and ~ P EM and E EM are the total energy and momentum of the fields. 2. Microscopic expression for magnetization Problem 6.7 from Jackson The microscopic current ~ j ( ~x,t ) is written as ~ j ( ~x,t ) = X j q j ~v j δ ( ~x - ~x j ( t )) . Just as for the charge density, this current can be broken up into a “free” (conduction)
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hw408 - Classical Electrodynamics Physics 6561 Homework 4....

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