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160 Chapter 4 c2 B1 = 1 j1 + 2E1 . t o t t2 Combining Eq. (4.6-9) with Eq. (4.6-8) results in ( E1)– 2E1=– 1 j1–1 2E1. oc2 t c2 t2 Assuming that the microwaves are plane waves that vary as E = E ei(kx t) j= j ei(kx– t), (4.6-9) (4.6-10) (4.6-11) (4.6-12) where k = 2 / and is the cyclic frequency 2 f , then Eq. (4.6-10) be- comes i 2 –k(k E1)+k2E1= oc2 j1+ c2 E1. (4.6-13) The electromagnetic waves are transverse waves, so k E1 = 0 and Eq. (4.6-13) becomes –i –c k E = j . (4.6-14) (222)11 o Since these waves are in the microwave frequency range, the ions are too massive to move on these fast time scales and the perturbed current j1 can come only from electron motion. The perturbed electron current density in a plasma is j1 = –neeve1, (4.6-15) where ne is the plasma density and ve1 is the perturbed electron velocity. If the applied magnetic field is zero or the perturbed electric field is parallel to the applied magnetic field (so called “O-waves”), the equation of motion for the perturbed electron motion is m ve1 =–eE1. (4.6-16) tPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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