In the region between the boundaries at in Fig. P5.6.3, the charge density is Fig. P5.6.3 where…

In the region between the boundaries at a1a9c39b adfe 410c 9b3a 6a8cb5e51f63in Fig. P5.6.3, the charge density is

4c1416bb e0bd 45a6 b4ee 196389d58bd1

Fig. P5.6.3

c2f7bf66 7edd 4508 9097 0e4b40db3f6a

where f3c00fbc d8f0 4fa1 b5d8 591a1c498837 and a591ed2f 4b1f 44dc ae8d 94e1d91963f1 are given constants. Electrodes at 47d5771b 6b3c 44d3 bf3c 7cd88363cd86 constrain the tangential electric field there to be

701b4cfa 1635 410a a090 d27ab21c90fe

The charge density might represent a traveling wave of space charge on a modulated particle beam, and the walls represent the traveling-wave structure which interacts with the beam. Thus, in a practical device, such as a traveling-wave amplifier designed to convert the kinetic energy of the moving charge to ac electrical energy available at the electrodes, the charge and potential distributions move to the right with the same velocity. This does not concern us, because we consider the interaction at one instant in time.

(a) Show that a particular solution is

528d7b19 3560 4964 a3a0 ede0d6f35656

(b) Show that the total potential is the sum of this solution and that solution to Laplace’s equation that makes the total solution satisfy the boundary conditions.

2accbb6d d554 4fe5 a174 7fd1bdc9bd43

(c) The force density (force per unit volume) acting on the charge is dccc404f 42cf 4c92 8374 6ba3662cecab. Show that the force 13664595 429d 492b a9d4 7c9bcee73c89 acting on a section of the charge of length in the x direction 06ab3310 32ed 4c2b 9372 37586deb1285 spanning the region 97595384 ce6f 4e31 a517 6629250e09c2 and unit length in the 329c5cc1 2f67 4cf1 bddc 557fb4e12419 direction is

c0d2337b 1e5c 4b31 8561 3a01b6f514fe

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