The cross-section of a two-dimensional system is shown in Fig. P7.9.9. The parallel plate…

The cross-section of a two-dimensional system is shown in Fig. P7.9.9. The parallel plate capacitor to the left of the plane afccfd84 1d7b 4e7d 8a9f b5bae421e1ae0 extends to 0b2ae9b5 44ec 4a8b 9224 8e90f118878c with the lower electrode at potential a1745bde 7014 4356 bc45 f71c811df108 and the upper one grounded. This upper electrode extends to the right to the plane 4ad4adc4 5e06 43a0 ac64 b287c61ba840, where it is bent downward to 0d24308a c76b 4bb0 9cf7 c2d7eeeff8eb and inward to the plane 5c456d24 3f03 4cd6 96a2 923599552f8e along the surface 0d24308a c76b 4bb0 9cf7 c2d7eeeff8eb. Region (a) is free space while region (b) to the left of the plane 5c456d24 3f03 4cd6 96a2 923599552f8e has uniform permittivity 093fe689 06f1 4a1c b9eb 1bde59c633bc and conductivity 78042420 11e2 4128 9ade 79f20cacfd5e. The applied voltage a1745bde 7014 4356 bc45 f71c811df108 is a step function of magnitude Vo.

(a) The voltage has been on for a long-time. What are the field and potential distributions in region (b)? Having determined f80049bd 5eda 4367 8417 a5e71a1a0f2d , what is the potential in region (a)?

(b) Now, 29f0e75f 570b 43ed ba29 590f0434f31b is to be found for c3a95885 4ccc 472d 8fe5 00871a2f1016. Example 6.6.3 illustrates the approach that can be used. Show that in the limit 6220af88 b15f 4c04 8954 bb5f1551979f becomes the result of part (a).

(c) In the special case where 093fe689 06f1 4a1c b9eb 1bde59c633bc = 2d833a3b 3455 45ca a567 0f8a5187c8b1, sketch the evolution of the field from the time just after the voltage is applied to the long-time limit of part (a).

Fig. P7.9.9

4fddaf76 3961 4272 a0db 6c9236249e5b   

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