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 0 extends to  with the lower electrode at potential  and the upper one grounded. This upper electrode extends to the right to the plane , where it is bent downward to  and inward to the plane  along the surface . Region (a) is free space while region (b) to the left of the plane  has uniform permittivity  and conductivity . The applied voltage  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  , what is the potential in region (a)?

(b) Now,  is to be found for . Example 6.6.3 illustrates the approach that can be used. Show that in the limit  becomes the result of part (a).

(c) In the special case where  = , 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

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