The system shown in cross-section by Fig. P7.5.6a extends to infinity in the and directions. The…

The system shown in cross-section by Fig. P7.5.6a extends to infinity in the 44a9ec95 6727 4b3f 89d6 becf7ff8770a and d582aa60 2ca9 4c48 bf58 17e5b8fff067 directions. The material of uniform conductivity e93475ad 60c4 4ce0 9ed3 cb272cb112c5 to the right is bounded at 68473e18 e24f 4279 82f7 cace32ab231a and dc2bc217 e9c2 409b 8dc7 53f33b36394e by electrodes at zero potential. The material of uniform conductivity 98213ebe 078c 403a a49a fb3a10325058 to the left is bounded in these planes by electrodes each at the potential 222d4d1a e6b0 4f63 b71a 13c6719adfa3. The approach to finding the fields is similar to that used in Example 6.6.3.

(a) What is 97bf1369 68e7 476f 9a80 3345b61ba883 as 3360a704 e32a 4597 b1bd 96ba41085377 and 642fc6a1 69b9 47df b3fa 68155b7bde87 as 181ff24a b4f9 482e b6c6 6335d651616f?

(b) Add to each of these solutions an infinite set such that the boundary conditions are satisfied in the planes 4a1eeb68 fbf8 4663 ae11 69cfa77d8174and dc2bc217 e9c2 409b 8dc7 53f33b36394e and as e843a4c6 5a4c 49ad 8c01 7ad7db421f84.

(c) What two boundary conditions relate 97bf1369 68e7 476f 9a80 3345b61ba883 to 642fc6a1 69b9 47df b3fa 68155b7bde87 in the plane ef24d57f a7d7 4ca8 a224 02a4fc4e93a5?

(d) Use these conditions to determine the coefficients in the infinite series, and hence find 3f3c2e3f b8bc 4a72 a5d7 46052e80f588 throughout the region between the electrodes.

Fig. P7.5.6a

2363f93e 3556 4b58 9ab4 45da5edc09ef

 (e) In the limits d9be9d3f 5033 48b3 b427 4d7eb1440467 (A numerical evaluation of the expressions for 3f3c2e3f b8bc 4a72 a5d7 46052e80f588 is not required.)

(f) Shown in Fig. P7.5.6b is a similar system but with the conductors bounded from above by free space. Repeat the steps (a) through (e) for the fields in the conducting layer.

Fig. P7.5.6b

4091896a 1e5b 4480 b06e d8c01e455f77

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