In the region shown in cross-section in Fig. P5.6.4, the charge density is where and are…

In the region b2c9cbff 57f5 451e 9ad4 20ad52e3fd0b shown in cross-section in Fig. P5.6.4, the charge density is

e07b5af2 32e3 446a bf26 85402d4d5b69

where 1c41e0fe e81b 474b 8d0f f265e3c3ebd6 and 5f5c3bf2 4ea1 41ac 9772 6714dd10f3b6 are constants. Electrodes at 519129ed 3e09 4587 a671 b79f50b58d4c constrain the potential there to be 5ac64d67 338f 40dc 9239 47b20c79cc71 cos97ec8a03 3f5d 4ecf bc81 df677f83c3de and 2be02252 50ed 451e ba3f 94e4163251b0 given constants42d084ee f650 4b2a be06 8545900bec1c, while an electrode at b0e3e32b 5dd3 4af9 8516 33939447af0a makes ecbf2cce 5c0d 42f5 9ddf c73b18bf4b98.

Fig. P5.6.4

be00958e c7a8 4dae ab37 f16d9e9391b3

(a) Find a particular solution that satisfies Poisson’s equation everywhere between the electrodes.

(b) What boundary conditions must the homogeneous solution satisfy at c92bbb96 b9f1 429d 897e 1de14b2a5aa9and b0e3e32b 5dd3 4af9 8516 33939447af0a?

(c) Find ab99c04f 2018 42a0 8db8 474bad54d366 in the region b2c9cbff 57f5 451e 9ad4 20ad52e3fd0b.

(d) The force density (force per unit volume) acting on the charge is 41e75d0f 2603 4bfc ba6a 577a7957ec33. Find the total force 24d57d17 8dab 444e 8844 5649a55ed5b4 acting on a section of the charge spanning the system from b0e3e32b 5dd3 4af9 8516 33939447af0a to 519129ed 3e09 4587 a671 b79f50b58d4c, of unit length in the 86286c73 5d8c 46ef 9bde 2f8222e6913a direction and of length 571c3242 427c 4fb2 8bbb acec02480e30 in the 47b48071 20ed 48c7 9591 99533835eba5 direction.

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