For a better understanding of the concepts underlying the derivation of the superposition…

For a better understanding of the concepts underlying the derivation of the superposition integral for Poisson’s equation, consider a hypothetical situation where a somewhat different equation is to be solved. The charge density is assumed in part to be a predetermined density 286d75af 96c0 4c88 bc92 ce0f079d6d23, and in part to be induced at a given point dd94138d 12e4 4a30 ad40 275bfa803381 in proportion to the potential itself at that same point. That is,

edee7181 6d20 4ba0 af0f 925eb300a051

(a) Show that the expression to be satisfied by 09bbd205 c820 4604 bbcd 300f9cc8fd60 is then not Poisson’s equation but rather

9792e6ec 8a22 4991 8c59 ed80f460f496

where 2e116c62 48eb 46b6 aca9 5b26e3b61a62now plays the role of 4e4a55e7 0982 4ebc 8e77 f09e953be05f.

(b) The first step in the derivation of the superposition integral is to find the response to a point source at the origin, defined such that

89631977 2284 4966 8741 fbfe75d7003b

Because the situation is then spherically symmetric, the desired response to this point source must be a function of r only. Thus, for this response, (b) becomes

6414f7e0 1431 4a12 930c 6ab08d1fde49

7b1dc02f 5799 48db a120 ad44480ee8ba

a70c9545 73ee 463e 8ad7 38d60a46fe47

a6157506 1d57 46ac 9e45 21aab842a87a

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