The charge distribution described in Prob. 4.4.3 is now at infinity (a) Show that the potential…

The charge distribution described in Prob. 4.4.3 is now at infinity 8a6de21f fb40 4e1a 89f8 f4671ae89aaa 

(a) Show that the potential in the neighborhood of the origin takes the form d712dd17 4691 4b4e b3aa 1b3510b5b12e 27e4f674e 9ad1 492f 9098 dbf67c0401002f7eb813c 2dba 4824 9a80 7ed0982fa60d. (b) How would you position the line charges so that in the limit where they moved to infinity, the potential would take the form of (4.1.18)?

The only charge is restricted to a square patch centered at the origin and lying in the 94244ad9 b2da 4ab2 9fa6 020bed8f185e plane, as shown in Fig. P4.5.1.

(a) Assume that the patch is very thin in the z direction compared to other dimensions of interest. Over its surface there is a given surface charge density 31d6162c a6c7 4d0b 9856 dca3db7bd09bs224eb35f f355 4777 9fbe 7fd17a1fa753. Express the potential 98d2c952 e8db 4b30 b39d 75f8ece58877 along the 996dae88 1438 4deb a664 18baae7af0fd axis for 3c215254 e4c0 43eb a307 99fb139982b9 in terms of a two-dimensional integral.

(b) For the particular surface charge distribution 31d6162c a6c7 4d0b 9856 dca3db7bd09bs ffa449d4 d694 4915 9649 cc07703496ebo|1719c920 7475 4af2 9237 a21be05dee12y|2004c09c ac25 4f2b a90c 522aac72fa4f2 where 31d6162c a6c7 4d0b 9856 dca3db7bd09bo and a are constants, determine 98d2c952 e8db 4b30 b39d 75f8ece58877 along the positive z axis.

(c) What is 98d2c952 e8db 4b30 b39d 75f8ece58877 at the origin?

(d) Show that 98d2c952 e8db 4b30 b39d 75f8ece58877 has a 996dae88 1438 4deb a664 18baae7af0fd dependence for 0ec3ddf8 311a 4d5d b238 8dfbe9275966 that is the same as for a point charge at the origin. In this limit, what is the equivalent point charge for the patch?

(e) What is db84ce04 07d8 488c 86c3 516d80b663ea along the positive 996dae88 1438 4deb a664 18baae7af0fd axis?

Fig. P4.5.1

a274ebc2 ebb3 41c3 8844 a89a8a5536c8

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