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  

(a) Show that the potential in the neighborhood of the origin takes the form  ²². (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  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 _s. Express the potential  along the  axis for  in terms of a two-dimensional integral.

(b) For the particular surface charge distribution _{s o}|_y|² where _o and a are constants, determine  along the positive z axis.

(c) What is  at the origin?

(d) Show that  has a  dependence for  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  along the positive  axis?

Fig. P4.5.1