This problem does not have spherical symmetry. However, we can still use the general solution to the Laplace equation in all regions where the Laplace equation V21? = 0 is valid. iii. Describe all regions where the Laplace equation V217 = O is valid. iv. Assuming the potential can have the above form in the region you specified in part iii), find the two leading non-zero A's and B's. (Hint: you already found the answer along the positive 2 axis (3 = 0), so your answer must match when you set S = 0. Any potential can be expanded into the form: 1 \"mono ole" "di ole" " uadru ole" V: (p+p2+%+m) 41:60. r r r v. What are the monopole and dipole moments of this system? vi. How would you expect the monopole and dipole moments to change if the charge distribution . d . . . . were shifted up by 3' so that IS was centred at the origin? Check to see If you get what you expect. Problem 1 Potential from a line of charge A uniform line charge density A extends from the origin to the point (0,0, d). 2 i. Find an expression for the electric potential along the positive 2 axis, V(z). Recall: 1 17' (11" V(F) = .0( ) 41:60 IF F'I A 4HEU large 2, (2 >> (1), the potential can be approximated by the first few terms of an expansion V(z) = A d 1n(1 + e), where E E - is a small number. 4E60 z Your answer to part i) should be V(z) = ln (1 + g) (If not, find out where you went wrong). At ii. Expand the potential V(z) into a Taylor series at the origin. Find the first two non-zero terms. Taylor expansion of f(x) about the origin: 00 f(x) = in) + f'(0)x + grow + = 2 n=1 f