# A thin dielectric rod, running from z=-a to z=+a, ...

A thin dielectric rod, running from z=-a to z=+a, carries thelinear charge
?=(?_0)cos(?z/a). Find the first non-zero termin the multipole expansion of the potential. Express the potentialin spherical cooardinates.

1. to solve we use the bermoulis equation
so from the figure in the text first we choosepoint 1 at the top of the tank and point 2 at the exit
from the tube
so now applying the bernoulis equation withv1 = 0 we get
(1 / 2) ? v22 =(P1 -P2) + ? g (y1 -y2)
but as the pressures are equal
P1 = P2
= 1 atmand
y1 = y2
= h
so on substitution we get
v2 = ?(2 g h)
(b)
in this case we use the bernolis equation withpoint 1 at the top of the tank and point 2 at the
highest point in the tube
from which we get
? g (y2 - y1) =(Patm - P2) + (1 / 2) ?(v12 - v22)
so when the siphon fails yto work the fluid willbe at rest at point 2 then we get
v2 = v1
= 0 and
y2 - y1 = ymax
we know that
P = Po + ? g h
Patm = P2 + ? gymax
as the minimum value of P2 is zero weget
ymax = Patm /? g
=Po / ? g