[SOLVED] Physics 311 Homework Set 8

1a) Consider Laplaces equation in spherical coordinates:
r2
V = 1
r2
@
@r

r2 @V
@r

+
1
r2 sin
@
@
sin
@V
@
+
1
r2 sin2
@2V
@2 = 0. (1)Find the general solution to Laplaces equation in spherical coordinates, for the case
where the electric potential depends only on r.b) Consider Laplaces equation in cylindrical coordinates:
r2
V = 1
s
@
@s

s
@V
@s

+
1
s2
@2V
@2 +
@2V
@z2 = 0. (2)Find the general solution to Laplaces equation in cylindrical coordinates, for the case
where the electric potential depends only on s.2. Find the potential in the infinite slot of Ex. 3.3 if the boundary at x = 0 consists of two
metal strips: one, from y = 0 to y = a/4, is held at a constant potential V0, and the
other, from y = a/4 to y = a, is at potential V0/4.3. A rectangular pipe, running parallel to the z-axis (from 1 to 1), has three grounded
metal sides at y = 0, y = b, and x = 0. The fourth side, at x = a, is maintained at a
specific potential V0(y).a) Develop a general formula for the potential within the pipe.b) Find the potential explicitly, for the case V0(y) = V0 (a constant).4. A cubical box (sides of length a) consists of five metal plates, which are welded together
and grounded. The plate at y = a is made of a separate sheet of metal, insulated from the
others, and held at a constant potential V0. Find the potential inside the box.