9-32. The problems below are concerned with obtaining a solution of Laplace’s
equation for temperature T. Chose an appropriate coordinate system and make
necessary assumptions about the solution in order to reduce the problem to a one-
dimensional Laplace equation.
(a) Find the steady-state temperature distribution along a bar of length Lassuming
that the sides of the bar are insulated and the ends are kept at temperatures T 0
and T 1 .This corresponds to solving d
(^2) T
dx^2
= 0,T(0) = T 0 and T(L) = T 1.
(b) Find the steady-state temperature distribution in a circular pipe where the inside
of the pipe has radius r 1 and temperature T 1 , and the outside of the pipe has
a radius r 2 and is maintained at a temperature T 2 .This corresponds to solving
1
rd
dr(
rdT
dr)= 0 such that T(r 1 ) = T 1 and T(r 2 ) = T 2
(c) Find the steady-state temperature distribution between two concentric spheres
of radii ρ 1 and ρ 2 , if the surface of the inner sphere is maintained at a temperature
T 1 ,whereas the outer sphere is maintained at a temperature T 2 .This corresponds
to solving^1
ρ^2d
dρ(
ρ^2 dT
dρ)= 0 such that T(ρ 1 ) = T 1 and T(ρ 2 ) = T 2.
(d) Find the steady-state temperature distribution between two infinite and parallel
plates z=z 1 and z=z 2 maintained, respectively, at temperatures of T 1 and T 2.
9-33. Find the potential function associated with the conservative vector field
F= 6xz ˆe 1 + 8yˆe 2 + 3 x^2 ˆe 3.9-34. Newton’s law of attraction states that two particles of masses m 1 and m 2
attract each other with a force which acts in the direction of the line joining the
two masses and whose magnitude is given by F=Gm 1 m 2 /r^2 ,where ris the distance
between the masses and Gis a universal constant.
(a) If mass m 1 is at the origin and mass m 2 is at a point (x, y, z ),find the vector force
of attraction of mass m 1 on mass m 2.
(b) If mass m 1 is at a fixed point P 1 (x 1 , y 1 , z 1 )and mass m 2 is at the point (x, y, z ),
find the vector force of attraction of mass m 1 on mass m 2.
9-35. Show that u=u(x, t) = f(x−ct)+ g(x+ct),f, g arbitrary functions, is a solution
of the wave equation
∂^2 u
∂x^2 =1
c^2∂^2 u