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CHAPTER 4

 

LAPLACE'S EQUATION

 

 

 

 

 

 

It is one of the most important partial differential equations and its solution is known as potential function or harmonic functions.

 

 

                                  in three dimension

 

 

                        in two dimension

 

 

                                  in one dimension

 

 

In steady-state case (u is independent of t) wave and heat equations change to Laplace's equation.

 

*  So steady-state solution of heat and wave equation is the solution of Laplace's equation also.

 

 

 

LAPLACE’S EQUATION IN :

 

 

-         Cartesian coordinates : (x,y,z)

 

 

…………………(1)

 

 

-         Cylindrical polar coordinates : (r,q , z)

 

 

   …………………..(2)

 

 

-         Spherical polar coordinates : (r,q,j)

 

 

 

   ……………….(3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


EXAMPLE 1

 

 

 

The upper plate of a parallel-plate capacitor is maintained at 100V, and the lower plate is maintained at 0V. The plates are 10 cm apart, and they are infinitely large. Find V in the parallel-plate region.

 

 

 

 

SOLUTION

 

 

 

Let us set up a coordinate system shown in figure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


So, the boundary conditions are

 

 

            V = 0V    at    y = 0    ………………(1)

 

 

and       V = 100V   at  y = 10 cm = 0.1 cm   …………………(2)

 

 

\      By symmetry V is a function of y only and is independent of x and z. Thus, this problem is to solve Laplace equation  i.e.,

 

 

 

            \       …………………..(3)

 

where A and B are constants.

 

 

Applying (1),    V(0) = 0 = B

 

 

Applying (2),    V(10) = 0.1          A + B = 100

 

                                                or,   A = 1000

 

 

\      From (3) the required charge V in parallel-plate is

 

 

V(y) = 1000y   Volt.

 

 

 

 

EXAMPLE 2

 

 

 

Find the temperature between two parallel plates  x = 0  and  x = d  having temperature 0°C and 100°C respectively.

 

 

 

 

SOLUTION

 

 

 

Let the temperature between two parallel plates be T.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


So, the boundary conditions are

 

 

            T = 0° C      at  x = 0

 

and       T = 100° C    at  x = d

 

 

\      T depends only on x and independent of y and z. Hence Laplace's equation

 

 

 

           

            becomes   

 

 

                        \   T(x) = Ax + B

 

           

 

            So, T(0) = 0 = B   and   T(d) = 100 = Ad + B

 

                                    or,   

 

 

\      The required temperature between the given parallel plates is

 

 

 

 

 

 

EXAMPLE 3

 

 

 

Find the temperature distribution in the region (i.e. cross-section of a solid cylinder) whose vertical portion of the boundary is at 20°C , the horizontal portion at 50°C , and the circular portion is insulated (as in figure).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


SOLUTION

 

 

 

Let T be the temperature distribution in the cross-section area of cylinder. If r be the radius of the circular area, then

 

 

                            so,      is any point on the circular boundary.

 

 

Since the circular portion is insulated, heat will not flow along r, so

 

 

                       

 

 

\      T depends only on q with boundary condition

 

 

T(0) = 50° C   and 

 

 

Hence Laplace's equation in polar coordinates becomes

 

 

                       

 

 

            or,       ……………………(1)

 

 

\     

 

 

and  

 

           

 

 

Hence from (1) the required temperature-distribution is,

 

 

           

 

 

or,       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Heat flows from x-axis along circles r = constant (dashes in figure) to the y-axis.

 

 

 

 

 

EXAMPLE 4

 

 

 

The inner and outer radii of two concentric, thin conducting spherical shells are a and b respectively. The space between the shells is filled with an insulating material. The inner shell is maintained at a potential V1 and the outer at a potential V2 (as figure). Determine the potential distribution in the insulating material by solving Laplace's equation in spherical polar coordinates.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


SOLUTION

 

 

 

Laplace's equation in spherical coordinates is

 

 

           

 

 

For a given fixed electric potential distribution

 

 

           

 

 

of a sphere of radius R, always

 

 

           

 

 

i.e.  u is independent of q.

 

 

The boundary conditions are

 

 

   ………………..(2)

 

 

Now, for symmetry  equation (1) becomes

 

 

               ……………………(3)

 

 

Again,