Computer Algebra Recipes

5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 233
> phi:=collect(phi,[cos(theta),sin(theta),r]);
μ
μ
C4 C2
C3 C2
Á := C4 C1 r + cos(μ)+ C3 C1 r + sin(μ)
r
r
To simplify the coe±cients in Á, Benny makes the following substitutions.
> phi:=subs(_C3=a/_C1,_C2=b*_C1/a,_C4=c/_C1,phi);
μ
Á := cr+ cb
μ
cos(μ)+ ar+
ar
b
sin(μ)
r
> phi:=algsubs(c*b=a*d,phi);
μ
Á := cos(μ) cr+ d
μ
+ ar+
r
b
sin(μ)
r
To determine the four unknown coe±cients a, b, c, andd, four boundary conditions
are required. On the inner boundary, r = 10 cm, the coe±cient of the
cos μ term must equal 15, while the coe±cient of the sin μ term must equal zero.
This yields two boundary conditions, labeled bc1 and bc2 .
> bc1:=eval(coeff(phi,cos(theta)),r=10)=15;
bc1 := 10 c + d
10 =15
> bc2:=eval(coeff(phi,sin(theta)),r=10)=0;
bc2 := 10 a + b
10 =0
On the outer boundary, r =20cm,thecoe±cientofthesinμterm must equal
30, while the coe±cient of the cos μ term is equal to zero. This yields two more
boundary conditions, bc3 and bc4 .
> bc3:=eval(coeff(phi,sin(theta)),r=20)=30;
bc3 := 20 a + b
20 =30
> bc4:=eval(coeff(phi,cos(theta)),r=20)=0;;
bc4 := 20 c + d
20 =0
The four boundary condition equations are solved for a, b, c, andd,
> coefficients:=solve(fbc1,bc2,bc3,bc4g,fa,b,c,dg);
½
coe±cients := d =200;a=2;c= ¡1
¾
;b= ¡200
2
which are assigned to produce the ¯nal solution Á to the potential problem.
> assign(coefficients): phi:=phi;
μ
Á := cos(μ) ¡ r
μ
200
+ + 2 r ¡
2 r
200
sin(μ)
r
As requested by Professor Nerd, Benny will now use Á to plot the equipotentials
in the annular region between r = 10 and 20. He ¯rst converts the potential
into Cartesian coordinates by substituting r = p x2 + y2 ,cosμ = x= p x2 + y2 ,
and sin μ = y= p x2 + y2 ,intoÁ.