Computer Algebra Recipes

222 CHAPTER 5. LINEAR PDE MODELS. PART 1
The horizontal curves are the equipotentials (constant v) and the vertical curves
are the electric ¯eld lines (constant u). The two sets of curves intersect at
right angles, as would be expected. The electric ¯eld lines also intersect the
cylindrical surface perpendicularly, because the conducting surface is also an
equipotential.
Instead of showing the electric ¯eld lines, one can plot the electric ¯eld
vectors. The electric ¯eld ~ E = ¡rv can be calculated in Cartesian coordinates
using the Gradient command.
> E:=-Gradient(v,'cartesian'[x,y]);
E := ¡
2 yx
(x 2 + y 2 ) 2 ex +
μ
¡1+
1
x2 ¡
+ y2 2 y 2
(x 2 + y 2 ) 2
Since we are interested only in the electric ¯eld outside the cylinder, two piecewise
functions are formed to calculate the x and y components of the electric
¯eld in the region x 2 + y 2 ¸ 1.
> Ex:=piecewise(x^2+y^2=1,E[1]):
> Ey:=piecewise(x^2+y^2=1,E[2]):
The fieldplot command is used to plot the electric ¯eld vectors as thick red
arrows, the grid being 12 £ 12. The arrows will point in the direction of ~ E,
their size being a measure of the magnitude, j ~ Ej.
> FP:=fieldplot([Ex,Ey],x=-2..2,y=-2..2,arrows=THICK,
grid=[12,12],color=red):
The equipotentials and electric ¯eld vectors are superimposed in the same picture,
a black-and-white rendition being shown in Figure 5.5.
> display(fCP(pw2,blue),FPg,labels=["x","y"],tickmarks=[2,2]);
y
2
–2 0 x 2
–2
Figure 5.5: Equipotentials and electric ¯eld vectors outside the cylinder.
ey