Potential of Halved Cylinder

Consider an infinitely long conducting cylinder of radius a with its axis coinciding with the z-axis. One half of the cylinder (cut the long way) (y > 0) is kept at a constant potential V₀, while the other half (y < 0) is

Figure 1.1

kept at a constant potential –V₀ (see Figure 1.1). Find the potential for all points inside the cylinder and the field E along the z-axis.

SOLUTION

This problem can be solved by several methods. We will use conformal mapping. Namely, we will try to find a function ω (z), where z = x + iy, to transform the curves of equal potential (in the cross section of the three dimensional body) into parallel straight lines in the u – v plane, where ω = u + iv = f (x, y) +ig (x, y) with both f and g satisfying the Laplace’s equation. We can easily find the solution for the potential problem in the ω plane, and because of the properties of a conformal mapping, the functions u = f (x, y) or v = g (x, y) will be a solution to the initial potential problem. For this problem, we can use the transformation

(1)

This will transform a circle R = a into two straight lines (see Figure 1.2). The upper half of the cylinder will go into v = π/2, and the lower half will

Figure 1.2

go into v = -π/2. So we have

(2)

Denote the argument of the natural log as ρe^iφ where ρ and φ are real. Then.

So v = φ. On the other hand,

(3)

For a complex number m + in = ρe^iφ, we have

(4)

(5)

Using (5), we obtain from (3)

or

So the potential which satisfies ϕ(v = -π/2) = -V₀, ϕ(v = π/2) = V₀ is

(6)

On the z-axis

A different solution to this problem may be found in Cronin, Greenberg.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو