Green’s Reciprocation Theorem

a) Prove Green’s reciprocation theorem: If ϕ is the potential due to a volume charge density ρ within a volume V and a surface charge density σ on the conducting surface S bounding the volume V, while ϕ' is the potential due to another charge distribution ρ' and σ' then

b) A point charge q is placed between two infinite grounded parallel conducting plates. If z₀ is the distance between q and the lower plate, find the total charge induced on the upper plate in terms of q, z₀, and l, where l is the distance between the plates (see Figure 1.1). Show your method clearly.

Figure 1.1

SOLUTION

a) We may prove the theorem by considering the volume integral of the following expression:

Integrating by parts in two ways, we have

(1)

Now,  (where n points opposite to the directed area of the surface S) and  so dividing (1) by 4π yields the desired result:

(2)

b) Let us introduce a second potential given by ϕ' = 2πσ' z, corresponding to a surface charge density on the upper plate of σ' and on the lower plate of 0 (see Figure 1.2). This introduced potential has no charge in the volume, and the real potential is zero on the plates so that the right-hand side of (2) is zero, yielding

where Q_i is the induced charge. So on the upper plate,

Figure 1.2

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