Charge and Dielectric

A charge e is situated at the point x = h > 0, y = z = 0 outside a homogeneous dielectric which fills the region x < 0 (see Figure 1.1).

Figure 1.1

a) Write the electric fields E(0⁺, y, z) and E(0^-, y, z)  just outside and just inside the dielectric in terms of the charge e and surface charge density σ_b of bound charges on the surface of the dielectric.

b) Express σ_b in terms of E(0^-, y, z). Denote by the dielectric constant of the dielectric.

c) By using the equations obtained in (a) and (b), show that

d) Calculate the electric field E' due σ_b to at the position (h, 0, 0) of the charge e. Show that it can be interpreted as the field of an image charge e' situated at the point (-h, 0, 0).

e) Show that the charge experiences the force

SOLUTION

a) The electric fields just above and below the dielectric due to the charge e and the surface charge σ_b(y, z) are, respectively,

(1)

(2)

b) From D = E + 4πP = εE, we have for the polarization P and its normal component to the boundary P_n

(3)

c) Using (2) in (3), we have

(4)

So

(5)

d) The field at the position of the charge due to the surface charge σ_b is

(6)

where dA is the area. We can rewrite this integral in cylindrical coordinates:

(7)

This can be interpreted as an image charge

(8)

at a distance 2h from the charge e (see Figure 1.2).

Figure 1.2

e) The force on the charge e is

(9)

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

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أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

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