Electromagnetism question (Stump book) - Dielectric

[Noob.com]

Ok this book is very common. It's the question number 6.5

A dielectric object that has a quasi-permanent polarization when the applied field is 0 is called an electret. Consider a uniformly polarized electret in the shape of a cylinder of height h and radius 10h. The polarization in the dielectric is P k^, where k^ is parallel to the cylinder axis.

Note: It's a big P not small p.

a. Sketch the electric field lines.
b. Calculate the electric field E at the center of the cylinder. because the raidus is large compared to the height, you may neglect edge effects.
c. Calculate the electric field E on the midplane of the cylinder, at distance 100h from the center. because the distance is large compared to the radius the dipole dominates the multipole expansion.

At least give me a hint how to start. Thank you! And tell me how does electret different from other normal dielectric materials.

______________________________________________________

This is out of the book.

A dielectric sphere is placed in an applied electric field. The radius of the sphere is a, and the dielectric constant is kappa (k). The electric field far from the sphere is Eo k^. Determine the dipole moment of the sphere, induced by the electric field.

Thanks!

 

[chrisk]

An electret is the electric equivalent of a permanent magnet. The polarization vector is proportional and in the same direction as the E field inside the electret. It can be thought of as a pair of circular plates at each end of the cylinder with uniform charge densities of opposite polarities. This should get you started.

 

[Noob.com]

So...
a. Electric field will be loop from top to bottom of the cyclinder. Inside will be arrow from bot to top. Correct? Since they both have the same direction.
b. How do you find the electric field inside the cylindrical? I know p = alpha * E
-- Gauss law, E * A = Charge density * Volume
==> E = Charge density / (2 * Epilon 0) h^
--Charge density = - (gradiant P)
? What can I do from here?

Problem 2: How do you start?

 

[chrisk]

Treating the ends of the electret as charged disks, you can find the electric field at any point on the axis using the definition for E and integrate the charge elements over the disk.

 

[Noob.com]

So basically it will be the same as point charge above the disk?

E = k * P * 20 * pi * h² * z / (100h²+z²)^3/2)

It's on the z direction and k = 1/(4*pi*ep0)

Since P = dipole moment / Volume

 

[chrisk]

The approach is to determine the bound surface charge density for each end of the electret and treat these ends as thin disks.

$$\sigma_b=\vec{P}\bullet\hat{n}$$

where

$$\hat{n}=\mbox{ unit vector normal to the surface and pointing outward.}$$

Now that the surface charge density is known, find E along the axis of the disk. Recall the P is the dipole moment per unit volume

$$P=\frac{n}{v}\mbox{ql}$$

$$dE=\mbox{K}\frac{\sigma{_b}rdrd\theta}{R^2}$$

where rdrd(theta) is a differential area of the disk radius and R² is the distance from the charge element to the point being evaluated. The polarization, P is the dipole moment per unit volume. A dimensional analysis gives

[P] = [1/L³][C][L] = [C]/[L²]