# Bound surface charge - hollow tube

1. Sep 3, 2014

### Roodles01

1. The problem statement, all variables and given/known data
A thick-walled cylindrical tube of dielectric material has internal radius a/2 and external radius 2a, and its polarization is given in cylindrical coordinates by the expression P = (A/r2) er.

Derive expressions for the net charge on the inner and outer surfaces of a length L of the tube and for the charge within the volume of length L of the tube.

2. Relevant equations
σb = P . n hat
where n hat is the outward-pointing unit vector normal to the surface and P is the polarization.
surface area of cylinder (ignoring ends) = 2∏rL

3. The attempt at a solution

σb total = σb inner + σb outer

σb outer = (A/(2a)2) * area of cylinder
σb outer = (A/(2a)2) * 2∏(2a)L
σb outer = ∏AL/a

Now the bit I can't seem to marry with the model answer . . .

σb inner = (A/(a/2)2) * 2∏(a)L
σb inner = (4A/a2) * 2∏(a)L
σb inner = 8∏AL/a

At least that's how I got it.

The model answer shows it to be
σb inner = 4∏AL/a

It has to be a problem I'm having with algebra again, but I can't see where.
Could you show me where I went wrong please.

Or am I right?

#### Attached Files:

• ###### Q1 01 01 edit.jpg
File size:
21.7 KB
Views:
106
Last edited: Sep 3, 2014
2. Sep 3, 2014

### Roodles01

Aaaaah!
Details.

Yes, the wrong bit is me not looking proplerly.
Sorry.

3. Sep 3, 2014

### BvU

Still has me puzzled. In the attachment I read $\vec P = \left ( A/r^3\right ) \, \vec e_r$ (in the text; in the picture it's hard to distinguish if it's a 3 or a 2).

Or is your exercise another one, with the a/2 instead of the a and the 2 instead of the 3 ?