Electric field of a cylindrical capacitor

[Eigenvaluable]

Homework Statement

Calculate the electric field of a cylindrical capacitor comprised of a smaller cylindrical conductor of radius $a$ enclosed within a larger cylindrical conductor of radius $b$ where $b>a$. The smaller cylinder has charge $+Q$ and the larger cylinder has charge $-Q$.

Homework Equations

$$\oint \vec{\mathbf{E}} \cdot \vec{\mathbf{dA}} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$

The Attempt at a Solution

I've already solved the problem, as follows:

Define a Gaussian surface in which the cylinder completely encloses the smaller cylinder and is completely within the larger cylinder, i.e. a Gaussian cylinder with radius $r$ where $a<r<b$.

Furthermore, let $L$ denote the length of the cylinder.

$$(2 \pi r L) E = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$
$$E = \frac{Q}{2 \pi r L \epsilon_0}$$

While this is trivially easy, this reference sheet assumes that $L >> b$ prior to calculating anything.

Why must this be done?

Thank you,
Eigenvaluable

 

[gneill]

Hello Eigenvaluable, Welcome to PF.

Homework Statement

Calculate the electric field of a cylindrical capacitor comprised of a smaller cylindrical conductor of radius $a$ enclosed within a larger cylindrical conductor of radius $b$ where $b>a$. The smaller cylinder has charge $+Q$ and the larger cylinder has charge $-Q$.

Homework Equations

$$\oint \vec{\mathbf{E}} \cdot \vec{\mathbf{dA}} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$

The Attempt at a Solution

I've already solved the problem, as follows:

Define a Gaussian surface in which the cylinder completely encloses the smaller cylinder and is completely within the larger cylinder, i.e. a Gaussian cylinder with radius $r$ where $a<r<b$.

Furthermore, let $L$ denote the length of the cylinder.

$$(2 \pi r L) E = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$
$$E = \frac{Q}{2 \pi r L \epsilon_0}$$

While this is trivially easy, this reference sheet assumes that $L >> b$ prior to calculating anything.

Why must this be done?

A similar stipulation is made in the analysis of parallel plate capacitors where the plate dimensions are taken to be much larger than the plate separation. Consider the assumptions being made about the field between the capacitor plates. Do they hold true everywhere?