# Energy per unit length of a cylindrical shell of charge

• zweebna
In summary: The total work is the sum of the work done on each radius, but since we are moving Q around, the total work done is different for each radius.
zweebna

## Homework Statement

"An infnitely long hollow cylinder of radius ##a## has surface charge density ##σ_a##.
It is surrounded by a coaxial hollow cylinder of radius ##b## with charge density ##σ_b##. The charge densities are such that the total confguration is electrically neutral. Using whatever method you choose, calculate the work per unit length needed to assemble the system.

## Homework Equations

$$\oint \vec{E} \cdot d\vec{a} = \frac{Q_{enc}}{\epsilon _0}$$
$$W = \frac{\epsilon _0}{2} \int E^2 d \tau$$

## The Attempt at a Solution

First of all, I assume that the charge density is uniform, which I think is a good assumption but I'm not sure.

Then I find ##\vec{E}##:
When ##r<a, \vec{E} = 0## (by symmetry)
When ##r>a, \vec{E} = 0## (given)

When ##a<r<b##: Use Gauss' Law $$\oint \vec{E} \cdot d\vec{a} = \frac{Q_{enc}}{\epsilon _0}$$
As my surface, I use a cylinder of radius r and length L:
$$\oint \vec{E} \cdot d\vec{a} = E2 \pi rL$$
and
$$Q_{enc} = \sigma_a 2 \pi a L$$
so
$$E2 \pi rL = \frac{\sigma_a 2 \pi a L}{\epsilon_0}$$
so
$$\vec{E} = \frac{\sigma_a a}{r\epsilon_0} \hat{r}$$

Now here's where I start to get into trouble. To find the energy I would use the equation for work above, with ##d\tau =rdrd\theta dz##, but how do I turn this into an energy per unit length? Integrating the ##dz## over all space gives an infinity obviously.
What I tried to do was just divide the whole thing by L and simply ignore the z differential, which I have no idea if I can do:
$$\frac{W}{L} = \frac{\epsilon_0}{2L} \int_{a}^{b} \int_{0}^{2\pi} (\frac{\sigma_a a}{r \epsilon_0})^2 rdrd\theta$$
Which leads to: $$\frac{W}{L} = \frac{\pi \sigma_a^2 a^2}{L \epsilon_0} ln(\frac{b}{a})$$

I'm not sure if this makes sense. Is it allowed for me to just ignore the z differential like that? I don't really see a way around it going to infinity.

I don't know if there is an easier way, but I supposed that at some time we have a charge +Q on some length of the inner cylinder and a charge -Q on the outer cylinder. Now consider moving ΔQ from the outer to the inner. We know the potential at each radius due to each cylinder, so we can compute the work done.

## What is the energy per unit length of a cylindrical shell of charge?

The energy per unit length of a cylindrical shell of charge is the amount of energy per unit length that is associated with the electric field created by a charged cylindrical shell. It is also known as the self-energy per unit length of the shell.

## How is the energy per unit length of a cylindrical shell of charge calculated?

The energy per unit length of a cylindrical shell of charge can be calculated using the formula: U = λ²/(2πε₀r), where λ is the linear charge density of the shell, ε₀ is the permittivity of free space, and r is the radius of the shell.

## What factors affect the energy per unit length of a cylindrical shell of charge?

The energy per unit length of a cylindrical shell of charge is affected by the linear charge density of the shell, the permittivity of free space, and the radius of the shell. Additionally, the direction and magnitude of the electric field produced by the shell also play a role in determining the energy per unit length.

## Does the energy per unit length of a cylindrical shell of charge vary along its length?

No, the energy per unit length of a cylindrical shell of charge is constant along the length of the shell. This is because the electric field produced by the shell is uniform in magnitude and direction at all points along its length.

## How is the energy per unit length of a cylindrical shell of charge related to the electric potential?

The energy per unit length of a cylindrical shell of charge is related to the electric potential by the formula: U = -∂V/∂r, where V is the electric potential and r is the distance from the center of the shell. This relationship shows that the energy per unit length is directly proportional to the rate of change of the electric potential with respect to the distance from the shell.

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