# E-field and charge density outside two coaxial cylinders

In summary, the problem involves finding the potential and electric field for a system consisting of an inner solid cylinder and an outer coaxial cylindrical shell with different voltages. The potential and electric field have been calculated for different regions, but the surface charge density at the outer surface of the outer cylinder cannot be determined using the methods tried so far. Using Gauss's law with a cylindrical surface around the outer cylinder may provide a solution, but choosing V = 0 at infinity may result in an infinite value for the surface charge density.
I am working on a problem of electrostatics, and I am having troubles in trying to figure out one part of it.

1. Homework Statement

It consists of an inner solid cylinder of radius ##a## with a voltage ##V_A##, and an outer coaxial cylindrical shell of inner radius ##b## and outer radius ##c## charged with a voltage ##V_B##. Find ##V(\mathbf{r})##, ##\mathbf{E}(\mathbf{r})## everywhere, and the surface change density at the surfaces of the conductors.

## Homework Equations

Maxwell's equations

## The Attempt at a Solution

I have calculated the potential field ##V## as well as the ##\mathbf{E}## field for ##r < a##, ##a < r < b##, and ##b < r < c##. I also calculated the surface charge density for the inner cylinder and the inner layer of the cylindrical shell. But can't figure out anything for ##c \leq r##.

I tried using Gauss's law, taking the origin of potential at infinity (##V(\infty) = 0##), finding the ##\mathbf{E}(\mathbf{r})## field for ##c<r## and calculating ##\displaystyle V_B = \int_{\infty}^{c} \mathrm{d}V = -\int_{\infty}^{c} \mathbf{E}(\mathbf{r} )\mathrm{d}\mathbf{r}## in order to find the value of surface charge density at ##r = c##. Also, I tried using that outside the shell we can assume the space is dielectric, so as ##-\nabla ^{2} V = 0##. None of this yields reasonable values, or at least I am doing something wrong in the procedure.

Procedure used:

• Using a Gauss cylinder of radius ##r>c##, ##Q_{in} = Q = 2\pi c L \sigma_{c}##, where ##L## is the length of the cylinder.
• Through Gauss's law, ##\Phi = E(r) Area = E \cdot 2 \pi r L = Q /\varepsilon_0 = \dfrac{2\pi c L \sigma_{c}}{\varepsilon_0}## ##\Rightarrow## ##\mathbf{E}(r) = \dfrac{\sigma_{c}}{\varepsilon_0}\dfrac{c}{r}\hat{r}##.
• Finding ##\sigma_c##: ##\displaystyle \left. V_B = \int_{\infty}^{c} \mathrm{d}V = -\int_{\infty}^{c} \mathbf{E} \mathrm{d}\mathbf{r} = - \dfrac{\sigma_c}{\varepsilon_0} c \ln r \, \right|_{\, \infty}^{\, c} = \infty##, which doesn't make sense.

Last edited:
There is something with the potential from an infinitely long line charge that makes it impossible to choose V = 0 at infinity. You bumped into that. But you can still use Gauss law for a a cylindrical surface around the outer cylinder as you did. Just don't go as far as infinity.

## 1. What is an electric field?

An electric field is a physical quantity that describes the force exerted on a charged particle by other charged particles. It is represented by a vector and can be thought of as the direction and strength of the force at a given point in space.

## 2. How is an electric field formed?

An electric field is formed when there is a difference in electric potential between two points in space. This potential difference causes the movement of electric charges, which creates the electric field.

## 3. What are coaxial cylinders?

Coaxial cylinders are two cylindrical conductors that share the same axis, with one cylinder surrounding the other. They are commonly used in electrical and electronic systems for their ability to transmit signals with minimal interference.

## 4. How is electric charge density related to the electric field outside two coaxial cylinders?

The electric charge density outside two coaxial cylinders is directly proportional to the electric field strength. As the electric field increases, the charge density also increases. Additionally, the charge density is inversely proportional to the distance from the cylinders, meaning it decreases as you move further away from the cylinders.

## 5. How are the properties of the electric field and charge density outside two coaxial cylinders affected by the cylinders' size and spacing?

The size and spacing of the coaxial cylinders affect the strength of the electric field and the charge density outside the cylinders. Generally, a larger cylinder and smaller spacing between the cylinders will result in a stronger electric field and higher charge density.

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