# Electric Field/Gauss' law of cylinder and shell

• ultrapowerpie
In summary, use Gauss' Law with a cylindrical Gaussian surface to find the electric field in terms of charge/unit length for a given charge density and radius of a cylindrical charge distribution.
ultrapowerpie

## Homework Statement

http://img244.imageshack.us/my.php?image=a1physicsmt8.png

Gauss' Law

## The Attempt at a Solution

It may be that I'm sick with a cold and can't think straight, but I"m not seeing any way to approach this problem using Gauss' law. I tried using a few prederived formulas, but like for the first part of this 5 part problem, I need the charge/length in order to find out E. I don't have that value, and I did try using rho as lambda, and got it wrong. I also have to find the charge at values inbetween the cylinder and the shell, outside the shell, and on the inner shell.

Some help would be nice, as I can't see how to do this by hand right now.

Consider a Gaussian cylinder with radius of 2 cm. This cylinder is within the green cylindrical charge distribution. Gauss's Law is

$$\oint_{S}\vec{E}\cdot\vec{dA}=\frac{1}{\epsilon_{0}}\int_{V}\rho dV$$

Use cylindrical coordinates, the given charge density, and the fact that the electric field is parallel to your Gaussian surface's normal vector to solve this.

Note that S is the surface area of your Gaussian cylinder and V is the volume within it.

Using a cylindrical Gaussian surface will give E in terms of charge/unit length. So, let the Gaussian surface be

$$2\pi\mbox{rl}$$

where l is a unit length. Then the charged enclosed per unit length within this surface is the volume within the surface times the charge density for r less than the radius of the inner conductor.

## 1. What is an electric field?

An electric field is a physical quantity that describes the influence of an electric charge on other charges in its vicinity. It is a vector quantity, meaning it has both magnitude and direction, and is measured in units of Newtons per Coulomb (N/C).

## 2. What is Gauss' law?

Gauss' law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the enclosed electric charge. It states that the electric flux through any closed surface is directly proportional to the charge enclosed by that surface, and is given by the equation ΦE = Qenc0, where ΦE is the electric flux, Qenc is the enclosed charge, and ε0 is the permittivity of free space.

## 3. How is Gauss' law applied to a cylinder or shell?

Gauss' law can be applied to a cylinder or shell by considering the electric field at a point outside or inside the cylinder/shell. For a point outside the cylinder/shell, the electric field is given by E = Q/(2πε0r), where Q is the total charge of the cylinder/shell and r is the distance from the center. For a point inside the cylinder/shell, the electric field is given by E = Q/(2πε0h), where h is the height of the cylinder/shell.

## 4. How does the electric field vary inside and outside a charged cylinder or shell?

Inside a charged cylinder or shell, the electric field is constant and directly proportional to the charge enclosed by the cylinder/shell. This means that the electric field is the same at all points inside the cylinder/shell. Outside the cylinder/shell, the electric field decreases as the distance from the center increases, following an inverse square law relationship.

## 5. Can Gauss' law be used for any shape or configuration of charge?

Yes, Gauss' law can be applied to any shape or configuration of charge, as long as the charge is enclosed by a closed surface. It is a fundamental law in electromagnetism and is used to calculate the electric field for various systems, including point charges, cylinders, and shells.

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