# Shell's Linear Charge density (electric field mistake?)

• blackbrawler
In summary, the conversation discusses the use of Gauss' Law to determine the linear charge density of a charged solid cylinder and cylindrical shell that are coaxial and have uniform surface charge densities. The conversation also mentions a graph of the electric field and confusion about using the equation for the outside of the shell. The solution manual uses a different method and gets a different result.
blackbrawler

## Homework Statement

Figure 23-37a shows a narrow charged solid cylinder that is
coaxial with a larger charged cylindrical shell. Both are noncon-ducting and thin and have uniform surface charge densities on their outer surfaces. Figure 23-37b gives the radial component E of the electric field versus radial distance r from the common axis, and Es = 3.0*103 N/C.What is the shell’s linear charge density?

## Homework Equations

Gauss' Law 2λ/4piεr = 2Kλ/r

## The Attempt at a Solution

Since I know the inside of the shell would be 0 I looked at the graph and determined that Eexternal = -2000N/C and r = .035m

so then I do -2000N/C = 2kλ/r

do some magic algebra

λ = -4 * 10-9 C/m

The solution manual shows something different and says that Eexternal = -3000N/C and that I should use Eexternal-Einternal to solve. The book gets λ′ = –5.8 × 10−9 C/m.

This really confuses me on many levels.

To start, just be really clear where you're putting you're Gaussian surfaces -- the first one is a cylinder that you put around the outside of the shell, and this is the equation you derive from it, correct?

$$E = \frac{\lambda}{2\pi \epsilon_{0} r}$$

I think the problem you face with using that equation is that when you use it for outside the shell, the charge you're enclosing includes both the charge on the inner and outer shells, so you can't solve for either directly. Looks like both shells have different linear charge densities, $\lambda_{inner}$ and $\lambda_{outer}$ -- which can you solve for directly with the above equation?

## What is Shell's Linear Charge Density?

Shell's Linear Charge Density is a measure of the distribution of electric charge along a one-dimensional line, such as a cylindrical shell. It is expressed in units of charge per unit length, such as coulombs per meter.

## How is Shell's Linear Charge Density calculated?

The formula for calculating Shell's Linear Charge Density is q/2πrL, where q is the total charge, r is the radius of the shell, and L is the length of the shell. This assumes that the charge is evenly distributed along the length of the shell.

## Is Shell's Linear Charge Density the same as electric field?

No, Shell's Linear Charge Density and electric field are two different concepts. Electric field is a measure of the strength and direction of the electric force at a given point in space, while Shell's Linear Charge Density is a measure of the distribution of charge along a line.

## What is the significance of Shell's Linear Charge Density?

Shell's Linear Charge Density is important in understanding the electric field created by a charged cylindrical shell. It helps in calculating the electric field at a point outside or inside the shell, and also in determining the charge enclosed by a Gaussian surface surrounding the shell.

## Can Shell's Linear Charge Density be negative?

Yes, Shell's Linear Charge Density can be negative if the charge distribution is not uniform along the length of the shell. This can happen if there are regions of excess positive or negative charge, resulting in an overall negative or positive linear charge density.

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