# E&M: Gauss' Law Surface Charge Density

• Wmdajt
Wmdajt
Homework Statement
A charge of uniform linear density 2.8nC/m is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius =4.8 cm, outer radius =9.40 cm). The net charge on the shell is zero. (a) What is the magnitude of the electric field 14.2 cm from the axis of the shell? What is the surface charge density on the (b) inner and (c) the outer surface of the shell?
Relevant Equations
E=λ/(2πε0r)
ε0ϕ = q
In this question I need to find the inner and outer charge density of the shell I did part A just fine, I used the formula for an electric field due to a line charge, but parts B and C is what's really confusing me. I'm not really sure how to go about it, I placed a spherical gaussian surface inside of the shell which gives me a flux of E*π*r^2. I didn't have a reason to attempt this, I was just seeing where it would take me. Any help would be appreciated, thanks!

Wmdajt said:
Homework Statement: A charge of uniform linear density 2.8nC/m is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius =4.8 cm, outer radius =9.40 cm). The net charge on the shell is zero. (a) What is the magnitude of the electric field 14.2 cm from the axis of the shell? What is the surface charge density on the (b) inner and (c) the outer surface of the shell?
Relevant Equations: E=λ/(2πε0r)
ε0ϕ = q

In this question I need to find the inner and outer charge density of the shell I did part A just fine, I used the formula for an electric field due to a line charge, but parts B and C is what's really confusing me. I'm not really sure how to go about it, I placed a spherical gaussian surface inside of the shell which gives me a flux of E*π*r^2. I didn't have a reason to attempt this, I was just seeing where it would take me. Any help would be appreciated, thanks!
What formula did you use? Please show your work and explain how you got E*π*r^2. A spherical Gaussian surface is not appropriate for a long wire. Do you see why?

kuruman said:
What formula did you use? Please show your work and explain how you got E*π*r^2. A spherical Gaussian surface is not appropriate for a long wire. Do you see why?
I just figured it out! I understand why a spherical Gaussian surface isn't appropriate, the way I drew my diagram was confusing me. I should've used a cylindrical surface. Negative charge is attracted to the inner wall of the shell equal to the amount of charge on the rod. From there I found the charge on the rod and divided it with the inner surface of the cylinder.

Wmdajt said:
From there I found the charge on the rod and divided it with the inner surface of the cylinder.
I don't understand this part. You are given the linear charge density on the rod. You do not explain how you found the charge density on the outer surface of the cylinder.

kuruman said:
I don't understand this part. You are given the linear charge density on the rod. You do not explain how you found the charge density on the outer surface of the cylinder.
Sorry, I wrote that in a bit of a hurry! I'll clarify.
The charge on the inner surface of the shell is the same as the opposite charge of the rod because the flux inside of a conductor must be zero.

qi = -q, Where qi is the charge of the inner shell surface and q is the charge of the rod

Charge density is defined as follows:
σ = qi/A
Charge is only on the surface of the shell due to it being a conductor, so we use the surface area of the body of a cylinder
σ = -q/(2πrh)

We clarified the outer shell charge is the same as the rod, we can find the charge using linear charge density
q = λh

Putting it all together in the equation above yields charge density of the outer surface of the shell
σ = -λ/(2πr)

The same formula is used for part C because the net charge in the shell is 0, so:
qo = qi = q, where qo is charge of the outer shell

## Related to E&M: Gauss' Law Surface Charge Density

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## What is Gauss' Law in electromagnetism?

Gauss' Law states that the net electric flux through any closed surface is equal to the charge enclosed within that surface divided by the permittivity of free space. Mathematically, it is expressed as ∮E·dA = Q_enclosed/ε₀, where E is the electric field, dA is a differential area on the closed surface, Q_enclosed is the total charge enclosed, and ε₀ is the permittivity of free space.

## How is surface charge density defined in the context of Gauss' Law?

Surface charge density, denoted by σ, is defined as the amount of electric charge per unit area on a surface. It is given by σ = Q/A, where Q is the total charge and A is the area over which the charge is distributed. In the context of Gauss' Law, surface charge density can be used to determine the electric field near a charged surface.

## How do you apply Gauss' Law to find the electric field due to a surface charge density?

To apply Gauss' Law to find the electric field due to a surface charge density, you choose a Gaussian surface that takes advantage of symmetry. For example, for a uniformly charged infinite plane, you would choose a cylindrical Gaussian surface with its axis perpendicular to the plane. By symmetry, the electric field is uniform and perpendicular to the surface. Using Gauss' Law, you can then solve for the electric field E as E = σ/(2ε₀).

## What are some common applications of Gauss' Law involving surface charge density?

Common applications of Gauss' Law involving surface charge density include calculating the electric field near charged conductors, determining the field inside and outside spherical shells, and analyzing the fields in capacitors. It is particularly useful in cases with high symmetry, such as spherical, cylindrical, or planar symmetry.

## Can Gauss' Law be used to determine the surface charge density on an irregularly shaped conductor?

While Gauss' Law is most straightforward to apply in cases with high symmetry, it can still be used for irregularly shaped conductors by considering small differential elements of the surface. However, the calculations can become complex, and numerical methods or approximations are often employed to determine the surface charge density in such cases.



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