Gauss's law for a cylinder

In summary: Expert SummarizerIn summary, to find the linear charge density on the inner cylinder, we can use the formula λ = ρ * A, where ρ is the given volume charge density and A is the cross-sectional area of the cylinder. To calculate the electric field for all values of R, we can use the formula E = k * Q / r^2, where Q is the charge on the cylinder given by Q = λ * L, and r is the distance from the cylinder. The electric field inside the nonconducting cylinder will be zero, while the electric field outside the cylinder will depend on the distance from the cylinder and the length of the cylinder.
  • #1
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Homework Statement


Suppose that the inner cylinder of the figure below is made of nonconducting material and carries a volume charge distribution given by ρ(R) = C/R, where C = 301 nC/m2. The outer cylinder is metallic and both cylinders are infinitely long.

23-36alt.gif

a.Find the charge per unit length (that is, the linear charge density) on the inner cylinder.

b.Calculate the electric field for all values of R.

Homework Equations


[tex]\Phi[/tex]=[tex]\frac{Q}{\epsilon_0{}}[/tex]


The Attempt at a Solution


a. I'm not quite sure how to find the linear charge density given the density for the volume which is in turn as a Area-density by the radius... it looked like a big conversion mess. I tried multiplying by .03m to see if that worked but apparently that's along the wrong lines of thinking. This is very basic but I'm just looking to see the relation between the linear and area density.

b.I would have thought that the metallic cylinder acted as a Faraday cage and would prevent charge from leaking through the outside but apparently that's not so. Also it mentioned that the inner rod is nonconducting so I am not sure whether the relation for uniform charge distribution inside a material still holds... if it does then the math i did earlier was wrong. I am confused as to why there is no field in the cylinder but there is on the outside of it? any clarification would be greatly appreciated. Numbers aren't exactly what I'm asking for I just want to understand the relationship for the fields in the cylinder and rod.

thank you very much for any and all help
 

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  • #2

Thank you for your post. I will try my best to assist you with your questions.

a. To find the linear charge density, we need to consider the definition of linear charge density, which is the amount of charge per unit length. In this case, we have a volume charge density, which is the amount of charge per unit volume. To find the linear charge density, we can use the formula:

λ = ρ * A,

where λ is the linear charge density, ρ is the volume charge density, and A is the cross-sectional area of the cylinder. In this case, the cross-sectional area of the cylinder is πR^2. Therefore, the linear charge density is:

λ = (301 nC/m^2) * (πR^2) = 301π nC/m

b. To calculate the electric field for all values of R, we can use the formula:

E = k * Q / r^2,

where E is the electric field, k is the Coulomb's constant, Q is the charge on the cylinder, and r is the distance from the cylinder. In this case, the charge on the cylinder is given by:

Q = λ * L,

where L is the length of the cylinder. Therefore, the electric field is:

E = k * (301π nC/m) * L / r^2

Note that the electric field inside the nonconducting cylinder will be zero, as there is no charge inside the cylinder. The electric field outside the cylinder will depend on the distance from the cylinder, r, and the length of the cylinder, L. I hope this helps to clarify the relationship between the fields in the cylinder and rod.
 
  • #3

Gauss's law for a cylinder states that the electric flux through a closed surface surrounding a cylinder is equal to the charge enclosed by that surface divided by the permittivity of free space. It is given by the equation Φ = Q/ε0.

a. To find the linear charge density on the inner cylinder, we can use the given volume charge distribution ρ(R) = C/R. We can rewrite this as ρ(R) = λ/πR^2, where λ is the linear charge density. We can equate this to the given charge distribution C/R and solve for λ. This gives us a linear charge density of λ = Cπ = 9.43 nC/m.

b. The electric field for all values of R can be calculated using the equation E = λ/2πε0R, where λ is the linear charge density and R is the distance from the axis of the cylinder. This equation holds for both conducting and nonconducting cylinders. As for the metallic cylinder acting as a Faraday cage, it does not prevent the electric field from existing outside of the cylinder. The electric field exists both inside and outside of the cylinder due to the non-uniform charge distribution on the inner cylinder. The metallic cylinder only shields the electric field inside of it from external influences.
 

What is Gauss's law for a cylinder?

Gauss's law for a cylinder is a fundamental law in electromagnetism that relates the electric flux through a closed cylindrical surface to the charge enclosed within that surface.

How is Gauss's law for a cylinder derived?

Gauss's law for a cylinder is derived from Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space. The derivation for a cylindrical surface involves using the symmetry of the cylinder to simplify the calculations.

What is the equation for Gauss's law for a cylinder?

The equation for Gauss's law for a cylinder is ΦE = Qenc0, where ΦE is the electric flux through the cylindrical surface, Qenc is the charge enclosed within that surface, and ε0 is the permittivity of free space.

What is the significance of Gauss's law for a cylinder?

Gauss's law for a cylinder is significant because it allows us to calculate the electric field at any point outside a charged cylinder by knowing the charge density and the radius of the cylinder. It also helps us understand the relationship between electric flux and charge, and the role of symmetry in simplifying calculations.

What are some real-world applications of Gauss's law for a cylinder?

Gauss's law for a cylinder has various applications in engineering and physics, such as in the design of cylindrical capacitors, calculation of electric fields in cylindrical conductors, and modeling of electromagnetic fields in cylindrical cavities. It is also used in the study of cylindrical sources of electric fields, such as charged cylindrical rods or wires.

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