Electric Field of an Infinitely Long Insulating Cylinder

[sicrayan]

Homework Statement

An infinitely long insulating cylinder of radius R has a volume charge density that varies with the radius as given by the following expression where ρ0, a, and b are positive constants and r is the distance from the axis of the cylinder.

Use Gauss's law to determine the magnitude of the electric field at the following radial distances. (Use the following as necessary: ε0, ρ0, a, b, r, and R.)
(a) r < R
(b) r > R

Homework Equations

gauss's law

 

[tiny-tim]

hi sicrayan!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[sicrayan]

Hi
Thank you for your answer.
My unsuccessful solution:
lets take the length l,
for r<R,

why?

 

[tiny-tim]

hi sicrayan!

no, you're taking the mass inside radius r as πr² times ρ(r) …

it isn't, you need to integrate

 

[rwisz]

: ∮E⃗ ⋅dA⃗ = Qenc/ε0
Electric field for a cylindrical charge distribution: E⃗ = (ρ0*r/2ε0)*(1-(a/r)^2) for r < a, E⃗ = ρ0*r/2ε0 for r > a

The electric field of an infinitely long insulating cylinder can be determined using Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space (ε0). In this case, the charge enclosed by the surface is the volume charge density (ρ0) multiplied by the volume of the cylinder (πR^2). Therefore, the electric field at any point outside the cylinder (r > R) can be calculated using the formula E = ρ0*r/2ε0, where r is the distance from the axis of the cylinder.

For points inside the cylinder (r < R), the electric field can be calculated using the formula E = (ρ0*r/2ε0)*(1-(a/r)^2), where a is the radius of the cylinder and r is the distance from the axis. This formula takes into account the varying charge density as given in the problem.

Therefore, to determine the magnitude of the electric field at specific radial distances, we can plug in the given values for ρ0, a, b, and R into the above equations. For example, for part (a) where r < R, we can use the formula E = (ρ0*r/2ε0)*(1-(a/r)^2) and plug in the given values to calculate the electric field at that distance. Similarly, for part (b) where r > R, we can use the formula E = ρ0*r/2ε0 and plug in the given values to calculate the electric field at that distance.

In conclusion, the electric field of an infinitely long insulating cylinder can be determined using Gauss's law and the given formula for cylindrical charge distribution. By plugging in the given values, we can calculate the electric field at any radial distance and gain a better understanding of the electric field around the cylinder.