Infinitely extended cylindrical region in free space has volume charge density

[thor89]

Homework Statement

An infinitely extended cylindrical region of radius a>0 situated in free space contains a volume charge density given by:
[
ρ(r)= volume charge density
ρo=constant=initial volume charge density
radius=a>0

ρ(r)=ρo(1+αr^2); r<=a
]
with ρ(r)=0 for r>a

Questions:
1. utilize gauss law together with the inherent symmetry of the problem to derive the resulting electrostatic field vector E(r) both inside and outside the cylinder

2. Use both Poisson’s and Laplace’s equations to
directly determine the electrostatic potential V(r) both inside
and outside the cylindrical region. From this potential function,
determine the electrostatic field vector E(r).

3. Determine the value of the parameter α for which
the electrostatic field vanishes everywhere in the region outside
the cylinder (r > a). Plot Er (r ) and V(r ) as a function of r for
this value of α.

Homework Equations

eo=epsilon-not

gauss' law : divergence of E(r) = ρ(r)/εo


closed∫{E.nda} = 1/εo*∫∫∫{V{ρ(r)d^3r}

The Attempt at a Solution

 

[Nate810]

1. By symmetry, the electric field vector E(r) is a radial vector pointing outward from the cylinder's center:E(r) = Er(r) * ˆr; r<=a We can then use Gauss' Law to solve for the electric field:divergence of E(r) = ρ(r)/εod/dr(Er(r)) = ρo(1+αr^2)/εoIntegrating both sides:Er(r) = (1/εo) * ∫ρo(1+αr^2)drEr(r) = (1/εo) * ρo * ∫(1+αr^2)dr Er(r) = (1/εo) * ρo * (r + (1/3α)r^3) Er(r) = (1/εo) * ρo * (1 + (1/3α)r^2) For r>a: Er(r) = 0 2. Poisson's equation: ∇^2V(r) = -ρ(r)/εo Laplace’s equation: ∇^2V(r) = 0 For r<=a: ∇^2V(r) = -ρo(1+αr^2)/εoFor r>a: ∇^2V(r) = 0 Integrating both sides: For r<=a: V(r) = (1/εo)*∫∫ρo(1+αr^2)drdθ V(r) = (1/εo)*ρo*∫∫(1+αr^2)drdθ V(r) = (1/εo)*ρo*(r + (1/3α)r^3) For r>a: